2) = P (z(l) > 2) ≤ P (|z(l) − E[z(l)]| > |2− E[z(l)]|) ≤ Var(z(l))\n(2− E[z(l)])2 (9)\nwhere the second inequality is established by Chebyshev’s inequality, here, zl is guaranteed to be nonnegative since it’s the output of the ReLU layer. We can further estimate the worst upper bound for the right hand side by considering the following optimization problem:\nsup E[z(l)],Var(z(l))\nVar(z(l))\n(2− E[z(l)])2\nsubject to (E[z(l)])2 +Var(z(l)) = E(l)\nAs z(l) is the rectified value, E[z(l)] > 0. One upper bound can be established as follows:\nsup E[z(l)],Var(z(l))\nVar(z(l))\n(2− E[z(l)])2 <\nE(l) (2− √ E(l))2\n(10)\nbecause E[z(l)] < √ E(l) and Var(z(l)) < E(l). Notice that the last term can be approximated to E(l) when E(l) is small enough, and we can establish our probability bound of overflowing the quantization range:\nP (|z(l)| > 2) < E(l) (2− √ E(l))2 ' E(l) 4 (11)\nFinally, consider the average overflow probability of z ∈ R from any layer:\nP (|z| > 2) = 1 L L∑ l=1 P (z(l) > 2) ' 1 4L L∑ l=1 ul−1E(1) ' E(1) 4L(1− u) (12)\nFormula (12) is derived because in deep neural networks, L is usually a large number allowing us to effectively ignore uL. Therefore, the overflow probability is determined by the input’s second moments E, the layer size L and the barrier parameter u. In practice, we can adjust both u and L to control the overflowing tails. In this work, E(1) is set around 1.0 because the input features are normalized, and L is around 50 in the ResNet-50. By choosing u = 0.1, the number of overflowing parameters can be controlled under 2.3%."},{"heading":"3.4 SPARSITY","text":"In order to achieve the goal of practical implementation on 8-bit hardware, we aggressively fix the range of quantization to z ∈ [−2, 2]. This leaves us with the task of mapping millions of parameters to this range. Therefore, we desire a sparse solution, and naturally, as is pointed out by Hoffer et al. (2018), we use L1 penalty | n| to encourage sparsity. Our unconstrained loss then becomes:\nminimize θ L(fθ(I)) + N∑ n=1 ψ̃t(|θn| − u) + γ N∑ n=1 | n| (13)\nwhere γ > 0 and is tuning variable, and ψ̃t is the log-barrier extension as proposed.\n3.5 µ-LAW QUANTIZATION CODEBOOK (MU8)\nEven after sparsifying the weight, we still are left will many parameters which are non-linear distributed. Luckily, they should all be small enough by now thanks to our log-barrier constraint. Uniform quantization (e.g. INT8) would cause huge information loss in this case. Inspired by µ-law’s (Deng & Doroslovacki, 2006) application in audio encoding, we construct a non-linear codebook accordingly, which could be implemented on 8-bit hardware directly. As is shown in Figure 1(right), we computed all possible values of FP8(1-4-3) or FP8(1-5-2)4used by Sun et al. (2019), and we can see our MU8 encoding are better at handling very small values e.g. FP8 (1-4- 3)’s smallest possible number is 1.9 × 10−3, FP8 (1-5-2) is 1.5 × 10−5, whereas our MU8 could handle 5×10−6 with the cost of getting sparser in larger numbers, which we have almost eliminated using log-barrier. Let’s denote our the quantization function as Qµ8(X) which take x as input and output quantized value x′. First, following the range of µ-law encoding, we set the input to Equation 15 x = z/2, since z ∈ [−2, 2] in this paper.\nF (x) = sgn(x) ln(1 + µ|x|) ln(1 + µ) −1 ≤ x ≤ 1 (14)\nwhere µ is a tuning variable, the larger µ is, the more non-linear it becomes, and sgn(x) is the sign function. Then, we get the y ∈ [−1, 1] as the output of F (x), then we perform Stochastic Rounding step introduced by Gupta et al. (2015). e.g. y× (27− 1) ∈ [15, 16], if y = 15.5/(27− 1), then P (ŷ = 16/(27 − 1)) = P (ŷ = 15/(27 − 1)) = 0.5; if y = 15.1/(27 − 1), then P ((ŷ = 15/(27 − 1)) = 0.9, P (ŷ = 16/(27 − 1)) = 0.1, as shown in the following equation.\nŷ ← Stochastic Rounding(y, LL,UL) = { LL w.p. 1− (y − LL) UL w.p. (y − LL) (15)\nwhere UL indicates upper limit and LL indicates lower limit. At last, we get the quantized value x′ via the inverse function of encoding function (15):\nx′ ← F−1(y) = sgn(y) 1 µ ((1 + µ)|y| − 1) (16)"},{"heading":"3.6 OTHER USEFUL TECHNIQUES","text":"Straight-through Estimator (STE): We adopt the same STE as Zhou et al. (2016) which is critical to the convergence of our models. The gradient of the quantization steps in the quantization model is based on the straight through estimator.\nChunked Updates: We also use the same gradient accumulation strategy as Sakr et al. (2019); Wang et al. (2018). During updates, since we have quantized the gradient and weight parameters using Qµ8(X), the gradient will be lower bounded by the smallest possible value of the MU8 encoding(10−6), which is small enough to preserve model accuracy within the 8-bit constraint. However, directly adding this small gradient to a large number (weight) in our MU8 quantization would cause this small gradient to be rounded-off and thus lose information due to the growing\n4FP8(1-4-3)’s largest value is [-240, 240], and FP8(1-5-2)’s is [-57344, 57344], they are better at handling long-tails than MU8, but we are not interested in large numbers in the long-tail.\nintervals of MU8 encoding at larger values. Performing chunked gradient updates, updating the weights only after k steps of gradient descent (in this work, we set k = 20), helps to accumulate the gradients to be large enough to be rounded up, aiding convergence."},{"heading":"4 EXPERIMENTAL RESULTS","text":"Table. 2 shows the performance of models trained with LogBTQ from scratch on the MNIST, CIFAR-10, and ImageNet datasets. Our performance on CIFAR-10 is even higher than the FP32 baseline, and our results using ResNet-50 trained from scratch on ImageNet are competitive against both of the works in comparison. This is particularly impressive given that LogBTQ uses strictly 8-bits to store gradients whereas previous work resorted to FP16 or hybrid 8/16. Our accuracy loss of ResNet-50 on ImageNet is only 2.48% which for many use cases is worth the benefit of keeping everything in only 8 bits of precision. For practical purposes, 2-5% accuracy loss should be tolerable if calculations can be kept strictly in 8 bits."},{"heading":"5 DISCUSSION","text":"Hou et al. (2019) also provides a bound for the quantized gradient of QAT models, giving the intuition that quantized gradients would slow down convergence, which we also observe in our training.\nFor each arithmetic operation, we perform our Qµ8(X) quantization right after each operation to ensure the number still falls into 8 bits. For out of range numbers, we clamp. We verify our assump-\ntions in our experiments: Figure 3 (left) shows a layer’s weight distribution during training, and we can see that they are nicely bounded by our log-barrier to within [-1.5, 1.5]. Figure 3 (right) shows the weight distribution is concentrating to smaller values during training.\nSince we only have 10−6 precision to handle the gradients, we are inevitably losing some accuracy compared with higher precision training schemes. FP16 can handle 10−8 precision and FP32 can handle 10−32 precision. As we can see in Table 2, LogBTQ can perform almost perfectly on easier tasks such as MNIST and CIFAR-10, but degrades when we are training on the more challenging ImageNet dataset. Our Mu8 encoding performs better than 1-5-2 and 1-4-3 with our LogBTQ quantization scheme, showing µ-law encoding can handle small fixed quantization range better than 1-5-2 or 1-4-3. As is pointed out by Sheng et al. (2018), MobileNet architecture has some layers that are unfriendly for quantization, e.g. An outlier in one channel could cause a huge quantization loss for the entire model due to an enlarged data range. We employed the same techniques proposed by Sheng et al. (2018), though MobileNet benchmark is still not as good as ResNet-50 due to the agressive reduction of parameters.\nOverall, it still depends on the use case of the deep learning models to consider the trade-off between accuracy and precision to choose which quantization scheme to adopt. As far as we are aware, LogBTQ is the first full 8-bit quantization scheme with competitive performance on a large-scale dataset such as ImageNet. This opens up the prospect of training fully-quantized models on lowprecision hardware."},{"heading":"6 CONCLUSION","text":"Motivated by the limitations of fake quantization, we propose Log Barrier Tail-bounded Quantization (LogBTQ) which introduces a log-barrier extension loss term that enforces soft constraints on the range of values that model parameters can take on at every operation. Our approach eliminates overflow in practice, sparsifying the weights and using µ-law non-uniform quantization, allowing for fully quantized 8-bit training of deep neural network models. By constraining the neural network parameters driven by theoretical motivations, this work enables the possibility for the first time fully-quantized training on low-precision hardware."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"As state-of-the-art deep learning models for vision, language understanding and speech grow increasingly large and computationally burdensome (He et al., 2017; Devlin et al., 2018; Karita et al., 2019), there is increasing antithetical demand, motivated by latency, security and privacy concerns, to perform training and inference in these models on smaller devices at the edge rather than in server farms in the cloud. Model quantization has emerged as a promising approach to enable deployment of deep learning models on edge devices that reduce energy, latency and storage requirements by performing floating-point computation in low precision (less than 32 bits).\\nThere are two primary strategies for quantization: Post-training approaches quantize the parameters of a model trained in full precision post-hoc, and tend to suffer a heavy penalty on accuracy since their inference graph differs substantially from training (Jacob et al., 2018). Quantization-aware training (QAT) (Bhuwalka et al., 2020) combats this discrepancy by simulating quantization during training, so that model parameters are learned that will work well when inference is performed in low precision. In this work, we focus on the latter setting, suitable for fully quantized training on low-precision (e.g. 8-bit) devices.\\nThough QAT results in quantized models that perform largely on par with their non-quantized counterparts, current state-of-the-art QAT methods (Wu et al., 2018; Wang et al., 2018; Bhuwalka et al., 2020) are not suitable for training on fully low-precision hardware because they employ fake quantization, meaning each operation is executed using 32- or 16-bit floating point arithmetic, and its output is quantized to lower precision, e.g. int8. This results in two key incompatibilities with fully low-precision training, and consequently deployment on real low-precision hardware. First, existing QAT approaches assume perfect sums in inner product operations, which means that the accumulators used to compute matrix multiplies (the acc row in Table 1) must be higher precision than the values being multiplied (other bit-precision rows in Table 1). This is to avoid losing res-\\nolution in low-precision additions, also known as swamping (Wang et al., 2018)1. Second, QAT commonly leverages dynamic quantization ranges per-layer, meaning the mapping between highand low-precision values varies by layer, carefully tuned as a function of the network architecture, optimization dynamics and data during training. While this practice results in higher quantized inference accuracy, it is also a challenge to low-precision training, since it is unclear how to tune those ranges when training on new data in the absence of high-precision arithmetic. These incompatibilities present a substantial hurdle to quantized training in practice. For example, an automotive electronics manufacturer may want to deploy a machine learning model on its 8-bit door lock or power window controller to adaptively fit the users’ habits. In this scenario, existing approaches for quantized training would fail (Sakr et al., 2019).\\nIn response, we propose a new approach for fully quantized training of neural network models, inspired by the barrier method from convex optimization (Boyd & Vandenberghe, 2004). Log Barrier Tail-bounded Quantization (LogBTQ) utilizes a log barrier extension loss (Kervadec et al., 2019) to constrain the output of the network, encouraging all model parameters and activations to stay within the same predefined range. The log barrier function itself is a smooth approximation of the indicator function, which is ideal for selecting the weights that are within the range of quantization (see Figure 1, left). By fixing a single quantization range throughout the network at the beginning of training, our approach both obviates the need for dynamic ranges, and the limits of the range are set so as to alleviate overflow2 in matrix multiply accumulations. We combine the log barrier extension loss with an L1 regularization term (Hoffer et al., 2018) to further reduce the total magnitude of parameters and activations in the model. To allow for gradients, which tend form a peaky distribution near extremely small values (Zhou et al., 2016; Jain et al., 2020), to be quantized using the same range as the rest of the network, we also adopt the nonlinear µ-law algorithm from audio applications (Deng & Doroslovacki, 2006) to construct a new MU8 codebook that better deals with “swamping” issues compared to the standard IEEE Float Standard. Experiments show that our approach achieves competitive results compared to state-of-art full-precision models on the MNIST, CIFAR-10 and ImageNet classification benchmarks, despite our models being trained end-to-end using only 8 bits of precision.\\n1Swamping: Accumulation of floating-point numbers, where the small magnitude value is ignored (or truncated) when it is added to the large magnitude sum.\\n2Overflowing: for the fixed-point accumulation where the accumulated value wraps around to the small value when it exceeds the largest value representable by the given accumulation precision.\"\n },\n {\n \"heading\": \"2 BACKGROUND AND RELATED WORK\",\n \"text\": \"\"\n },\n {\n \"heading\": \"2.1 POST-TRAINING QUANTIZATION\",\n \"text\": \"There was been a recent surge of interest in quantization research. In 2020 alone, there were a number of important developments in post-training quantization. Rusci et al. (2020); Jain et al. (2020); Esser et al. (2020); Uhlich et al. (2020) proposed learning-based approaches for determining the quantization ranges of activation and weights at low precision. Stock et al. (2020) advocates preserving the quality of the reconstruction of the network outputs rather than its weights. They all show excellent performance compared to full-precision models after quantization. Sakr & Shanbhag (2019) presented a detailed analysis of reduced precision training for a feedforward network that accounts for both the forward and backward passes, demonstrating that precision can be greatly reduced throughout the network computations while largely preserving training quality. Our work share the same intuition of preferring small predetermined dynamic range (PDR) and small clipping rate3. However, Sakr & Shanbhag (2019)’s approach requires the network first be trained to convergence at full 32 bit precision, which is a significant limitation. In this paper, we focus on training rather than inference on low-precision hardware, therefore, we do not assume access to a full-precision high-performing model as a starting point.\"\n },\n {\n \"heading\": \"2.2 QUANTIZATION-AWARE TRAINING\",\n \"text\": \"Pioneering works in this domain (Zhou et al., 2016; Courbariaux et al., 2015) looked at quantizing model weights, activations, gradients to lower precision to accelerate neural network training. The terminology quantization aware training (QAT) was first introduced by Jacob et al. (2018). QAT incoporates quantization error as noise during training and as part of the overall loss, which the optimization algorithm tries to minimize. Hence, the model learns parameters that are more robust to quantization, but QAT is not meant to be performed entirely in low precision, it aims to learn parameters that will work well for low-precision inference. More recently, several works further pursued the goal of enabling fully low-precision training (Wu et al., 2018; Wang et al., 2018; Das et al., 2018; Sun et al., 2019). As shown in Table 1, most existing work employs fake quantization, resorting to higher precision values to compensate for the swamping issue, especially during gradient accumulation. Mixed-precision quantization (Das et al., 2018; Wang et al., 2018; Zhang et al., 2020a), which quantizes a neural network using multiple bit precisions across layers, still relies on higher-precision gradients to preserve model accuracy. This means it is difficult, if not impossible, to implement these approaches on low-bit (e.g. 8-bit) hardware.\\nMost similar to our work, Sun et al. (2019) claim it is possible to do every step in low precision, but the quantization range for the layers in their work is very carefully chosen empirically, which presents great difficulty if we were to train models from scratch on low-precision hardware. Their method also requires a copy of the quantization error (residual) in FP16(1-6-9) (hence 8/16 in Ta-\\n3see Appendix B of Sakr & Shanbhag (2019) explaining PDR; refer to their Criterion 2 about clipping rate.\\nble 1). In addition to the 9-bit mantissa, the exponent bit in their floating point format would need to be manually modified to store the residual due to its small value.\\nIn this paper, we propose a new quantization scheme: log-barrier tail-bounded quantization (LogBTQ) that can perform fully end-to-end low precision training, suitable for deployment on lowprecision hardware. Our major contributions are the following:\\n1. We apply a log barrier extension loss to soft-threshold the values of network weights and activations to constrain all the values to be small. Our quantization scheme also enables global fixed-range quantization which together significantly alleviates the overflow issue caused by large numbers and dynamic range.\\n2. We add an L1 loss term to encourage sparsity and further reduce overflow.\\n3. We propose µ-law quantization (MU8) instead of INT8, FP8(1-4-3) or FP8(1-5-2) to construct a more accurate codebook that better compensates for the peaky concentration of network parameters around small values.\"\n },\n {\n \"heading\": \"3 LOG BARRIER TAIL-BOUNDED QUANTIZATION (LOGBTQ)\",\n \"text\": \"The overall diagram of our quantization scheme is shown in Figure 2 (left). Figure 2 (right) shows the backward pass, where we quantize everything at each layer and all operations including input x, weights w, activations a, errors e, and gradients g (including the gradient accumulation step). We denote all these values as the set Z = {x,w, a, e, g}. In this work, different from previous works (Sakr & Shanbhag, 2019; Zhang et al., 2020b) that used adaptive quantization range, we adopt a globally fixed quantization range for every element z ∈ Z, and set z ∈ [−2, 2]. We do not need to adjust the range and precision during training as in other quantization work that relies on layer-wise dynamic ranges. This would greatly reduce the overhead for implementation on hardware.\"\n },\n {\n \"heading\": \"3.1 CONSTRAINED FORMULATION\",\n \"text\": \"Let D = {I1, ...IN} denote the labeled set of N training images, and f denote the neural network model, θ here denotes all the parameters of the neural network including weights w. For task, such as image classification, we are usually solving such an optimization problem: min\\nθ L(fθ(I)) where\\nL is the loss function of our neural network training objective. In this work, we use the typical cross-entropy loss, and since we are interested in constraining the quantization threshold, we are effectively performing constrained optimization in such a form:\\nminimize θ L (fθ(I)) subject to |θn| ≤ u, n = 1, . . . , N. (1)\\nWith u our desired barrier (perturbation). In practice, we set u = 0.1 to ensure we can represent as much information as possible within our quantization range (Figure 1, left). This setting is also explained further in Section 3.3.\"\n },\n {\n \"heading\": \"3.2 LOG-BARRIER EXTENSION FUNCTION\",\n \"text\": \"Theoretically, problem (1) should be best solved by the log barrier method which is an interior point method that perfectly handles inequality constraints. (Tibshirani, 2019; Boyd & Vandenberghe, 2004): In phase I, we would perform Lagrangian-dual optimization to find the feasible points:\\nmaximize λ minimize θ L (x, λ) = L (fθ(I)) + N∑ n=1 λn(|θn| − u))\\nsubject to λ 0, n = 1, . . . , N.\\n(2)\\nwhere λ ∈ R1×N+ is the Lagrangian multiplier (dual variable). After we find a feasible set of network parameters, we can use the barrier method to solve equation (1) as an unconstrained problem:\\nminimize θ L(fθ(I)) + N∑ n=1 ψt(|θn| − u) (3)\\nsolving problem (3) is Phase II, where ψt is the standard log-barrier function: ψt = − 1t log(−z). As t approaches infinity, the approximation becomes closer to the indicator function. Also, for any value of t, if any of the constraints is violated, the value of the barrier approaches infinity. However, a huge limitation in applicability to practical problems such as ours is that the domain of Eq. (3) must be the set of feasible points. The canonical barrier method above is also prohibitively computationally expensive given there are millions of parameters in the network, and we need to alternate the training between primal and dual and do projected gradient ascent for the dual variable.\\nWe are not particularly concerned with the weak duality gap to lower-bound the optimal solution in this work, instead, we are interested in the property of the barrier method to handle inequality constraints. Therefore, inspired by Kervadec et al. (2019), we formulate quantization as an unconstrained loss to approximate the constrained optimization problem:\\nminimize θ L(fθ(I)) + N∑ n=1 ψ̃t(|θn| − u) (4)\\nwhere ψ̃t is the log-barrier extension, which is convex, continuous, and twice-differentiable:\\nψ̃t(z) =\\n{ − 1t log(−z) z ≤ − 1 t2\\ntz − 1t log( 1 t2 ) + 1 t otherwise\\n(5)\\nin our case, the input z to the log-barrier extension is the same z we defined in the beginning of Section 3, and t is the scaling parameter. It basically shares the same property with the standard logbarrier function, when t approaches +∞, our log-barrier extension would approach a hard indicator: H(z) = 0 if z ≤ 0 and +∞ otherwise. But its domain is not restricted to the feasible points only. This removes the demanding requirement for explicit Lagrangian optimization.\\nWe are doing approximated Lagrangian optimization with implicit dual variables. Our strict positive gradient of ψ̃t will get higher when z approaches violation and effectively push back into the feasible value set. Because our penalty does not serve as a strict barrier of the feasible set, there is possibility of overflow. However, our goal is to achieve the practical goal of fully-quantized training on low-bit hardware, as long as the majority of values stay within a high confidence interval, the approach will work in practice, as we show in experimental results (§4). Recall the scenario in Figure 1(left).\"\n },\n {\n \"heading\": \"3.3 TAIL BOUND OF DISTRIBUTION\",\n \"text\": \"In this section, we demonstrate that the probability of overflowing the quantization range can be controlled for a standard ResNet model. The ResNet model has L layers, each layer contains a CNN operation and ReLU as an activation. Consider at layer l, for each output element O ∈ R in the\\nlayer, a CNN operation with the kernel size k and channel size c can be viewed as a rectified dot product between the input feature I ∈ Rck2with w ∈ Rck2 .\\nO = ReLU( ck2∑ i=1 wiIi) (6)\\nSuppose wi ∼ N (0, σ2(l)) and σ 2 (l) is dependent on the layer l and the parameter u we choose.\\nThe second moment of any element z(l+1) ∈ R from layer l + 1 can be connected with the second moment of element z(l) from layer l as follows:\\nE[z2(l+1)] = ck2\\n2 σ2(l)E[z 2 (l)] (7)\\nIn the He initialization, we could set σw = √ 2 ck2 to cancel the first two terms. In this work, we want to control σ2(l) in order to reduce the chance of overflow. We choose σ 2 (l) to depend on u which\\nwas defined in optimization problem (1): σ(l) = √ 2u ck2 . Then, we can simplify Eq. (7):\\nE[z2(l+1)] = uE[z 2 (l)] (8)\\nSuppose the input feature to the first layer has second moments E(1) = E[z21 ], then the second moments at layer l can be estimated as: E(l) = E[z2(l)] = u\\nl−1E(1). Next, for each element z(l) at layer l, its tail distribution outside the quantization region can be bounded by\\nP (|z(l)| > 2) = P (z(l) > 2) ≤ P (|z(l) − E[z(l)]| > |2− E[z(l)]|) ≤ Var(z(l))\\n(2− E[z(l)])2 (9)\\nwhere the second inequality is established by Chebyshev’s inequality, here, zl is guaranteed to be nonnegative since it’s the output of the ReLU layer. We can further estimate the worst upper bound for the right hand side by considering the following optimization problem:\\nsup E[z(l)],Var(z(l))\\nVar(z(l))\\n(2− E[z(l)])2\\nsubject to (E[z(l)])2 +Var(z(l)) = E(l)\\nAs z(l) is the rectified value, E[z(l)] > 0. One upper bound can be established as follows:\\nsup E[z(l)],Var(z(l))\\nVar(z(l))\\n(2− E[z(l)])2 <\\nE(l) (2− √ E(l))2\\n(10)\\nbecause E[z(l)] < √ E(l) and Var(z(l)) < E(l). Notice that the last term can be approximated to E(l) when E(l) is small enough, and we can establish our probability bound of overflowing the quantization range:\\nP (|z(l)| > 2) < E(l) (2− √ E(l))2 ' E(l) 4 (11)\\nFinally, consider the average overflow probability of z ∈ R from any layer:\\nP (|z| > 2) = 1 L L∑ l=1 P (z(l) > 2) ' 1 4L L∑ l=1 ul−1E(1) ' E(1) 4L(1− u) (12)\\nFormula (12) is derived because in deep neural networks, L is usually a large number allowing us to effectively ignore uL. Therefore, the overflow probability is determined by the input’s second moments E, the layer size L and the barrier parameter u. In practice, we can adjust both u and L to control the overflowing tails. In this work, E(1) is set around 1.0 because the input features are normalized, and L is around 50 in the ResNet-50. By choosing u = 0.1, the number of overflowing parameters can be controlled under 2.3%.\"\n },\n {\n \"heading\": \"3.4 SPARSITY\",\n \"text\": \"In order to achieve the goal of practical implementation on 8-bit hardware, we aggressively fix the range of quantization to z ∈ [−2, 2]. This leaves us with the task of mapping millions of parameters to this range. Therefore, we desire a sparse solution, and naturally, as is pointed out by Hoffer et al. (2018), we use L1 penalty | n| to encourage sparsity. Our unconstrained loss then becomes:\\nminimize θ L(fθ(I)) + N∑ n=1 ψ̃t(|θn| − u) + γ N∑ n=1 | n| (13)\\nwhere γ > 0 and is tuning variable, and ψ̃t is the log-barrier extension as proposed.\\n3.5 µ-LAW QUANTIZATION CODEBOOK (MU8)\\nEven after sparsifying the weight, we still are left will many parameters which are non-linear distributed. Luckily, they should all be small enough by now thanks to our log-barrier constraint. Uniform quantization (e.g. INT8) would cause huge information loss in this case. Inspired by µ-law’s (Deng & Doroslovacki, 2006) application in audio encoding, we construct a non-linear codebook accordingly, which could be implemented on 8-bit hardware directly. As is shown in Figure 1(right), we computed all possible values of FP8(1-4-3) or FP8(1-5-2)4used by Sun et al. (2019), and we can see our MU8 encoding are better at handling very small values e.g. FP8 (1-4- 3)’s smallest possible number is 1.9 × 10−3, FP8 (1-5-2) is 1.5 × 10−5, whereas our MU8 could handle 5×10−6 with the cost of getting sparser in larger numbers, which we have almost eliminated using log-barrier. Let’s denote our the quantization function as Qµ8(X) which take x as input and output quantized value x′. First, following the range of µ-law encoding, we set the input to Equation 15 x = z/2, since z ∈ [−2, 2] in this paper.\\nF (x) = sgn(x) ln(1 + µ|x|) ln(1 + µ) −1 ≤ x ≤ 1 (14)\\nwhere µ is a tuning variable, the larger µ is, the more non-linear it becomes, and sgn(x) is the sign function. Then, we get the y ∈ [−1, 1] as the output of F (x), then we perform Stochastic Rounding step introduced by Gupta et al. (2015). e.g. y× (27− 1) ∈ [15, 16], if y = 15.5/(27− 1), then P (ŷ = 16/(27 − 1)) = P (ŷ = 15/(27 − 1)) = 0.5; if y = 15.1/(27 − 1), then P ((ŷ = 15/(27 − 1)) = 0.9, P (ŷ = 16/(27 − 1)) = 0.1, as shown in the following equation.\\nŷ ← Stochastic Rounding(y, LL,UL) = { LL w.p. 1− (y − LL) UL w.p. (y − LL) (15)\\nwhere UL indicates upper limit and LL indicates lower limit. At last, we get the quantized value x′ via the inverse function of encoding function (15):\\nx′ ← F−1(y) = sgn(y) 1 µ ((1 + µ)|y| − 1) (16)\"\n },\n {\n \"heading\": \"3.6 OTHER USEFUL TECHNIQUES\",\n \"text\": \"Straight-through Estimator (STE): We adopt the same STE as Zhou et al. (2016) which is critical to the convergence of our models. The gradient of the quantization steps in the quantization model is based on the straight through estimator.\\nChunked Updates: We also use the same gradient accumulation strategy as Sakr et al. (2019); Wang et al. (2018). During updates, since we have quantized the gradient and weight parameters using Qµ8(X), the gradient will be lower bounded by the smallest possible value of the MU8 encoding(10−6), which is small enough to preserve model accuracy within the 8-bit constraint. However, directly adding this small gradient to a large number (weight) in our MU8 quantization would cause this small gradient to be rounded-off and thus lose information due to the growing\\n4FP8(1-4-3)’s largest value is [-240, 240], and FP8(1-5-2)’s is [-57344, 57344], they are better at handling long-tails than MU8, but we are not interested in large numbers in the long-tail.\\nintervals of MU8 encoding at larger values. Performing chunked gradient updates, updating the weights only after k steps of gradient descent (in this work, we set k = 20), helps to accumulate the gradients to be large enough to be rounded up, aiding convergence.\"\n },\n {\n \"heading\": \"4 EXPERIMENTAL RESULTS\",\n \"text\": \"Table. 2 shows the performance of models trained with LogBTQ from scratch on the MNIST, CIFAR-10, and ImageNet datasets. Our performance on CIFAR-10 is even higher than the FP32 baseline, and our results using ResNet-50 trained from scratch on ImageNet are competitive against both of the works in comparison. This is particularly impressive given that LogBTQ uses strictly 8-bits to store gradients whereas previous work resorted to FP16 or hybrid 8/16. Our accuracy loss of ResNet-50 on ImageNet is only 2.48% which for many use cases is worth the benefit of keeping everything in only 8 bits of precision. For practical purposes, 2-5% accuracy loss should be tolerable if calculations can be kept strictly in 8 bits.\"\n },\n {\n \"heading\": \"5 DISCUSSION\",\n \"text\": \"Hou et al. (2019) also provides a bound for the quantized gradient of QAT models, giving the intuition that quantized gradients would slow down convergence, which we also observe in our training.\\nFor each arithmetic operation, we perform our Qµ8(X) quantization right after each operation to ensure the number still falls into 8 bits. For out of range numbers, we clamp. We verify our assump-\\ntions in our experiments: Figure 3 (left) shows a layer’s weight distribution during training, and we can see that they are nicely bounded by our log-barrier to within [-1.5, 1.5]. Figure 3 (right) shows the weight distribution is concentrating to smaller values during training.\\nSince we only have 10−6 precision to handle the gradients, we are inevitably losing some accuracy compared with higher precision training schemes. FP16 can handle 10−8 precision and FP32 can handle 10−32 precision. As we can see in Table 2, LogBTQ can perform almost perfectly on easier tasks such as MNIST and CIFAR-10, but degrades when we are training on the more challenging ImageNet dataset. Our Mu8 encoding performs better than 1-5-2 and 1-4-3 with our LogBTQ quantization scheme, showing µ-law encoding can handle small fixed quantization range better than 1-5-2 or 1-4-3. As is pointed out by Sheng et al. (2018), MobileNet architecture has some layers that are unfriendly for quantization, e.g. An outlier in one channel could cause a huge quantization loss for the entire model due to an enlarged data range. We employed the same techniques proposed by Sheng et al. (2018), though MobileNet benchmark is still not as good as ResNet-50 due to the agressive reduction of parameters.\\nOverall, it still depends on the use case of the deep learning models to consider the trade-off between accuracy and precision to choose which quantization scheme to adopt. As far as we are aware, LogBTQ is the first full 8-bit quantization scheme with competitive performance on a large-scale dataset such as ImageNet. This opens up the prospect of training fully-quantized models on lowprecision hardware.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"Motivated by the limitations of fake quantization, we propose Log Barrier Tail-bounded Quantization (LogBTQ) which introduces a log-barrier extension loss term that enforces soft constraints on the range of values that model parameters can take on at every operation. Our approach eliminates overflow in practice, sparsifying the weights and using µ-law non-uniform quantization, allowing for fully quantized 8-bit training of deep neural network models. By constraining the neural network parameters driven by theoretical motivations, this work enables the possibility for the first time fully-quantized training on low-precision hardware.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":88,"cells":{"paper_id":{"kind":"string","value":"SP:d9155553fae947cc53d87a221fdd1d57b44f5ec6"},"summaries":{"kind":"list like","value":["I read this paper with great interest. The authors propose an easy-to-understand, easy-to-implement baseline method for detecting when inputs to a ML model is out of distribution. The method involves augmenting the training dataset with an out of distribution dataset and adding an additional class in the classification layer for out of distribution. The paper describes several experiments—some in computer vision, some in NLP—and then compares them to other OOD techniques. The results are comparable to other techniques, although the proposed technique is definitely simpler."],"string":"[\n \"I read this paper with great interest. The authors propose an easy-to-understand, easy-to-implement baseline method for detecting when inputs to a ML model is out of distribution. The method involves augmenting the training dataset with an out of distribution dataset and adding an additional class in the classification layer for out of distribution. The paper describes several experiments—some in computer vision, some in NLP—and then compares them to other OOD techniques. The results are comparable to other techniques, although the proposed technique is definitely simpler.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Refraining from confidently predicting when faced with categories of inputs different from those seen during training is an important requirement for the safe deployment of deep learning systems. While simple to state, this has been a particularly challenging problem in deep learning, where models often end up making overconfident predictions in such situations. In this work we present a simple, but highly effective approach to deal with out-of-distribution detection that uses the principle of abstention: when encountering a sample from an unseen class, the desired behavior is to abstain from predicting. Our approach uses a network with an extra abstention class and is trained on a dataset that is augmented with an uncurated set that consists of a large number of out-of-distribution (OoD) samples that are assigned the label of the abstention class; the model is then trained to learn an effective discriminator between in and out-of-distribution samples. We compare this relatively simple approach against a wide variety of more complex methods that have been proposed both for out-of-distribution detection as well as uncertainty modeling in deep learning, and empirically demonstrate its effectiveness on a wide variety of of benchmarks and deep architectures for image recognition and text classification, often outperforming existing approaches by significant margins. Given the simplicity and effectiveness of this method, we propose that this approach be used as a new additional baseline for future work in this domain."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Loïc Barrault","Fethi Bougares","Lucia Specia","Chiraag Lala","Desmond Elliott","Stella Frank"],"title":"Findings of the third shared task on multimodal machine translation","venue":"In Proceedings of the Third Conference on Machine Translation: Shared Task Papers,","year":2018},{"authors":["Abhijit Bendale","Terrance E Boult"],"title":"Towards open set deep networks","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2016},{"authors":["TE Boult","S Cruz","AR Dhamija","M Gunther","J Henrydoss","WJ Scheirer"],"title":"Learning and the unknown: Surveying steps toward open world recognition","venue":"In Proceedings of the AAAI Conference on Artificial Intelligence,","year":2019},{"authors":["Samuel R. Bowman","Gabor Angeli","Christopher Potts","Christopher D. Manning"],"title":"A large annotated corpus for learning natural language inference","venue":"In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing (EMNLP). Association for Computational Linguistics,","year":2015},{"authors":["Terrance DeVries","Graham W Taylor"],"title":"Learning confidence for out-of-distribution detection in neural networks","venue":"arXiv preprint arXiv:1802.04865,","year":2018},{"authors":["Akshay Raj Dhamija","Manuel Günther","Terrance Boult"],"title":"Reducing network agnostophobia","venue":"In Advances in Neural Information Processing Systems,","year":2018},{"authors":["Tom Fawcett"],"title":"An introduction to roc analysis","venue":"Pattern recognition letters,","year":2006},{"authors":["Yarin Gal","Zoubin Ghahramani"],"title":"Dropout as a bayesian approximation: Representing model uncertainty in deep learning","venue":"In international conference on machine learning,","year":2016},{"authors":["ZongYuan Ge","Sergey Demyanov","Zetao Chen","Rahil Garnavi"],"title":"Generative openmax for multiclass open set classification","venue":"arXiv preprint arXiv:1707.07418,","year":2017},{"authors":["Ian Goodfellow","Jean Pouget-Abadie","Mehdi Mirza","Bing Xu","David Warde-Farley","Sherjil Ozair","Aaron Courville","Yoshua Bengio"],"title":"Generative adversarial nets","venue":"In Advances in neural information processing systems,","year":2014},{"authors":["Chuan Guo","Geoff Pleiss","Yu Sun","Kilian Q Weinberger"],"title":"On calibration of modern neural networks","venue":"arXiv preprint arXiv:1706.04599,","year":2017},{"authors":["Dan Hendrycks","Kevin Gimpel"],"title":"A baseline for detecting misclassified and out-of-distribution examples in neural networks","venue":"arXiv preprint arXiv:1610.02136,","year":2016},{"authors":["Dan Hendrycks","Mantas Mazeika","Thomas Dietterich"],"title":"Deep anomaly detection with outlier exposure","venue":"arXiv preprint arXiv:1812.04606,","year":2018},{"authors":["Yen-Chang Hsu","Yilin Shen","Hongxia Jin","Zsolt Kira"],"title":"Generalized odin: Detecting out-ofdistribution image without learning from out-of-distribution data","venue":"arXiv preprint arXiv:2002.11297,","year":2020},{"authors":["Diederik P Kingma","Max Welling"],"title":"Auto-encoding variational bayes","venue":"arXiv preprint arXiv:1312.6114,","year":2013},{"authors":["Alex Krizhevsky","Geoffrey Hinton"],"title":"Learning multiple layers of features from tiny images","venue":"Technical report, Citeseer,","year":2009},{"authors":["Balaji Lakshminarayanan","Alexander Pritzel","Charles Blundell"],"title":"Simple and scalable predictive uncertainty estimation using deep ensembles","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["Ken Lang"],"title":"Newsweeder: Learning to filter netnews","venue":"In Proceedings of the Twelfth International Conference on Machine Learning,","year":1995},{"authors":["Kimin Lee","Honglak Lee","Kibok Lee","Jinwoo Shin"],"title":"Training confidence-calibrated classifiers for detecting out-of-distribution samples","venue":"arXiv preprint arXiv:1711.09325,","year":2017},{"authors":["Kimin Lee","Kibok Lee","Honglak Lee","Jinwoo Shin"],"title":"A simple unified framework for detecting out-of-distribution samples and adversarial attacks","venue":"In Advances in Neural Information Processing Systems,","year":2018},{"authors":["Shiyu Liang","Yixuan Li","R Srikant"],"title":"Enhancing the reliability of out-of-distribution image detection in neural networks. 2018","venue":"URL http://arxiv.org/abs/1706.02690","year":2018},{"authors":["Andrew L. Maas","Raymond E. Daly","Peter T. Pham","Dan Huang","Andrew Y. Ng","Christopher Potts"],"title":"Learning word vectors for sentiment analysis","venue":"In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies,","year":2011},{"authors":["Wesley J Maddox","Pavel Izmailov","Timur Garipov","Dmitry P Vetrov","Andrew Gordon Wilson"],"title":"A simple baseline for bayesian uncertainty in deep learning","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Stephan Mandt","Matthew D Hoffman","David M Blei"],"title":"Stochastic gradient descent as approximate bayesian inference","venue":"The Journal of Machine Learning Research,","year":2017},{"authors":["Lawrence Neal","Matthew Olson","Xiaoli Fern","Weng-Keen Wong","Fuxin Li"],"title":"Open set learning with counterfactual images","venue":"In Proceedings of the European Conference on Computer Vision (ECCV),","year":2018},{"authors":["Yuval Netzer","Tao Wang","Adam Coates","Alessandro Bissacco","Bo Wu","Andrew Y Ng"],"title":"Reading digits in natural images with unsupervised feature learning","venue":null,"year":2011},{"authors":["Anh Nguyen","Jason Yosinski","Jeff Clune"],"title":"Deep neural networks are easily fooled: High confidence predictions for unrecognizable images","venue":"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,","year":2015},{"authors":["Kazuki Osawa","Siddharth Swaroop","Mohammad Emtiyaz E Khan","Anirudh Jain","Runa Eschenhagen","Richard E Turner","Rio Yokota"],"title":"Practical deep learning with bayesian principles","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Adam Paszke","Sam Gross","Francisco Massa","Adam Lerer","James Bradbury","Gregory Chanan","Trevor Killeen","Zeming Lin","Natalia Gimelshein","Luca Antiga"],"title":"Pytorch: An imperative style, high-performance deep learning library","venue":"In Advances in neural information processing systems,","year":2019},{"authors":["Nitish Srivastava","Geoffrey Hinton","Alex Krizhevsky","Ilya Sutskever","Ruslan Salakhutdinov"],"title":"Dropout: A simple way to prevent neural networks from overfitting","venue":"Journal of Machine Learning Research,","year":2014},{"authors":["Sunil Thulasidasan","Gopinath Chennupati","Jeff A Bilmes","Tanmoy Bhattacharya","Sarah Michalak"],"title":"On mixup training: Improved calibration and predictive uncertainty for deep neural networks","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Antonio Torralba","Rob Fergus","William T Freeman"],"title":"80 million tiny images: A large data set for nonparametric object and scene recognition","venue":"IEEE transactions on pattern analysis and machine intelligence,","year":1958},{"authors":["Sachin Vernekar","Ashish Gaurav","Vahdat Abdelzad","Taylor Denouden","Rick Salay","Krzysztof Czarnecki"],"title":"Out-of-distribution detection in classifiers via generation","venue":null,"year":1910},{"authors":["Apoorv Vyas","Nataraj Jammalamadaka","Xia Zhu","Dipankar Das","Bharat Kaul","Theodore L Willke"],"title":"Out-of-distribution detection using an ensemble of self supervised leave-out classifiers","venue":"In Proceedings of the European Conference on Computer Vision (ECCV),","year":2018},{"authors":["Fisher Yu","Ari Seff","Yinda Zhang","Shuran Song","Thomas Funkhouser","Jianxiong Xiao"],"title":"Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop","venue":"arXiv preprint arXiv:1506.03365,","year":2015},{"authors":["Hongyi Zhang","Moustapha Cisse","Yann N Dauphin","David Lopez-Paz"],"title":"mixup: Beyond empirical risk minimization","venue":"arXiv preprint arXiv:1710.09412,","year":2017},{"authors":["Xiang Zhang","Junbo Zhao","Yann LeCun"],"title":"Character-level Convolutional Networks for Text Classification","venue":"[cs],","year":2015},{"authors":["Bolei Zhou","Agata Lapedriza","Aditya Khosla","Aude Oliva","Antonio Torralba"],"title":"Places: A 10 million image database for scene recognition","venue":"IEEE transactions on pattern analysis and machine intelligence,","year":2017}],"string":"[\n {\n \"authors\": [\n \"Loïc Barrault\",\n \"Fethi Bougares\",\n \"Lucia Specia\",\n \"Chiraag Lala\",\n \"Desmond Elliott\",\n \"Stella Frank\"\n ],\n \"title\": \"Findings of the third shared task on multimodal machine translation\",\n \"venue\": \"In Proceedings of the Third Conference on Machine Translation: Shared Task Papers,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Abhijit Bendale\",\n \"Terrance E Boult\"\n ],\n \"title\": \"Towards open set deep networks\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"TE Boult\",\n \"S Cruz\",\n \"AR Dhamija\",\n \"M Gunther\",\n \"J Henrydoss\",\n \"WJ Scheirer\"\n ],\n \"title\": \"Learning and the unknown: Surveying steps toward open world recognition\",\n \"venue\": \"In Proceedings of the AAAI Conference on Artificial Intelligence,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Samuel R. Bowman\",\n \"Gabor Angeli\",\n \"Christopher Potts\",\n \"Christopher D. Manning\"\n ],\n \"title\": \"A large annotated corpus for learning natural language inference\",\n \"venue\": \"In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing (EMNLP). Association for Computational Linguistics,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Terrance DeVries\",\n \"Graham W Taylor\"\n ],\n \"title\": \"Learning confidence for out-of-distribution detection in neural networks\",\n \"venue\": \"arXiv preprint arXiv:1802.04865,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Akshay Raj Dhamija\",\n \"Manuel Günther\",\n \"Terrance Boult\"\n ],\n \"title\": \"Reducing network agnostophobia\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Tom Fawcett\"\n ],\n \"title\": \"An introduction to roc analysis\",\n \"venue\": \"Pattern recognition letters,\",\n \"year\": 2006\n },\n {\n \"authors\": [\n \"Yarin Gal\",\n \"Zoubin Ghahramani\"\n ],\n \"title\": \"Dropout as a bayesian approximation: Representing model uncertainty in deep learning\",\n \"venue\": \"In international conference on machine learning,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"ZongYuan Ge\",\n \"Sergey Demyanov\",\n \"Zetao Chen\",\n \"Rahil Garnavi\"\n ],\n \"title\": \"Generative openmax for multiclass open set classification\",\n \"venue\": \"arXiv preprint arXiv:1707.07418,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Ian Goodfellow\",\n \"Jean Pouget-Abadie\",\n \"Mehdi Mirza\",\n \"Bing Xu\",\n \"David Warde-Farley\",\n \"Sherjil Ozair\",\n \"Aaron Courville\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Generative adversarial nets\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Chuan Guo\",\n \"Geoff Pleiss\",\n \"Yu Sun\",\n \"Kilian Q Weinberger\"\n ],\n \"title\": \"On calibration of modern neural networks\",\n \"venue\": \"arXiv preprint arXiv:1706.04599,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Dan Hendrycks\",\n \"Kevin Gimpel\"\n ],\n \"title\": \"A baseline for detecting misclassified and out-of-distribution examples in neural networks\",\n \"venue\": \"arXiv preprint arXiv:1610.02136,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Dan Hendrycks\",\n \"Mantas Mazeika\",\n \"Thomas Dietterich\"\n ],\n \"title\": \"Deep anomaly detection with outlier exposure\",\n \"venue\": \"arXiv preprint arXiv:1812.04606,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Yen-Chang Hsu\",\n \"Yilin Shen\",\n \"Hongxia Jin\",\n \"Zsolt Kira\"\n ],\n \"title\": \"Generalized odin: Detecting out-ofdistribution image without learning from out-of-distribution data\",\n \"venue\": \"arXiv preprint arXiv:2002.11297,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Diederik P Kingma\",\n \"Max Welling\"\n ],\n \"title\": \"Auto-encoding variational bayes\",\n \"venue\": \"arXiv preprint arXiv:1312.6114,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Alex Krizhevsky\",\n \"Geoffrey Hinton\"\n ],\n \"title\": \"Learning multiple layers of features from tiny images\",\n \"venue\": \"Technical report, Citeseer,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Balaji Lakshminarayanan\",\n \"Alexander Pritzel\",\n \"Charles Blundell\"\n ],\n \"title\": \"Simple and scalable predictive uncertainty estimation using deep ensembles\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Ken Lang\"\n ],\n \"title\": \"Newsweeder: Learning to filter netnews\",\n \"venue\": \"In Proceedings of the Twelfth International Conference on Machine Learning,\",\n \"year\": 1995\n },\n {\n \"authors\": [\n \"Kimin Lee\",\n \"Honglak Lee\",\n \"Kibok Lee\",\n \"Jinwoo Shin\"\n ],\n \"title\": \"Training confidence-calibrated classifiers for detecting out-of-distribution samples\",\n \"venue\": \"arXiv preprint arXiv:1711.09325,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Kimin Lee\",\n \"Kibok Lee\",\n \"Honglak Lee\",\n \"Jinwoo Shin\"\n ],\n \"title\": \"A simple unified framework for detecting out-of-distribution samples and adversarial attacks\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Shiyu Liang\",\n \"Yixuan Li\",\n \"R Srikant\"\n ],\n \"title\": \"Enhancing the reliability of out-of-distribution image detection in neural networks. 2018\",\n \"venue\": \"URL http://arxiv.org/abs/1706.02690\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Andrew L. Maas\",\n \"Raymond E. Daly\",\n \"Peter T. Pham\",\n \"Dan Huang\",\n \"Andrew Y. Ng\",\n \"Christopher Potts\"\n ],\n \"title\": \"Learning word vectors for sentiment analysis\",\n \"venue\": \"In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Wesley J Maddox\",\n \"Pavel Izmailov\",\n \"Timur Garipov\",\n \"Dmitry P Vetrov\",\n \"Andrew Gordon Wilson\"\n ],\n \"title\": \"A simple baseline for bayesian uncertainty in deep learning\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Stephan Mandt\",\n \"Matthew D Hoffman\",\n \"David M Blei\"\n ],\n \"title\": \"Stochastic gradient descent as approximate bayesian inference\",\n \"venue\": \"The Journal of Machine Learning Research,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Lawrence Neal\",\n \"Matthew Olson\",\n \"Xiaoli Fern\",\n \"Weng-Keen Wong\",\n \"Fuxin Li\"\n ],\n \"title\": \"Open set learning with counterfactual images\",\n \"venue\": \"In Proceedings of the European Conference on Computer Vision (ECCV),\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Yuval Netzer\",\n \"Tao Wang\",\n \"Adam Coates\",\n \"Alessandro Bissacco\",\n \"Bo Wu\",\n \"Andrew Y Ng\"\n ],\n \"title\": \"Reading digits in natural images with unsupervised feature learning\",\n \"venue\": null,\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Anh Nguyen\",\n \"Jason Yosinski\",\n \"Jeff Clune\"\n ],\n \"title\": \"Deep neural networks are easily fooled: High confidence predictions for unrecognizable images\",\n \"venue\": \"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Kazuki Osawa\",\n \"Siddharth Swaroop\",\n \"Mohammad Emtiyaz E Khan\",\n \"Anirudh Jain\",\n \"Runa Eschenhagen\",\n \"Richard E Turner\",\n \"Rio Yokota\"\n ],\n \"title\": \"Practical deep learning with bayesian principles\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Adam Paszke\",\n \"Sam Gross\",\n \"Francisco Massa\",\n \"Adam Lerer\",\n \"James Bradbury\",\n \"Gregory Chanan\",\n \"Trevor Killeen\",\n \"Zeming Lin\",\n \"Natalia Gimelshein\",\n \"Luca Antiga\"\n ],\n \"title\": \"Pytorch: An imperative style, high-performance deep learning library\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Nitish Srivastava\",\n \"Geoffrey Hinton\",\n \"Alex Krizhevsky\",\n \"Ilya Sutskever\",\n \"Ruslan Salakhutdinov\"\n ],\n \"title\": \"Dropout: A simple way to prevent neural networks from overfitting\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Sunil Thulasidasan\",\n \"Gopinath Chennupati\",\n \"Jeff A Bilmes\",\n \"Tanmoy Bhattacharya\",\n \"Sarah Michalak\"\n ],\n \"title\": \"On mixup training: Improved calibration and predictive uncertainty for deep neural networks\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Antonio Torralba\",\n \"Rob Fergus\",\n \"William T Freeman\"\n ],\n \"title\": \"80 million tiny images: A large data set for nonparametric object and scene recognition\",\n \"venue\": \"IEEE transactions on pattern analysis and machine intelligence,\",\n \"year\": 1958\n },\n {\n \"authors\": [\n \"Sachin Vernekar\",\n \"Ashish Gaurav\",\n \"Vahdat Abdelzad\",\n \"Taylor Denouden\",\n \"Rick Salay\",\n \"Krzysztof Czarnecki\"\n ],\n \"title\": \"Out-of-distribution detection in classifiers via generation\",\n \"venue\": null,\n \"year\": 1910\n },\n {\n \"authors\": [\n \"Apoorv Vyas\",\n \"Nataraj Jammalamadaka\",\n \"Xia Zhu\",\n \"Dipankar Das\",\n \"Bharat Kaul\",\n \"Theodore L Willke\"\n ],\n \"title\": \"Out-of-distribution detection using an ensemble of self supervised leave-out classifiers\",\n \"venue\": \"In Proceedings of the European Conference on Computer Vision (ECCV),\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Fisher Yu\",\n \"Ari Seff\",\n \"Yinda Zhang\",\n \"Shuran Song\",\n \"Thomas Funkhouser\",\n \"Jianxiong Xiao\"\n ],\n \"title\": \"Lsun: Construction of a large-scale image dataset using deep learning with humans in the loop\",\n \"venue\": \"arXiv preprint arXiv:1506.03365,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Hongyi Zhang\",\n \"Moustapha Cisse\",\n \"Yann N Dauphin\",\n \"David Lopez-Paz\"\n ],\n \"title\": \"mixup: Beyond empirical risk minimization\",\n \"venue\": \"arXiv preprint arXiv:1710.09412,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Xiang Zhang\",\n \"Junbo Zhao\",\n \"Yann LeCun\"\n ],\n \"title\": \"Character-level Convolutional Networks for Text Classification\",\n \"venue\": \"[cs],\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Bolei Zhou\",\n \"Agata Lapedriza\",\n \"Aditya Khosla\",\n \"Aude Oliva\",\n \"Antonio Torralba\"\n ],\n \"title\": \"Places: A 10 million image database for scene recognition\",\n \"venue\": \"IEEE transactions on pattern analysis and machine intelligence,\",\n \"year\": 2017\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION AND RELATED WORK","text":"Most of supervised machine learning has been developed with the assumption that the distribution of classes seen at train and test time are the same. However, the real-world is unpredictable and open-ended, and making machine learning systems robust to the presence of unknown categories and out-of-distribution samples has become increasingly essential for their safe deployment. While refraining from predicting when uncertain should be intuitively obvious to humans, the peculiarities of DNNs makes them overconfident to unknown inputs Nguyen et al. (2015) and makes this a challenging problem to solve in deep learning.\nA very active sub-field of deep learning, known as out-of-distribution (OoD) detection, has emerged in recent years that attempts to impart to deep neural networks the quality of \"knowing when it doesn’t know\". The most straight-forward approach in this regard is based on using the DNNs output as a proxy for predictive confidence. For example, a simple baseline for detecting OoD samples using thresholded softmax scores was presented in Hendrycks & Gimpel (2016). where the authors provided empirical evidence that for DNN classifiers, in-distribution predictions do tend to have higher winning scores than OoD samples, thus empirically justifying the use of softmax thresholding as a useful baseline. However this approach is vulnerable to the pathologies discussed in Nguyen et al. (2015). Subsequently, increasingly sophisticated methods have been developed to attack the OoD problem. Liang et al. (2018) introduced a detection technique that involves perturbing the inputs in the direction of increasing the confidence of the network’s predictions on a given input, based on the observation that the magnitude of gradients on in-distribution data tend to be larger than for OoD data. The method proposed in Lee et al. (2018) also involves input perturbation, but confidence in this case was measured by the Mahalanobis distance score using the computed mean and covariance of the pre-softmax scores. A drawback of such methods, however, is that it introduces a number of\nhyperparameters that need to be tuned on the OoD dataset, which is infeasible in many real-world scenarios as one does not often know in advance the properties of unknown classes. A modified version of the perturbation approach was recently proposed in in Hsu et al. (2020) that circumvents some of these issues, though one still needs to ascertain an ideal perturbation magnitude, which might not generalize from one OoD set to the other.\nGiven that one might expect a classifier to be more uncertain when faced with OoD data, many methods developed for estimating uncertainty for DNN predictions have also been used for OoD detection. A useful baseline in this regard is the temperature scaling method of Guo et al. (2017) that was was proposed for calibrating DNN predictions on in-distribution data and has been observed to also serve as a useful OoD detector in some scenarios. Further, label smoothing techniques like mixup Zhang et al. (2017) have also been shown to be able to improve OoD detection performance in DNNs Thulasidasan et al. (2019). An ensemble-of-deep models approach, that is also augmented with adversarial examples during training, described in Lakshminarayanan et al. (2017) was also shown to improve predictive uncertainty and succesfully applied to OoD detection.\nIn the Bayesian realm, methods such as Maddox et al. (2019) and Osawa et al. (2019) have also been used for OoD detection, though at increased computational cost. However, it has been argued that for OoD detection, Bayesian priors on the data are not completely justified since one does not have access to the prior of the open-set Boult et al. (2019). Nevertheless, simple approaches like dropout – which have been shown to be equivalent to deep gaussian processes Gal & Ghahramani (2016) – have been used as baselines for OoD detection.\nTraining the model to recognize unknown classes by using data from categories that do not overlap with classes of interest has been shown to be quite effective for out-of-distribution detection and a slew of methods that use additional data for discriminating between ID and OD data have been proposed. DeVries & Taylor (2018) describes a method ithat uses a separate confidence branch and misclassified training data samples that serve as a proxy for OoD samples. In the outlier exposure technique described in Hendrycks et al. (2018), the predictions on natural outlier images used in training are regularized against the uniform distribution to encourage high-entropy posteriors on outlier samples. An approach that uses an extra-class for outlier samples is described in Neal et al. (2018), where instead of natural outliers, counterfactual images that lie just outside the class boundaries of known classes are generated using a GAN and assigned the extra class label. A similar approach using generative samples for the extra class, but using a conditional Variational Auto-Encoders Kingma & Welling (2013) for generation, is described in Vernekar et al. (2019). A method to force a DNN to produce high-entropy (i.e., low confidence) predictions and suppress the magnitude of feature activations for OoD samples was discussed in Dhamija et al. (2018), where, arguing that methods that use an extra background class for OoD samples force all such samples to lie in one region of the feature space, the work also forces separation by suppressing the activation magnitudes of samples from unknown classes\nThe above works have shown that the use of known OoD samples (or known unknowns) often generalizes well to unknown unknown samples. Ineed, even though the space of unknown classes is potentially infinite, and one can never know in advance the myriad of inputs that can occur during test time, empirically this approach has been shown to work. The abstention method that we describe in the next section borrows ideas from many of the above methods: as in Hendrycks et al. (2018), we uses additional samples of real images and text from non-overlapping categories to train the model to abstain, but instead of entropy regularization over OoD samples, out method uses an extra abstention class. While it has been sometimes argued in the literature that that using an additional abstention (or rejection) class is not an effective approach for OoD detection Dhamija et al. (2018); Lee et al. (2017), comprehensive experiments we conduct in this work demonstrate that this is not the case. Indeed, we find that such an approach is not only simple but also highly effective for OoD detection, often outperforming existing methods that are more complicated and involve tuning of multiple hyperparameters. The main contributions of this work are as follows:\n• To the best of our knowledge, this is the first work to comprehensively demonstrate the efficacy of using an extra abstention (or rejection class) in combination with outlier training data for effective OoD detection. • In addition to being effective, our method is also simple: we introduce no additional\nhyperparameters in the loss function, and train with regular cross entropy. From a practical standpoint, this is especially useful for deep learning practitioners who might not wish\nto make modifications to the loss function while training deep models. In addition, since outlier data is simply an additional training class, no architectural modifications to existing networks are needed. • Due to the simplicity and effectiveness of this method, we argue that this approach be\nconsidered a strong baseline for comparing new methods in the field of OoD detection."},{"heading":"2 OUT-OF-DISTRIBUTION DETECTION WITH AN ABSTAINING CLASSIFIER","text":"(DAC)\nOur approach uses a DNN trained with an extra abstention class for detecting out-of-distribution and novel samples; from here on, we will refer to this as the deep abstaining classifier (DAC). We augment our training set of in-distribution samples (Din) with an auxiliary dataset of known out-of-distribution samples (D̃out), that are known to be mostly disjoint from the main training set (we will use Dout to denote unknown out-of-distribution samples that we use for testing). We assign the training label of K + 1 to all the outlier samples in D̃out (where K is the number of known classes) and train with cross-entropy; the minimization problem then becomes:\nmin θ E(x,y)∼Din [− logPθ(y = ŷ|x)] + Ex,y∼D̃out [− logPθ(y = K + 1|x)] (1)\nwhere θ are the weights of the neural network. This is somewhat similar to the approaches described in Hendrycks et al. (2018) as well as in Lee et al. (2017), with the main difference being that in those methods, an extra class is not used; instead predictions on outliers are regularized against the uniform distribution. Further the loss on the outlier samples is weighted by a hyperparameter λ which has to be tuned; in contrast, our approach does not introduce any additional hyperparameters.\nIn our experiments, we find that the presence of an abstention class that is used to capture the mass in D̃out significantly increases the ability to detect Dout during testing. For example, in Figure 1, we show the distribution of the winning logits (pre-softmax activations) in a regular DNN (left). For the same experimental setup, the abstention logit of the DAC produces near-perfect separation of the in and outof-distribution logits indicating that using an abstention class for mapping outliers can be a\nvery effective approach to OoD detection. Theoretically, it might be argued that the abstention class might only capture data that is aligned with the weight vector of that class, and thus this approach might fail to detect the myriad of OoD inputs that might span the entire input region. Comprehensive experiments over a wide variety of benchmarks described in the subsequent section, however, empirically demonstrate that while the detection is not perfect, it performs very well, and indeed, much better than more complicated approaches.\nOnce the model is trained, we use a simple thresholding mechanism for detection. Concretely, the detector, g(x) : X → 0, 1 assigns label 1 (OoD) if the softmax score of the abstention class, i.e., pK+1(x) is above some threshold δ, and label 0, otherwise:\ng(x) = { 1 if pK+1(x) ≥ δ 0 otherwise (2)\nLike in other methods, the threshold δ has to be determined based on acceptable risk that might be specific to the application. However, using performance metrics like area under the ROC curve (AUROC), we can determine threshold-independent performance of various methods, and we use this as one of our evaluation metrics in all our experiments."},{"heading":"3 EXPERIMENTS","text":"The experiments we describe here can be divided into two sets: in the first set, we compare against methods that are explicitly designed for OoD detection, while in the second category, we compare\nagainst methods that are known to improve predictive uncertainty in deep learning. In both cases, we report results over a variety of architectures to demonstrate the efficacy of our method."},{"heading":"3.1 DATASETS","text":"For all computer vision experiments, we use CIFAR-10 and CIFAR-100 Krizhevsky & Hinton (2009) as the in-distribution datasets, in addition to augmenting our training set with 100K unlabeled samples from the Tiny Images dataset Torralba et al. (2008). For the out-of-distribution datasets, we test on the following:\n• SVHN Netzer et al. (2011), a large set of 32×32 color images of house numbers, comprising of ten classes of digits 0− 9. We use a subset of the 26K images in the test set. • LSUN Yu et al. (2015), the Large-scale Scene Understanding dataset, comprising of 10\ndifferent types of scenes.\n• Places365 Zhou et al. (2017), a large collection of pictures of scenes that fall into one of 365 classes.\n• Tiny ImageNet tin (2017) (not to be confused with Tiny Images) which consists of images belonging to 200 categories that are a subset of ImageNet categories. The images are 64×64 color, which we scale down to 32× 32 when testing. • Gaussian A synthetically generated dataset consisting of 32× 32 random Gaussian noise\nimages, where each pixel is sampled from an i.i.d Gaussian distribution.\nFor the NLP experiments, we use 20 Newsgroup Lang (1995), TREC Sherman, and SST Socher et al. (2013) datasets as our in-distribution datasets, which are the same as those used by Hendrycks et al. (2018) to facilitate direct comparison. We use the 50-category version of TREC, and for SST, we use binarized labels where neutral samples are removed. For out OoD training data, we use unlabeled samples from Wikitext2 by assigning them to the abstention class. We test our model on the following OoD datasets:\n• SNLI Bowman et al. (2015) is a dataset of predicates and hypotheses for natural language inference. We use the hypotheses for testing .\n• IMDB Maas et al. (2011) is a sentiment classification dataset of movie reviews, with similar statistics to those of SST.\n• Multi30K Barrault et al. (2018) is a dataset of English-German image descriptions, of which we use the English descriptions.\n• WMT16 Bojar et al. (2016) is a dataset of English-German language pairs designed for machine translation task. We use the English portion of the test set from WMT16.\n• Yelp Zhang et al. (2015) is a dataset of restaurant reviews."},{"heading":"3.2 COMPARISON AGAINST OOD METHODS","text":"In this section, we compare against a slew of recent state-of-the-art methods that have been explicitly designed for OoD detection. For the image experiments, we compare against the following:\n• Deep Outlier Exposure, as described in Hendrycks et al. (2018) and discussed in Section 1 • Ensemble of Leave-out Classifiers Vyas et al. (2018) where each classifier is trained by\nleaving out a random subset of training data (which is treated as OoD data), and the rest is treated as ID data.\n• ODIN, as described in Liang et al. (2018) and discussed in Section 1. ODIN uses input perturbation and temperature scaling to differentiate between ID and OoD samples.\n• Deep Mahalanobis Detector, proposed in Lee et al. (2018) which estimates the classconditional distribution over hidden layer features of a deep model using Gaussian discriminant analysis and a Mahalanobis distance based confidence-score for thresholding, and further, similar to ODIN, uses input perturbation while testing.\n• OpenMax, as described in Bendale & Boult (2016) for novel category detection. This method uses mean activation vectors of ID classes observed during training followed by Weibull fitting to determine if a given sample is novel or out-of-distribution.\nFor all of the above methods, we use published results when available, keeping the architecture and datasets the same as in the experiments described in the respective papers. For the NLP experiments, we only compare against the published results in Hendrycks et al. (2018). For OpenMax, we re-implement the authors’ published algorithm using the PyTorch framework Paszke et al. (2019)."},{"heading":"3.2.1 METRICS","text":"Following established practices in the literature, we use the following metrics to measure detection performance of our method:\n• AUROC or Area Under the Receiver Operating Characteristic curve depicts the relationship between the True Positive Rate (TPR) (also known as Recall)and the False Positive Rate (FPR) and can be interpreted as the probability that a positive example is assigned a higher detection score than a negative example Fawcett (2006). Unlike 0/1 accuracy, the AUROC has the desirable property that it is not affected by class imbalance1. • FPR at 95% TPR which is the probability that a negative sample is misclassified as a\npositive sample when the TPR (or recall) on the positive samples is 95%.\nIn work that we compare against, the out-of-distribution samples are treated as the positive class, so we do the same here, and treat the in-distribution samples as the negative class."},{"heading":"3.2.2 RESULTS","text":"Detailed results against the various OoD methods are shown in Tables 1 through 3 for vision and language respectively, where we have a clear trend: in almost all cases, the DAC outperforms the other methods, often by significant margins especially when the in-distribution data is more complex, as is the case with CIFAR-100. While the Outlier Exposure method Hendrycks et al. (2018) (shown at the top in Table 1) is conceptually similar to ours, the presence of an extra abstention class in our model often bestows significant performance advantages. Further, we do not need to tune a separate hyperparameter which determines the weight of the outlier loss, as done in Hendrycks et al. (2018).\nIn fact, the simplicity of our method is one of its striking features: we do not introduce any additional hyperparameters in our approach, which makes it significantly easier to implement than methods such as ODIN and the Mahalanobis detector; these methods need to be tuned separately on each OoD dataset, which is usually not possible as one does not have access to the distribution of unseen classes in advance. Indeed, when performance of these methods is tested without tuning on the OoD test set, the DAC significantly outperforms methods such as the Mahalanobis detector (shown at the bottom of Table 1). We also show the performance against the OpenMax approach of Bendale & Boult (2016) in Table 2 and in every case, the DAC outperforms OpenMax by significant margins.\nWhile the abstention approach uses an extra class and OoD samples while training, and thus does incur some training overhead, it is significantly less expensive during test time, as the forward pass is no different from that of a regular DNN. In contrast, methods like ODIN and the Mahalanobis detector require gradient calculation with respect to the input in order to apply the input perturbation; the DAC approach thus offers a computationally simpler alternative. Also, even though the DAC approach introduces additional network parameters in the final linear layers (due to the presence of an extra abstention class), and thus might be more prone to overfitting, we find that this to be not the case as evidenced by the generalization of OoD performance to different types of test datasets."},{"heading":"3.3 COMPARISON AGAINST UNCERTAINTY-BASED METHODS","text":"Next we perform experiments to compare the OoD detection performance of the DAC against various methods that have been proposed for improving predictive uncertainty in deep learning. In these cases,\n1An alternate area-under-the-curve metric, known as Area under Precision Recall Curve, or AUPRC, is used when the size of the negative class is high compared to the positive class. We do not report AUPRC here, as we keep our in-distribution and out-of-distribution sets balanced in these experiments.\nhttps://github.com/hendrycks/outlier-exposure\none expects that such methods will cause the DNN to predict with less confidence when presented with inputs from a different distribution or from novel categories; we compare against the following methods:\n• Softmax Thresholding This is the simplest baseline, where OoD samples are detected by thresholding on the winning softmax score; scores falling below a threshold are rejected.\n• Entropy Thresholding Another simple baseline, where OoD samples are rejected if the Shannon entropy calculated over the softmax posteriors is above a certain threshold.\n• MonteCarlo Dropout A Bayesian inspired approach proposed in Gal & Ghahramani (2016) for improving the predictive uncertainty for deep learning. We found a dropout probability of p = 0.5 to perform well, and use 100 forward passes per sample during the prediction.\n• Temperature Scaling, which improves DNN calibration as described in Guo et al. (2017). The scaling temperature T is tuned on a held-out subset of the validation set of the indistribution data.\n• Mixup As shown in Thulasidasan et al. (2019), Mixup can be an effective OoD detector, so we also use this as one of our baselines.\n• Deep Ensembles which was introduced in Lakshminarayanan et al. (2017) for improving uncertainty estimates for both classification and regression. In this approach, multiple versions of the same model are trained using different random initializations, and while training, adversarial samples are generated to improve model robustness. We use an ensemble size of 5 as suggested in their paper.\n• SWAG, as described in Maddox et al. (2019), which is a Bayesian approach to deep learning and exploits the fact that SGD itself can be viewed as approximate Bayesian inference Mandt et al. (2017). We use an ensemble size of 30 as proposed in the original paper."},{"heading":"3.3.1 RESULTS","text":"Detailed results are shown in Table 4, where the best performing method for each metric is shown in bold. The DAC is the only method in this set of experiments that uses an augmented dataset, and as is clearly evident from the results, this confers a significant advantage over the other methods in most cases. Calibration methods like temperature scaling, while producing well calibrated scores on in-distribution data, end up reducing the confidence on in-distribution data as well, and thus losing discriminative power between the two types of data. We also note here that many of the methods listed in the table, like temperature scaling and deep ensembles, can be combined with the\nabstention approach. Indeed, the addition of an extra abstention class and training with OoD data is compatible with most uncertainty modeling techniques in deep learning; we leave the exploration of such combination approaches for future work."},{"heading":"4 CONCLUSION","text":"We presented a simple, but highly effective method for open set and out-of-distribution detection that clearly demonstrated the efficacy of using an extra abstention class and augmenting the training set with outliers. While previous work has shown the efficacy of outlier exposure Hendrycks et al. (2018), here we demonstrated an alternative approach for exploiting outlier data that further improves upon existing methods, while also being simpler to implement compared to many of the other methods. The ease of implementation, absence of additional hyperparameter tuning and computational efficiency during testing makes this a very viable approach for improving out-of-distribution and novel category detection in real-world deployments; we hope that this will also serve as an effective baseline for comparing future work in this domain."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION AND RELATED WORK\",\n \"text\": \"Most of supervised machine learning has been developed with the assumption that the distribution of classes seen at train and test time are the same. However, the real-world is unpredictable and open-ended, and making machine learning systems robust to the presence of unknown categories and out-of-distribution samples has become increasingly essential for their safe deployment. While refraining from predicting when uncertain should be intuitively obvious to humans, the peculiarities of DNNs makes them overconfident to unknown inputs Nguyen et al. (2015) and makes this a challenging problem to solve in deep learning.\\nA very active sub-field of deep learning, known as out-of-distribution (OoD) detection, has emerged in recent years that attempts to impart to deep neural networks the quality of \\\"knowing when it doesn’t know\\\". The most straight-forward approach in this regard is based on using the DNNs output as a proxy for predictive confidence. For example, a simple baseline for detecting OoD samples using thresholded softmax scores was presented in Hendrycks & Gimpel (2016). where the authors provided empirical evidence that for DNN classifiers, in-distribution predictions do tend to have higher winning scores than OoD samples, thus empirically justifying the use of softmax thresholding as a useful baseline. However this approach is vulnerable to the pathologies discussed in Nguyen et al. (2015). Subsequently, increasingly sophisticated methods have been developed to attack the OoD problem. Liang et al. (2018) introduced a detection technique that involves perturbing the inputs in the direction of increasing the confidence of the network’s predictions on a given input, based on the observation that the magnitude of gradients on in-distribution data tend to be larger than for OoD data. The method proposed in Lee et al. (2018) also involves input perturbation, but confidence in this case was measured by the Mahalanobis distance score using the computed mean and covariance of the pre-softmax scores. A drawback of such methods, however, is that it introduces a number of\\nhyperparameters that need to be tuned on the OoD dataset, which is infeasible in many real-world scenarios as one does not often know in advance the properties of unknown classes. A modified version of the perturbation approach was recently proposed in in Hsu et al. (2020) that circumvents some of these issues, though one still needs to ascertain an ideal perturbation magnitude, which might not generalize from one OoD set to the other.\\nGiven that one might expect a classifier to be more uncertain when faced with OoD data, many methods developed for estimating uncertainty for DNN predictions have also been used for OoD detection. A useful baseline in this regard is the temperature scaling method of Guo et al. (2017) that was was proposed for calibrating DNN predictions on in-distribution data and has been observed to also serve as a useful OoD detector in some scenarios. Further, label smoothing techniques like mixup Zhang et al. (2017) have also been shown to be able to improve OoD detection performance in DNNs Thulasidasan et al. (2019). An ensemble-of-deep models approach, that is also augmented with adversarial examples during training, described in Lakshminarayanan et al. (2017) was also shown to improve predictive uncertainty and succesfully applied to OoD detection.\\nIn the Bayesian realm, methods such as Maddox et al. (2019) and Osawa et al. (2019) have also been used for OoD detection, though at increased computational cost. However, it has been argued that for OoD detection, Bayesian priors on the data are not completely justified since one does not have access to the prior of the open-set Boult et al. (2019). Nevertheless, simple approaches like dropout – which have been shown to be equivalent to deep gaussian processes Gal & Ghahramani (2016) – have been used as baselines for OoD detection.\\nTraining the model to recognize unknown classes by using data from categories that do not overlap with classes of interest has been shown to be quite effective for out-of-distribution detection and a slew of methods that use additional data for discriminating between ID and OD data have been proposed. DeVries & Taylor (2018) describes a method ithat uses a separate confidence branch and misclassified training data samples that serve as a proxy for OoD samples. In the outlier exposure technique described in Hendrycks et al. (2018), the predictions on natural outlier images used in training are regularized against the uniform distribution to encourage high-entropy posteriors on outlier samples. An approach that uses an extra-class for outlier samples is described in Neal et al. (2018), where instead of natural outliers, counterfactual images that lie just outside the class boundaries of known classes are generated using a GAN and assigned the extra class label. A similar approach using generative samples for the extra class, but using a conditional Variational Auto-Encoders Kingma & Welling (2013) for generation, is described in Vernekar et al. (2019). A method to force a DNN to produce high-entropy (i.e., low confidence) predictions and suppress the magnitude of feature activations for OoD samples was discussed in Dhamija et al. (2018), where, arguing that methods that use an extra background class for OoD samples force all such samples to lie in one region of the feature space, the work also forces separation by suppressing the activation magnitudes of samples from unknown classes\\nThe above works have shown that the use of known OoD samples (or known unknowns) often generalizes well to unknown unknown samples. Ineed, even though the space of unknown classes is potentially infinite, and one can never know in advance the myriad of inputs that can occur during test time, empirically this approach has been shown to work. The abstention method that we describe in the next section borrows ideas from many of the above methods: as in Hendrycks et al. (2018), we uses additional samples of real images and text from non-overlapping categories to train the model to abstain, but instead of entropy regularization over OoD samples, out method uses an extra abstention class. While it has been sometimes argued in the literature that that using an additional abstention (or rejection) class is not an effective approach for OoD detection Dhamija et al. (2018); Lee et al. (2017), comprehensive experiments we conduct in this work demonstrate that this is not the case. Indeed, we find that such an approach is not only simple but also highly effective for OoD detection, often outperforming existing methods that are more complicated and involve tuning of multiple hyperparameters. The main contributions of this work are as follows:\\n• To the best of our knowledge, this is the first work to comprehensively demonstrate the efficacy of using an extra abstention (or rejection class) in combination with outlier training data for effective OoD detection. • In addition to being effective, our method is also simple: we introduce no additional\\nhyperparameters in the loss function, and train with regular cross entropy. From a practical standpoint, this is especially useful for deep learning practitioners who might not wish\\nto make modifications to the loss function while training deep models. In addition, since outlier data is simply an additional training class, no architectural modifications to existing networks are needed. • Due to the simplicity and effectiveness of this method, we argue that this approach be\\nconsidered a strong baseline for comparing new methods in the field of OoD detection.\"\n },\n {\n \"heading\": \"2 OUT-OF-DISTRIBUTION DETECTION WITH AN ABSTAINING CLASSIFIER\",\n \"text\": \"(DAC)\\nOur approach uses a DNN trained with an extra abstention class for detecting out-of-distribution and novel samples; from here on, we will refer to this as the deep abstaining classifier (DAC). We augment our training set of in-distribution samples (Din) with an auxiliary dataset of known out-of-distribution samples (D̃out), that are known to be mostly disjoint from the main training set (we will use Dout to denote unknown out-of-distribution samples that we use for testing). We assign the training label of K + 1 to all the outlier samples in D̃out (where K is the number of known classes) and train with cross-entropy; the minimization problem then becomes:\\nmin θ E(x,y)∼Din [− logPθ(y = ŷ|x)] + Ex,y∼D̃out [− logPθ(y = K + 1|x)] (1)\\nwhere θ are the weights of the neural network. This is somewhat similar to the approaches described in Hendrycks et al. (2018) as well as in Lee et al. (2017), with the main difference being that in those methods, an extra class is not used; instead predictions on outliers are regularized against the uniform distribution. Further the loss on the outlier samples is weighted by a hyperparameter λ which has to be tuned; in contrast, our approach does not introduce any additional hyperparameters.\\nIn our experiments, we find that the presence of an abstention class that is used to capture the mass in D̃out significantly increases the ability to detect Dout during testing. For example, in Figure 1, we show the distribution of the winning logits (pre-softmax activations) in a regular DNN (left). For the same experimental setup, the abstention logit of the DAC produces near-perfect separation of the in and outof-distribution logits indicating that using an abstention class for mapping outliers can be a\\nvery effective approach to OoD detection. Theoretically, it might be argued that the abstention class might only capture data that is aligned with the weight vector of that class, and thus this approach might fail to detect the myriad of OoD inputs that might span the entire input region. Comprehensive experiments over a wide variety of benchmarks described in the subsequent section, however, empirically demonstrate that while the detection is not perfect, it performs very well, and indeed, much better than more complicated approaches.\\nOnce the model is trained, we use a simple thresholding mechanism for detection. Concretely, the detector, g(x) : X → 0, 1 assigns label 1 (OoD) if the softmax score of the abstention class, i.e., pK+1(x) is above some threshold δ, and label 0, otherwise:\\ng(x) = { 1 if pK+1(x) ≥ δ 0 otherwise (2)\\nLike in other methods, the threshold δ has to be determined based on acceptable risk that might be specific to the application. However, using performance metrics like area under the ROC curve (AUROC), we can determine threshold-independent performance of various methods, and we use this as one of our evaluation metrics in all our experiments.\"\n },\n {\n \"heading\": \"3 EXPERIMENTS\",\n \"text\": \"The experiments we describe here can be divided into two sets: in the first set, we compare against methods that are explicitly designed for OoD detection, while in the second category, we compare\\nagainst methods that are known to improve predictive uncertainty in deep learning. In both cases, we report results over a variety of architectures to demonstrate the efficacy of our method.\"\n },\n {\n \"heading\": \"3.1 DATASETS\",\n \"text\": \"For all computer vision experiments, we use CIFAR-10 and CIFAR-100 Krizhevsky & Hinton (2009) as the in-distribution datasets, in addition to augmenting our training set with 100K unlabeled samples from the Tiny Images dataset Torralba et al. (2008). For the out-of-distribution datasets, we test on the following:\\n• SVHN Netzer et al. (2011), a large set of 32×32 color images of house numbers, comprising of ten classes of digits 0− 9. We use a subset of the 26K images in the test set. • LSUN Yu et al. (2015), the Large-scale Scene Understanding dataset, comprising of 10\\ndifferent types of scenes.\\n• Places365 Zhou et al. (2017), a large collection of pictures of scenes that fall into one of 365 classes.\\n• Tiny ImageNet tin (2017) (not to be confused with Tiny Images) which consists of images belonging to 200 categories that are a subset of ImageNet categories. The images are 64×64 color, which we scale down to 32× 32 when testing. • Gaussian A synthetically generated dataset consisting of 32× 32 random Gaussian noise\\nimages, where each pixel is sampled from an i.i.d Gaussian distribution.\\nFor the NLP experiments, we use 20 Newsgroup Lang (1995), TREC Sherman, and SST Socher et al. (2013) datasets as our in-distribution datasets, which are the same as those used by Hendrycks et al. (2018) to facilitate direct comparison. We use the 50-category version of TREC, and for SST, we use binarized labels where neutral samples are removed. For out OoD training data, we use unlabeled samples from Wikitext2 by assigning them to the abstention class. We test our model on the following OoD datasets:\\n• SNLI Bowman et al. (2015) is a dataset of predicates and hypotheses for natural language inference. We use the hypotheses for testing .\\n• IMDB Maas et al. (2011) is a sentiment classification dataset of movie reviews, with similar statistics to those of SST.\\n• Multi30K Barrault et al. (2018) is a dataset of English-German image descriptions, of which we use the English descriptions.\\n• WMT16 Bojar et al. (2016) is a dataset of English-German language pairs designed for machine translation task. We use the English portion of the test set from WMT16.\\n• Yelp Zhang et al. (2015) is a dataset of restaurant reviews.\"\n },\n {\n \"heading\": \"3.2 COMPARISON AGAINST OOD METHODS\",\n \"text\": \"In this section, we compare against a slew of recent state-of-the-art methods that have been explicitly designed for OoD detection. For the image experiments, we compare against the following:\\n• Deep Outlier Exposure, as described in Hendrycks et al. (2018) and discussed in Section 1 • Ensemble of Leave-out Classifiers Vyas et al. (2018) where each classifier is trained by\\nleaving out a random subset of training data (which is treated as OoD data), and the rest is treated as ID data.\\n• ODIN, as described in Liang et al. (2018) and discussed in Section 1. ODIN uses input perturbation and temperature scaling to differentiate between ID and OoD samples.\\n• Deep Mahalanobis Detector, proposed in Lee et al. (2018) which estimates the classconditional distribution over hidden layer features of a deep model using Gaussian discriminant analysis and a Mahalanobis distance based confidence-score for thresholding, and further, similar to ODIN, uses input perturbation while testing.\\n• OpenMax, as described in Bendale & Boult (2016) for novel category detection. This method uses mean activation vectors of ID classes observed during training followed by Weibull fitting to determine if a given sample is novel or out-of-distribution.\\nFor all of the above methods, we use published results when available, keeping the architecture and datasets the same as in the experiments described in the respective papers. For the NLP experiments, we only compare against the published results in Hendrycks et al. (2018). For OpenMax, we re-implement the authors’ published algorithm using the PyTorch framework Paszke et al. (2019).\"\n },\n {\n \"heading\": \"3.2.1 METRICS\",\n \"text\": \"Following established practices in the literature, we use the following metrics to measure detection performance of our method:\\n• AUROC or Area Under the Receiver Operating Characteristic curve depicts the relationship between the True Positive Rate (TPR) (also known as Recall)and the False Positive Rate (FPR) and can be interpreted as the probability that a positive example is assigned a higher detection score than a negative example Fawcett (2006). Unlike 0/1 accuracy, the AUROC has the desirable property that it is not affected by class imbalance1. • FPR at 95% TPR which is the probability that a negative sample is misclassified as a\\npositive sample when the TPR (or recall) on the positive samples is 95%.\\nIn work that we compare against, the out-of-distribution samples are treated as the positive class, so we do the same here, and treat the in-distribution samples as the negative class.\"\n },\n {\n \"heading\": \"3.2.2 RESULTS\",\n \"text\": \"Detailed results against the various OoD methods are shown in Tables 1 through 3 for vision and language respectively, where we have a clear trend: in almost all cases, the DAC outperforms the other methods, often by significant margins especially when the in-distribution data is more complex, as is the case with CIFAR-100. While the Outlier Exposure method Hendrycks et al. (2018) (shown at the top in Table 1) is conceptually similar to ours, the presence of an extra abstention class in our model often bestows significant performance advantages. Further, we do not need to tune a separate hyperparameter which determines the weight of the outlier loss, as done in Hendrycks et al. (2018).\\nIn fact, the simplicity of our method is one of its striking features: we do not introduce any additional hyperparameters in our approach, which makes it significantly easier to implement than methods such as ODIN and the Mahalanobis detector; these methods need to be tuned separately on each OoD dataset, which is usually not possible as one does not have access to the distribution of unseen classes in advance. Indeed, when performance of these methods is tested without tuning on the OoD test set, the DAC significantly outperforms methods such as the Mahalanobis detector (shown at the bottom of Table 1). We also show the performance against the OpenMax approach of Bendale & Boult (2016) in Table 2 and in every case, the DAC outperforms OpenMax by significant margins.\\nWhile the abstention approach uses an extra class and OoD samples while training, and thus does incur some training overhead, it is significantly less expensive during test time, as the forward pass is no different from that of a regular DNN. In contrast, methods like ODIN and the Mahalanobis detector require gradient calculation with respect to the input in order to apply the input perturbation; the DAC approach thus offers a computationally simpler alternative. Also, even though the DAC approach introduces additional network parameters in the final linear layers (due to the presence of an extra abstention class), and thus might be more prone to overfitting, we find that this to be not the case as evidenced by the generalization of OoD performance to different types of test datasets.\"\n },\n {\n \"heading\": \"3.3 COMPARISON AGAINST UNCERTAINTY-BASED METHODS\",\n \"text\": \"Next we perform experiments to compare the OoD detection performance of the DAC against various methods that have been proposed for improving predictive uncertainty in deep learning. In these cases,\\n1An alternate area-under-the-curve metric, known as Area under Precision Recall Curve, or AUPRC, is used when the size of the negative class is high compared to the positive class. We do not report AUPRC here, as we keep our in-distribution and out-of-distribution sets balanced in these experiments.\\nhttps://github.com/hendrycks/outlier-exposure\\none expects that such methods will cause the DNN to predict with less confidence when presented with inputs from a different distribution or from novel categories; we compare against the following methods:\\n• Softmax Thresholding This is the simplest baseline, where OoD samples are detected by thresholding on the winning softmax score; scores falling below a threshold are rejected.\\n• Entropy Thresholding Another simple baseline, where OoD samples are rejected if the Shannon entropy calculated over the softmax posteriors is above a certain threshold.\\n• MonteCarlo Dropout A Bayesian inspired approach proposed in Gal & Ghahramani (2016) for improving the predictive uncertainty for deep learning. We found a dropout probability of p = 0.5 to perform well, and use 100 forward passes per sample during the prediction.\\n• Temperature Scaling, which improves DNN calibration as described in Guo et al. (2017). The scaling temperature T is tuned on a held-out subset of the validation set of the indistribution data.\\n• Mixup As shown in Thulasidasan et al. (2019), Mixup can be an effective OoD detector, so we also use this as one of our baselines.\\n• Deep Ensembles which was introduced in Lakshminarayanan et al. (2017) for improving uncertainty estimates for both classification and regression. In this approach, multiple versions of the same model are trained using different random initializations, and while training, adversarial samples are generated to improve model robustness. We use an ensemble size of 5 as suggested in their paper.\\n• SWAG, as described in Maddox et al. (2019), which is a Bayesian approach to deep learning and exploits the fact that SGD itself can be viewed as approximate Bayesian inference Mandt et al. (2017). We use an ensemble size of 30 as proposed in the original paper.\"\n },\n {\n \"heading\": \"3.3.1 RESULTS\",\n \"text\": \"Detailed results are shown in Table 4, where the best performing method for each metric is shown in bold. The DAC is the only method in this set of experiments that uses an augmented dataset, and as is clearly evident from the results, this confers a significant advantage over the other methods in most cases. Calibration methods like temperature scaling, while producing well calibrated scores on in-distribution data, end up reducing the confidence on in-distribution data as well, and thus losing discriminative power between the two types of data. We also note here that many of the methods listed in the table, like temperature scaling and deep ensembles, can be combined with the\\nabstention approach. Indeed, the addition of an extra abstention class and training with OoD data is compatible with most uncertainty modeling techniques in deep learning; we leave the exploration of such combination approaches for future work.\"\n },\n {\n \"heading\": \"4 CONCLUSION\",\n \"text\": \"We presented a simple, but highly effective method for open set and out-of-distribution detection that clearly demonstrated the efficacy of using an extra abstention class and augmenting the training set with outliers. While previous work has shown the efficacy of outlier exposure Hendrycks et al. (2018), here we demonstrated an alternative approach for exploiting outlier data that further improves upon existing methods, while also being simpler to implement compared to many of the other methods. The ease of implementation, absence of additional hyperparameter tuning and computational efficiency during testing makes this a very viable approach for improving out-of-distribution and novel category detection in real-world deployments; we hope that this will also serve as an effective baseline for comparing future work in this domain.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":89,"cells":{"paper_id":{"kind":"string","value":"SP:70b8c75426f18a3dc4a359c8a8cd7dd2076953a0"},"summaries":{"kind":"list like","value":["The authors propose to address the robustness over outliers for optimal transport (OT). They propose a new formulation based on penalizing the contaminated probability measures by a signed measure (which shares a close relation with unbalanced OT). The authors further derive an equivalent formulation by adjusting the cost matrix for the corresponding standard OT. Empirically, the authors evaluate their proposed approach on a toy example of robust mean estimation and outlier detection for data collection. "],"string":"[\n \"The authors propose to address the robustness over outliers for optimal transport (OT). They propose a new formulation based on penalizing the contaminated probability measures by a signed measure (which shares a close relation with unbalanced OT). The authors further derive an equivalent formulation by adjusting the cost matrix for the corresponding standard OT. Empirically, the authors evaluate their proposed approach on a toy example of robust mean estimation and outlier detection for data collection. \"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Optimal transport (OT) provides a way of measuring distances between distributions that depends on the geometry of the sample space. In light of recent advances in solving the OT problem, OT distances are widely used as loss functions in minimum distance estimation. Despite its prevalence and advantages, however, OT is extremely sensitive to outliers. A single adversarially-picked outlier can increase OT distance arbitrarily. To address this issue, in this work we propose an outlier-robust OT formulation. Our formulation is convex but challenging to scale at a first glance. We proceed by deriving an equivalent formulation based on cost truncation that is easy to incorporate into modern stochastic algorithms for regularized OT. We demonstrate our model applied to mean estimation under the Huber contamination model in simulation as well as outlier detection on real data."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["David Alvarez-Melis","Tommi S Jaakkola"],"title":"Gromov-Wasserstein alignment of word embedding spaces","venue":null,"year":2018},{"authors":["Yogesh Balaji","Rama Chellappa","Soheil Feizi"],"title":"Robust optimal transport with applications in generative modeling and domain adaptation","venue":"Advances in Neural Information Processing Systems,","year":2020},{"authors":["Federico Bassetti","Antonella Bodini","Eugenio Regazzini"],"title":"On minimum Kantorovich distance estimators","venue":"Statistics & Probability Letters,","year":2006},{"authors":["Luis A Caffarelli","Robert J McCann"],"title":"Free boundaries in optimal transport and Monge-Ampere obstacle problems","venue":"Annals of Mathematics,","year":2010},{"authors":["Gao Chao","Yao Yuan","Zhu Weizhi"],"title":"Robust estimation via generative adversarial networks","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["Lenaic Chizat","Gabriel Peyré","Bernhard Schmitzer","François-Xavier Vialard"],"title":"Scaling algorithms for unbalanced optimal transport problems","venue":"Mathematics of Computation,","year":2018},{"authors":["Lenaı̈c Chizat"],"title":"Unbalanced optimal transport: Models, numerical methods, applications","venue":"Numerical Analysis [math.NA]. Université Paris sciences et lettres,","year":2017},{"authors":["Nicolas Courty","Rémi Flamary","Devis Tuia"],"title":"Domain adaptation with regularized optimal transport","venue":"In Joint European Conference on Machine Learning and Knowledge Discovery in Databases,","year":2014},{"authors":["Nicolas Courty","Rémi Flamary","Amaury Habrard","Alain Rakotomamonjy"],"title":"Joint distribution optimal transportation for domain adaptation","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["Marco Cuturi"],"title":"Sinkhorn Distances: Lightspeed Computation of Optimal Transport","venue":"Advances in Neural Information Processing Systems","year":2013},{"authors":["Steven Diamond","Stephen Boyd"],"title":"CVXPY: A Python-embedded modeling language for convex optimization","venue":"Journal of Machine Learning Research,","year":2016},{"authors":["Alessio Figalli"],"title":"The optimal partial transport problem","venue":"Archive for Rational Mechanics and Analysis,","year":2010},{"authors":["Rémi Flamary","Nicolas Courty"],"title":"POT Python optimal transport library, 2017","venue":"URL https: //pythonot.github.io/","year":2017},{"authors":["Aude Genevay","Marco Cuturi","Gabriel Peyré","Francis Bach"],"title":"Stochastic optimization for largescale optimal transport","venue":"In Advances in Neural Information Processing Systems,","year":2016},{"authors":["Nhat Ho","XuanLong Nguyen","Mikhail Yurochkin","Hung Hai Bui","Viet Huynh","Dinh Phung"],"title":"Multilevel clustering via Wasserstein means","venue":"In International Conference on Machine Learning,","year":2017},{"authors":["Gao Huang","Chuan Guo","Matt J Kusner","Yu Sun","Fei Sha","Kilian Q Weinberger"],"title":"Supervised word mover’s distance","venue":"In Advances in Neural Information Processing Systems,","year":2016},{"authors":["Peter J. Huber","Elvezio Ronchetti"],"title":"Robust Statistics. Wiley Series in Probability and Statistics","venue":"2nd ed edition,","year":2009},{"authors":["Leonid Vitalievich Kantorovich"],"title":"On the translocation of masses","venue":"In Dokl. Akad. Nauk. USSR (NS),","year":1942},{"authors":["Matt Kusner","Yu Sun","Nicholas Kolkin","Kilian Weinberger"],"title":"From Word Embeddings To Document Distances","venue":"In International Conference on Machine Learning,","year":2015},{"authors":["Matthias Liero","Alexander Mielke","Giuseppe Savaré"],"title":"Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures","venue":"Inventiones Mathematicae,","year":2018},{"authors":["Sebastian Nowozin","Botond Cseke","Ryota Tomioka"],"title":"f-GAN: Training generative neural samplers using variational divergence minimization","venue":"In Advances in Neural Information Processing Systems,","year":2016},{"authors":["Ofir Pele","Michael Werman"],"title":"Fast and robust earth mover’s distances","venue":"IEEE 12th International Conference on Computer Vision,","year":2009},{"authors":["Gabriel Peyré","Marco Cuturi"],"title":"Computational Optimal Transport","venue":"[stat],","year":2018},{"authors":["Benedetto Piccoli","Francesco Rossi"],"title":"Generalized Wasserstein distance and its application to transport equations with source","venue":"Archive for Rational Mechanics and Analysis,","year":2014},{"authors":["Yanyao Shen","Sujay Sanghavi"],"title":"Learning with bad training data via iterative trimmed loss minimization","venue":"In International Conference on Machine Learning,","year":2019},{"authors":["Justin Solomon","Fernando De Goes","Gabriel Peyré","Marco Cuturi","Adrian Butscher","Andy Nguyen","Tao Du","Leonidas Guibas"],"title":"Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains","venue":"ACM Transactions on Graphics (TOG),","year":2015},{"authors":["Sanvesh Srivastava","Cheng Li","David B. Dunson"],"title":"Scalable Bayes via Barycenter in Wasserstein Space","venue":"[stat],","year":2018},{"authors":["Guillaume Staerman","Pierre Laforgue","Pavlo Mozharovskyi","Florence d’Alché Buc"],"title":"When OT meets MOM: Robust estimation of Wasserstein distance","venue":null,"year":2006},{"authors":["Kaiwen Wu","Gavin Weiguang Ding","Ruitong Huang","Yaoliang Yu"],"title":"On Minimax Optimality of GANs for Robust Mean Estimation","venue":"In International Conference on Artificial Intelligence and Statistics,","year":2020},{"authors":["Mikhail Yurochkin","Sebastian Claici","Edward Chien","Farzaneh Mirzazadeh","Justin Solomon"],"title":"Hierarchical Optimal Transport for Document Representation","venue":null,"year":2019},{"authors":[],"title":"limn→∞WC((P + S)n, νn)→ W (P + S,Q)","venue":"(Villani,","year":2009}],"string":"[\n {\n \"authors\": [\n \"David Alvarez-Melis\",\n \"Tommi S Jaakkola\"\n ],\n \"title\": \"Gromov-Wasserstein alignment of word embedding spaces\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Yogesh Balaji\",\n \"Rama Chellappa\",\n \"Soheil Feizi\"\n ],\n \"title\": \"Robust optimal transport with applications in generative modeling and domain adaptation\",\n \"venue\": \"Advances in Neural Information Processing Systems,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Federico Bassetti\",\n \"Antonella Bodini\",\n \"Eugenio Regazzini\"\n ],\n \"title\": \"On minimum Kantorovich distance estimators\",\n \"venue\": \"Statistics & Probability Letters,\",\n \"year\": 2006\n },\n {\n \"authors\": [\n \"Luis A Caffarelli\",\n \"Robert J McCann\"\n ],\n \"title\": \"Free boundaries in optimal transport and Monge-Ampere obstacle problems\",\n \"venue\": \"Annals of Mathematics,\",\n \"year\": 2010\n },\n {\n \"authors\": [\n \"Gao Chao\",\n \"Yao Yuan\",\n \"Zhu Weizhi\"\n ],\n \"title\": \"Robust estimation via generative adversarial networks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Lenaic Chizat\",\n \"Gabriel Peyré\",\n \"Bernhard Schmitzer\",\n \"François-Xavier Vialard\"\n ],\n \"title\": \"Scaling algorithms for unbalanced optimal transport problems\",\n \"venue\": \"Mathematics of Computation,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Lenaı̈c Chizat\"\n ],\n \"title\": \"Unbalanced optimal transport: Models, numerical methods, applications\",\n \"venue\": \"Numerical Analysis [math.NA]. Université Paris sciences et lettres,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Nicolas Courty\",\n \"Rémi Flamary\",\n \"Devis Tuia\"\n ],\n \"title\": \"Domain adaptation with regularized optimal transport\",\n \"venue\": \"In Joint European Conference on Machine Learning and Knowledge Discovery in Databases,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Nicolas Courty\",\n \"Rémi Flamary\",\n \"Amaury Habrard\",\n \"Alain Rakotomamonjy\"\n ],\n \"title\": \"Joint distribution optimal transportation for domain adaptation\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Marco Cuturi\"\n ],\n \"title\": \"Sinkhorn Distances: Lightspeed Computation of Optimal Transport\",\n \"venue\": \"Advances in Neural Information Processing Systems\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Steven Diamond\",\n \"Stephen Boyd\"\n ],\n \"title\": \"CVXPY: A Python-embedded modeling language for convex optimization\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Alessio Figalli\"\n ],\n \"title\": \"The optimal partial transport problem\",\n \"venue\": \"Archive for Rational Mechanics and Analysis,\",\n \"year\": 2010\n },\n {\n \"authors\": [\n \"Rémi Flamary\",\n \"Nicolas Courty\"\n ],\n \"title\": \"POT Python optimal transport library, 2017\",\n \"venue\": \"URL https: //pythonot.github.io/\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Aude Genevay\",\n \"Marco Cuturi\",\n \"Gabriel Peyré\",\n \"Francis Bach\"\n ],\n \"title\": \"Stochastic optimization for largescale optimal transport\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Nhat Ho\",\n \"XuanLong Nguyen\",\n \"Mikhail Yurochkin\",\n \"Hung Hai Bui\",\n \"Viet Huynh\",\n \"Dinh Phung\"\n ],\n \"title\": \"Multilevel clustering via Wasserstein means\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Gao Huang\",\n \"Chuan Guo\",\n \"Matt J Kusner\",\n \"Yu Sun\",\n \"Fei Sha\",\n \"Kilian Q Weinberger\"\n ],\n \"title\": \"Supervised word mover’s distance\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Peter J. Huber\",\n \"Elvezio Ronchetti\"\n ],\n \"title\": \"Robust Statistics. Wiley Series in Probability and Statistics\",\n \"venue\": \"2nd ed edition,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Leonid Vitalievich Kantorovich\"\n ],\n \"title\": \"On the translocation of masses\",\n \"venue\": \"In Dokl. Akad. Nauk. USSR (NS),\",\n \"year\": 1942\n },\n {\n \"authors\": [\n \"Matt Kusner\",\n \"Yu Sun\",\n \"Nicholas Kolkin\",\n \"Kilian Weinberger\"\n ],\n \"title\": \"From Word Embeddings To Document Distances\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Matthias Liero\",\n \"Alexander Mielke\",\n \"Giuseppe Savaré\"\n ],\n \"title\": \"Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures\",\n \"venue\": \"Inventiones Mathematicae,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Sebastian Nowozin\",\n \"Botond Cseke\",\n \"Ryota Tomioka\"\n ],\n \"title\": \"f-GAN: Training generative neural samplers using variational divergence minimization\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Ofir Pele\",\n \"Michael Werman\"\n ],\n \"title\": \"Fast and robust earth mover’s distances\",\n \"venue\": \"IEEE 12th International Conference on Computer Vision,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Gabriel Peyré\",\n \"Marco Cuturi\"\n ],\n \"title\": \"Computational Optimal Transport\",\n \"venue\": \"[stat],\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Benedetto Piccoli\",\n \"Francesco Rossi\"\n ],\n \"title\": \"Generalized Wasserstein distance and its application to transport equations with source\",\n \"venue\": \"Archive for Rational Mechanics and Analysis,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Yanyao Shen\",\n \"Sujay Sanghavi\"\n ],\n \"title\": \"Learning with bad training data via iterative trimmed loss minimization\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Justin Solomon\",\n \"Fernando De Goes\",\n \"Gabriel Peyré\",\n \"Marco Cuturi\",\n \"Adrian Butscher\",\n \"Andy Nguyen\",\n \"Tao Du\",\n \"Leonidas Guibas\"\n ],\n \"title\": \"Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains\",\n \"venue\": \"ACM Transactions on Graphics (TOG),\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Sanvesh Srivastava\",\n \"Cheng Li\",\n \"David B. Dunson\"\n ],\n \"title\": \"Scalable Bayes via Barycenter in Wasserstein Space\",\n \"venue\": \"[stat],\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Guillaume Staerman\",\n \"Pierre Laforgue\",\n \"Pavlo Mozharovskyi\",\n \"Florence d’Alché Buc\"\n ],\n \"title\": \"When OT meets MOM: Robust estimation of Wasserstein distance\",\n \"venue\": null,\n \"year\": 2006\n },\n {\n \"authors\": [\n \"Kaiwen Wu\",\n \"Gavin Weiguang Ding\",\n \"Ruitong Huang\",\n \"Yaoliang Yu\"\n ],\n \"title\": \"On Minimax Optimality of GANs for Robust Mean Estimation\",\n \"venue\": \"In International Conference on Artificial Intelligence and Statistics,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Mikhail Yurochkin\",\n \"Sebastian Claici\",\n \"Edward Chien\",\n \"Farzaneh Mirzazadeh\",\n \"Justin Solomon\"\n ],\n \"title\": \"Hierarchical Optimal Transport for Document Representation\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [],\n \"title\": \"limn→∞WC((P + S)n, νn)→ W (P + S,Q)\",\n \"venue\": \"(Villani,\",\n \"year\": 2009\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Optimal transport is a fundamental problem in applied mathematics. In its original form (Monge, 1781), the problem entails finding the minimum cost way to transport mass from a prescribed probability distribution µ on X to another prescribed distribution ν on X . Kantorovich (1942) relaxed Monge’s formulation of the optimal transport problem to obtain the Kantorovich formulation:\nOT(µ, ν) , min Π∈F(µ,ν)\nE(X1,X2)∼Π [ c(X1, X2) ] , (1.1)\nwhere F(µ, ν) is the set of couplings between µ and ν (probability distributions on X × X whose marginals are µ and ν) and c is a cost function, where we typically assume c(x, y) ≥ 0 and c(x, x) = 0. Compared to other notions of distance between probability distributions, optimal transport uniquely depends on the geometry of the sample space.\nRecent advancements in optimization for optimal transport (Cuturi, 2013; Solomon et al., 2015; Genevay et al., 2016; Seguy et al., 2018) enabled its broad adaptation in machine learning applications where geometry of the data is important. See (Peyré & Cuturi, 2018) for a survey. Optimal transport has found applications in natural language processing (Kusner et al., 2015; Huang et al., 2016; Alvarez-Melis & Jaakkola, 2018; Yurochkin et al., 2019), generative modeling (Arjovsky et al., 2017), clustering (Ho et al., 2017), domain adaptation (Courty et al., 2014; 2017), large-scale Bayesian modeling (Srivastava et al., 2018), and many other domains.\nMany applications use OT as a loss in an optimization problem of the form:\nθ ∈ arg minθ∈Θ OT(µn, νθ), (1.2) where {νθ}θ∈Θ is a collection of parametric models, µn is the empirical distribution of the samples. Such estimators are called minimum Kantorovich estimators (MKE) (Bassetti et al., 2006). They are popular alternatives to likelihood-based estimators, especially in generative modeling. For example, when OT(·, ·) is the Wasserstein-1 distance and νθ is a generator parameterized by a neural network with weights θ, equation 1.2 corresponds to the Wasserstein GAN (Arjovsky et al., 2017).\nOne drawback of optimal transport is its sensitivity to outliers. Because all the mass in µ must be transported to ν, a small fraction of outliers can have an outsized impact on the optimal transport problem. For statistics and machine learning applications in which the data is corrupted or noisy, this is a major issue. For example, the poor performance of Wasserstein GANs in the presence of outliers was noted in the recent works on outlier-robust generative learning with f -divergence GANs (Chao et al., 2018; Wu et al., 2020). The problem of outlier-robustness in MKE has not been studied, with the exception of two concurrent works (Staerman et al., 2020; Balaji et al., 2020).\nIn this paper, we propose a modification of OT to address its sensitivity to outliers. Our formulation can be used as a loss in equation 1.2 so that it is robust to a small fraction of outliers in the data. To keep things simple, we consider the -contamination model (Huber & Ronchetti, 2009). Let νθ0 be a member of a parametric model {νθ : θ ∈ Θ} and let\nµ = (1− )νθ0 + ν̃,\nwhere µ is the data-generating distribution, > 0 is the fraction of outliers, and ν̃ is the distribution of the outliers. Although the fraction of outliers is capped at , the value of the outliers is arbitrary, so the outliers may have an arbitrarily large impact on the optimal transport problem. Our goal is to modify the optimal transport problem so that it is more robust to outliers. We have in mind the downstream application of learning θ0 from (samples from) µ in the -contamination model. Our main contributions are as follows:\n1. We propose a robust OT formulation that is suitable for statistical estimation in the - contamination model using MKE. 2. We show that our formulation is equivalent to the original OT problem with a clipped transport cost. This connection enables us to leverage the voluminous literature on computational optimal transport to develop efficient algorithm to perform MKE robust to outliers. 3. Our formulation enables a new application of optimal transport: outlier detection in data."},{"heading":"2 PROBLEM FORMULATION","text":""},{"heading":"2.1 ROBUST OT FOR MKE","text":"To promote outlier-robustness in MKE, we need to allow the corresponding OT problem to ignore the outliers in the data distribution µ. The -contamination model imposes a cap on the fraction of outliers, so it is not hard to see that ‖µ − νθ0‖TV ≤ , where ‖ · ‖TV is the total-variation norm defined as ‖µ‖TV = ∫ 1 2 |µ(dx)|. This suggests we solve a TV-constrained/regularized version of equation 1.2. The constrained version\nmin θ∈Θ,µ̃\nOT(µ̃, νθ)\nsubject to ‖µ− µ̃‖TV ≤\nsuffers from identification issues. In particular, it cannot distinguish between “clean” distributions within TV distance of νθ0 . This makes it unsuitable as a loss function for statistical estimation, because it cannot lead to a consistent estimator. However, its regularized counterpart\nmin θ∈Θ,s\nOT(µ+ s, νθ) + λ‖s‖TV, (2.1)\nwhere λ > 0 is a regularization parameter, does not suffer from this issue. In the rest of this paper, we work with the TV-regularized formulation equation 2.1.\nThe main idea of our formulation is to allow for modifications of µ, while penalizing their magnitude and ensuring that the modified µ is still a probability measure. Below we formulate this intuition in an optimization problem titled ROBOT (ROBust Optimal Transport):\nFormulation 1:\nROBOT(µ, ν) = minΠ∈F+(Rd×Rd) s∈F(Rd) ∫ C(x, y) Π(dx,dy) + λ‖s‖TV subject to ∫ B×Rd Π(dx, dy) = ∫ B (µ(dx) + s(dx)) ≥ 0 ∀ B ∈ B(Rd) (Borel σ-algebra)∫ Rd×C Π(dx,dy) = ∫ C ν(dy) ∀ C ∈ B(Rd)∫ s(dx) = 0.\n(2.2)\nHere F(Rd) denotes the set of all signed measures with finite total variation on Rd, F+(Rd × Rd) is the set of all measures with finite total variation on Rd × Rd. The first and the last constraints ensure that µ + s is a valid probability measure, while λ‖s‖TV penalizes the amount of modifications in µ. It is worth noting that we can identify exact locations of outliers in µ by inspecting µ+ s, i.e. if µ(x) + s(x) = 0, then x got eliminated and is an outlier.\nROBOT, unlike classical OT, guarantees that an adversarially picked outliers can not increase the distance arbitrarily. Let µ̃ = (1− )µ+ µc, i.e. µ̃ is µ contaminated with outliers from µc, and let ν be an arbitrary measure (in MKE, µ̃ is the contaminated data and ν is the model we learn). Adversary can arbitrarily increase OT(µ̃, ν) by manipulating the outlier distribution µc. For ROBOT we have the following bound:\nTheorem 2.1. Let µ̃ = (1− )µ+ µc for some ∈ [0, 1), then\nROBOT(µ̃, ν) ≤ (OT(µ, ν) + λ ‖µ− µc‖TV) ∧ λ‖µ̃− ν‖TV ∧OT(µ̃, ν). (2.3)\nThis bound has two key takeaways: since TV norm of any two distributions is bounded by 1, adversary can not increase ROBOT(µ̃, ν) arbitrarily; in the absence of outliers, ROBOT is bounded by classical OT. See Appendix C for the proof.\nRelated work We note connection between equation 2.2 and unbalanced OT (UOT) (Chizat., 2017; Chizat et al., 2018). UOT is typically formulated by replacing TV norm with KL(µ + s|µ) and adding an analogous term for ν. Chizat et al. (2018) studied entropy regularized UOT with various divergences penalizing marginal violations. Optimization problems similar to equation 2.2 have also been considered outside of the ML literature (Piccoli & Rossi, 2014; Liero et al., 2018). We are unaware of prior applications of UOT to outlier-robustness, but it was studied in the concurrent work of Balaji et al. (2020). Another relevant variation of OT is partial OT (Figalli, 2010; Caffarelli & McCann, 2010). It may also be considered for outlier-robustness, but it has a drawback of forcing mass destruction rather than adjusting marginals to ignore outliers when they are present. A concurrent work by Staerman et al. (2020) took a different path: they replaced the expectation in the Wasserstein-1 dual with a median-of-means to promote robustness. It is unclear what is the corresponding primal, making it hard to interpret as an optimal transport problem.\nA major challenge with the aforementioned methods, including our Formulation 1, is the difficulty of the optimization problem. This is especially the case for MKEs, where a transport problem has to be solved in every iteration to obtain the gradient of the model parameters. Chizat et al. (2018) proposed a Sinkhorn-like algorithm for entropy regularized UOT, but it is not amenable to stochastic optimization. Balaji et al. (2020) proposed a stochastic optimization algorithm based on the UOT dual, but it requires two additional neural networks (total of four including dual potentials) to parameterize modified marginal distributions (i.e., µ + s and analogous one for ν). Optimizing with a median-of-means in the objective function as in (Staerman et al., 2020) is also challenging. The key contribution of our work is a formulation equivalent to equation 2.2, which is easily compatible with the large body of classical OT optimization techniques (Cuturi, 2013; Solomon et al., 2015; Genevay et al., 2016; Seguy et al., 2018).\nMore efficient equivalent formulation At a first glance, there are two issues with equation 2.2: it appears asymmetric and it is unclear if it can be optimized efficiently. Below we present an equivalent formulation that is free of these issues:\nFormulation 2:\nROBOT(µ, ν) = minΠ∈F+(Rd×Rd) ∫ Cλ(x, y)Π(dx, dy) subject to ∫ B×Rd Π(dx, dy) = ∫ B µ(dx) ∀ B ∈ B(Rd)∫ Rd×C Π(dx, dy) = ∫ C ν(dy) ∀ C ∈ B(Rd), (2.4)\nwhere Cλ is the truncated cost function defined as Cλ(x, y) = C(x, y) ∧ 2λ. Looking at equation 2.4, it is not apparent that it adds robustness to MKE, but it is symmetric, easy to combine\nwith entropic regularization by simply truncating the cost, and benefits from stochastic optimization algorithms (Genevay et al., 2016; Seguy et al., 2018). This formulation also has a distant relation to the idea of loss truncation for achieving robustness (Shen & Sanghavi, 2019). Pele & Werman (2009) considered the Earth Mover Distance (discrete OT) with truncated cost to achieve computational improvements; they also mentioned its potential to promote robustness against outlier noise but did not explore this direction.\nIn Section 3, we establish equivalence between the two ROBOT formulations, equation 2.2 and equation 2.4. This equivalence allows us to obtain an efficient algorithm based on equation 2.4 for robust MKE. We also provide a simple procedure for computing optimal s in equation 2.2 from the solution of equation 2.4, enabling a new OT application: outlier detection. We verify the effectiveness of robust MKE and outlier detection in our experiments in Section 4. Before presenting the equivalence proof, we formulate the discrete analogs of the two ROBOT formulations for their practical value."},{"heading":"2.2 DISCRETE ROBOT FORMULATIONS","text":"In practice we typically encounter samples from the distributions, rather then the distributions themselves. Sampling is also built into stochastic optimization. In this subsection, we present the discrete versions of the ROBOT formulations. The key detail is that, in equation 2.2, µ, ν and s are all supported on Rd, while in the discrete case the empirical measures µn ∈ ∆n−1 and νm ∈ ∆m−1 are supported on a set of points (∆r is the unit probability simplex in Rr). As a result, to formulate a discrete version of equation 2.2, we need to augment µn and νm with each others’ supports. To be precise, let supp(µn) = {X1, . . . , Xn} and supp(νm) = {Y1, . . . , Ym}. Define C = {Z1, Z2, . . . , Zm+n} = {X1, . . . , Xn, Y1, . . . , Ym}. Then discrete analog of equation 2.2 is Formulation 1 (discrete):\nROBOT(µn, νm) = minΠ∈R(m+n)×(m+n) s∈Rm+n 〈Caug,Π〉+ λ [‖s1‖1 + ‖t1‖1] subject to Π1m+n = [ µn + s1 t1 ] , Π>1m+n = [ 0 νm ] Π 0, 1>m+ns = 0, (2.5)\nwhere Caug ∈ R(m+n)×(m+n) is the augmented cost function Caug,i,j = c(Zi, Zj) (c is the ground cost, e.g., squared Euclidean distance), s = (s1, t1) and 1r is the vector all ones in Rr. The TV norm got replaced with its discrete analog, the L1 norm. Similarly to its continuous counterpart, the optimization problem is harder than the typical OT due to additional constraint optimization variable s and increased cost matrix size.\nThe discrete analog of equation 2.4 is straightforward:\nFormulation 2 (discrete):\nROBOT(µn, νm) = { minΠ∈Rn×m 〈Cλ,Π〉 subject to Π1n = µn, Π >1m = νm, Π 0, (2.6)\nwhereCλ,i,j = c(Xi, Yj)∧2λ. As in the continuous case, it is easy to adapt modern (regularized) OT solvers without any computational overhead. As in the continuous case, formulations of equation 2.5 and equation 2.6 are equivalent. It is also possible to recover s of equation 2.5 from the solution of equation 2.6 to perform outlier detection.\nTwo-sided formulation So far we have assumed that one of the input distributions does not have outliers, which is the setting of MKE, where the clean distribution corresponds to the model we learn. In some applications, both distributions may be corrupted. To address this case, we provide an equivalent two-sided formulation, analogous to UOT with TV norm:\nFormulation 3 (two-sided):\nROBOT(µn, νm) = min Π∈R(m+n)×(m+n) s1∈Rm+n, s2∈Rm+n 〈Caug,Π〉+ λ [‖s1‖1 + ‖t1‖1 + ‖s2‖1 + ‖t2‖1] subject to Π1m+n = [ µn + s1 t1 ] , Π>1m+n = [ s2 νm + t2 ] Π 0, 1>m+ns1 = 0, 1>m+ns2 = 0. (2.7) where s1 = (s>1 , t > 1 ) > and s2 = (s>2 , t > 2 ) >."},{"heading":"3 EQUIVALENCE OF THE ROBOT FORMULATIONS","text":"In this section we present our main theorem, which demonstrates the equivalence between two formulations of the robust optimal transport:\nTheorem 3.1. For any two measures µ and ν, ROBOT(µ, ν) has same value for both the formulations, i.e., Formulation 1 is equivalent to Formulation 2 both for continuous and discrete case. Moreover, we can recover optimal coupling of one formulation from the other.\nBelow we sketch the proof of this theorem and highlight some important techniques used in the proof. We focus on the discrete case as it is more intuitive and has concrete practical implications in our experiments. A complete proof can be found in Appendix A. Please also see Appendix A.2 for the proof of equivalence between Formulations 1, 2 and 3 in the discrete case."},{"heading":"3.1 PROOF SKETCH","text":"In the remainder of this section we consider the discrete case, i.e., equation 2.5 for Formulation 1 (F1) and equation 2.6 for Formulation 2 (F2). Suppose Π∗2 is an optimal solution of F2. Then we construct a feasible solution Π∗1, s ∗ 1 = (s ∗ 1, t ∗ 1) of F1 based on Π ∗ 2 with the same value of the objective function as F2 and claim that (Π∗1, s ∗ 1) is an optimal solution. We prove the claim by contradiction: if (Π∗1, s ∗ 1) is not optimal, then there exists another pair (Π̃1, s̃1) which is optimal for F1 with strictly less objective value. We then construct another feasible solution Π∗2,new of Formulation 2 which has the same objective value as of (Π̃1, s̃1) for F1. This implies Π∗2,new has strictly less objective value for F2 than Π∗2, which is a contradiction.\nThe two main pillars of this proof are (1) to construct a feasible solution of F1 starting from a feasible solution of F2 and (2) to show that the solution constructed is indeed optimal for F1. Hence step (1) gives a recipe to construct an optimal solution of F1 starting from an optimal solution of F2. We elaborate the first point in the next subsection, which has practical implications for outlier detection. The other point is more technical; interested readers may go through the proof in Appendix A.1.\nAlgorithm 1 Generating optimal solution of F1 from F2\n1: Start with Π∗2 ∈ Rn×m, an optimal solution of Formulation 2. 2: Create an augmented matrix Π ∈ Rm+n×m+n with all 0. Divide Π into four blocks:\nΠ = Π11︸︷︷︸ n×n Π12︸︷︷︸ n×m\nΠ21︸︷︷︸ m×n Π22︸︷︷︸ m×m 3: Set Π12 ← Π∗2 and collect all the indices I = {(i, j) : Ci,j > 2λ}. 4: Set Π12(i, j)← 0 for (i, j) ∈ I. 5: Set Π22(j, j)← ∑n i=1 Π ∗ 2(i, j)1(i,j)∈I for all 1 ≤ j ≤ m and set Π∗1 ← Π.\n6: Set s∗1(i) ≤ ∑m j=1 Π ∗ 2(i, j)1(i,j)∈I for all 1 ≤ i ≤ n. 7: Set t∗1(j) = Π22(j, j) for all 1 ≤ j ≤ m. 8: return Π∗1, s ∗ 1, t ∗ 1."},{"heading":"3.2 GOING FROM FORMULATION 2 TO FORMULATION 1","text":"Let Π∗2 (respectively Π ∗ 1) be an optimal solution of F2 (respectively F1). Recall that Π ∗ 1 has dimension (m + n) × (m + n). From the column sum constraint in F1, we need to take the first n columns of Π∗1 to be exactly 0, whereas the last m columns must sum up to νm. For any matrix A, we denote by A[(a : b)× (c : d)] the submatrix consisting of rows from a to b and columns from c to d. Our main idea is to put a modified version of Π∗2 in Π ∗ 1[(1 : n) × (n + 1 : m + n)] and make Π∗1[(n+1 : m+n)×(n+1 : m+n)] diagonal. First we describe how to modify Π∗2. Observe that, if for some (i, j) Ci,j > 2λ, we expectXi ∈ supp(µn) to be an outlier resulting in high transportation cost, which is why we truncate the cost in F2. Therefore, to get an optimal solution of F1, we make the corresponding value of optimal plan 0 and dump the mass into the corresponding slack variable t∗1 in the diagonal of the bottom right submatrix. This changes the row sum, which is taken care of by s∗1. But, as we are not moving this mass outside the corresponding column, the column sum of Π∗1[(1 : (m + n)) : ((n + 1) : (m + n))] remains same as column sum of Π ∗ 2, which is νn. We summarize this procedure in Algorithm 1.\nExample. In Figure 1, we provide an example to visualize the construction. On the left, we have Π∗2, an optimal solution of Formulation 2. The blue triangles denote the positions where the corresponding cost value is ≤ 2λ, and light-green squares denote the positions where the corresponding value of the cost matrix is> 2λ. To construct an optimal solution Π∗1 of Formulation 1 from this Π ∗ 2, we first create an augmented matrix of size 6× 6. We keep all the entries of of left 6× 3 sub-matrix as 0 (in this picture blank elements indicate 0). On the right submatrix, we put Π∗2 into the top-right block, but remove the masses from light-green squares, i.e. where cost value is > 2λ, and put it in the diagonal entries of the bottom right block as shown in Figure 1. This mass contributes to the slack variables s1 and t1, and this augmented matrix along with s1, t1 give us an optimal solution of Formulation 1."},{"heading":"3.3 OUTLIER DETECTION WITH ROBOT","text":"Our construction algorithm has practical consequences for outlier detection. Suppose we have two datasets, a clean dataset νm (i.e., has no outliers) and an outlier-contaminated dataset µn. We can detect the outliers in µn without directly solving costly Formulation 1 by following Algorithm 2. In this algorithm, λ is a regularization parameter that can be chosen via cross-validation or heuristically (see Section 4.2 for an example). In Section 4.2, we use this algorithm to perform outlier detection on image data.\nAlgorithm 2 Outlier detection in contaminated data 1: Start with µn (contaminted data) and νm (clean data). 2: Solve Formulation 2 and obtain Π∗2 using a suitable value of λ. 3: Use Algorithm 1 to obtain Π∗1, s ∗ 1, t ∗ 1 from Π ∗ 2.\n4: Find I, the set of all the indices where µn + s∗1 = 0. 5: Return I as the indices of outliers in µn.\nTable 1: Robust mean estimation with GANs using different distribution divergences. True mean is η0 = 05; sample size n = 1000; contamination proportion = 0.2. We report results over 30 experiment restarts.\nContamination JS Loss SH Loss RKL Loss ROBOT UOT\nN (0.1 · 15, I5) 0.09 ± 0.03 0.11 ± 0.03 0.115 ± 0.03 0.1 ± 0.03 0.1 ± 0.04 N (0.5 · 15, I5) 0.23 ± 0.04 0.24 ± 0.05 0.24 ± 0.05 0.117 ± 0.03 0.2 ± 0.04 N (1 · 15, I5) 0.43 ± 0.05 0.43 ± 0.06 0.43 ± 0.06 0.261 ± 0.06 0.25 ± 0.05 N (2 · 15, I5) 0.67 ± 0.07 0.67 ± 0.08 0.67 ± 0.08 0.106 ± 0.03 0.1 ± 0.03\n(a) Varying proportion of contamination (b) Varying outlier distribution mean\nFigure 2: Empirical study of regularization hyperparameter λ sensitivity"},{"heading":"4 EMPIRICAL STUDIES","text":"To evaluate effectiveness of ROBOT, we consider the task of robust mean estimation under the Huber contamination model. The data is generated from (1 − )N (η0, Id) + N (η1, Id) and the goal is to estimate η0. Prior work has advocated for using f -divergence GANs (Chao et al., 2018; Wu et al., 2020) for this problem and pointed out inefficiencies of Wasserstein GAN in the presence of outliers. We show that our robust OT formulation allows us to estimate the uncontaminated mean η0 comparably or better than a variety of f -divergence GANs. We also use this simulated setup to study sensitivity to the regularization hyperparameter λ.\nIn our second experiment, we present a new application of optimal transport enabled by ROBOT. Suppose we have collected a curated dataset νm (i.e., we know that it has no outliers)—such data collection is expensive, and we want to benefit from it to automate subsequent data collection. Let µn be a second dataset collected “in the wild,” i.e., it may or may not have outliers. We demonstrate how ROBOT can be used to identify outliers in µn using the curated dataset νm."},{"heading":"4.1 ROBUST MEAN ESTIMATION","text":"Following Wu et al. (2020), we consider a simple generator of the form gθ(x) = x + θ, x ∼ N (0, Id), d is the data dimension. The basic idea of robust mean estimation with GANs is to minimize various distributional divergences between samples from gθ and observed data simulated from (1 − )N (η0, Id) + N (η1, Id). The goal is to estimate η0 with θ. To efficiently implement ROBOT GAN, we use a standard min-max optimization approach: solve the inner max (ROBOT) and use gradient descent for the outer min parameter. To solve ROBOT, it is straightforward to adopt any of the prior stochastic regularized OT solvers: the only modification is the truncation of the cost entries as in equation 2.6. We use the stochastic algorithm for semi-discrete regularized OT from\n(Genevay et al., 2016, Algorithm 2). We summarize ROBOT GAN in Algorithm 3. Line 5 - Line 10 perform the inner optimization where we solve entropy regularized OT dual with truncated cost and Line 11 - Line 12 perform gradient update of θ.\nAlgorithm 3 ROBOT GAN\n1: Input: robustness regularizion λ, entropic regularization α, data distribution µn ∈ ∆n−1, supp(µn) = X = [X1, . . . , Xn], steps sizes τ and γ 2: Initialize: Initialize θ = θinit, set number of iterations M and L, i = 0, v = ṽ = 0. 3: for j = 1, . . . ,M do 4: Generate z̃ ∼ N (0, Id) and set z = z̃ + θ. 5: Set the cost vector c ∈ Rn as c(k) = c(Xk, z) ∧ 2λ for k = 1, . . . , n. 6: for i = 1, . . . , L do . solve entropy regularized OT dual 7: Set h← ṽ−cα and do the normalized exponential transformation u← eh\n〈1,eh〉 . 8: Calculate the gradient∇ṽ← µn − u. 9: Update ṽ← ṽ + γ∇ṽ and v← (1/(j + i))ṽ + (j + i− 1/(j + i))v.\n10: Do the same transformation of v as in Step 7, i.e. set h← v−cα and set Π← eh\n〈1,eh〉 . 11: Set Π(k) = 0 for k such that C(Xk, z) > 2λ for k = 1, . . . , n. 12: Calculate gradient with respect to θ as ∇θ = 2 [ z ∑ k Π(k)−X>Π ] 13: Update θ ← θ − τ∇θ. 14: Ouput: θ\nFor the f -divergence GANs (Nowozin et al., 2016) we use the code of Wu et al. (2020) for GANs with Jensen-Shannon (JS) loss, squared Hellinger (SH) loss and Reverse Kullback-Leibler (RKL) loss. For the exact expression of these divergences see Table 1 of Wu et al. (2020). We report estimation error measured by the Euclidean distance between true uncontaminated mean η0 and estimated mean θ for various contamination distributions in Table 1. ROBOT GAN performs well across all considered contamination distributions. As the difference between true mean η0 and contamination mean η1 increases, the estimation error of all methods tends to increase. However, when it becomes easier to distinguish outliers from clean samples, i.e., η1 = 2 · 15, performance of ROBOT noticeably improves.\nWe also compared to the Sinkhorn-based UOT algorithm (Chizat et al., 2018) available in the Python Optimal Transport (POT) library (Flamary & Courty, 2017); to obtain a UOT GAN, we modified steps 5-11 of Algorithm 3 for computing Π. Unsurprisingly, both ROBOT and UOT perform similarly: recall equivalence to Formulation 3, which is similar to UOT with TV norm. The key insight of our work is the equivalence to classical OT with truncated cost, that greatly simplifies optimization and allows to use existing stochastic OT algorithms. In this experiment, the sample size n = 1000 is sufficiently small for the Sinkhorn-based UOT POT implementation to be effective, but it breaks in the experiment we present in Section 4.2. We also tried the code of Balaji et al. (2020) based on CVXPY (Diamond & Boyd, 2016), but it is too slow even for the n = 1000 sample size.\nIn the previous experiment, we set λ = 0.5. Now we demonstrate empirically that there is a broad range of λ values performing well. In Figure 2a, we study sensitivity of λ under various contamination proportions holding η0 = 15 and η1 = 5 · 15 fixed. Horizontal lines correspond to λ = ∞, i.e., vanilla OT. The key observations are: there is a wide range of λ efficient at all contamination proportions, and ROBOT is always at least as good as vanilla OT (even when there is no contamination = 0). In Figure 2b, we present a similar study varying the mean of the contamination distribution and holding = 0.2 fixed. We see that as the contamination distribution gets closer to the true distribution, it becomes harder to pick a good λ, but the performance is always at least as good as the vanilla OT (horizontal lines)."},{"heading":"4.2 OUTLIER DETECTION FOR DATA COLLECTION","text":"Our robust OT formulation equation 2.5 enables outlier identification. Let νm be a clean dataset and µn potentially contaminated with outliers. Recall that ROBOT allows modification of one of the input distributions to eliminate potential outliers. We can identify outliers in µn as follows: if µn(i)+s ∗ 1(i) = 0, thenXi, the ith point in µn, is an outlier. Instead of directly solving equation 2.5,\nwhich may be inefficient, we use our equivalence results and solve an easier optimization problem equation 2.6, followed by recovering s to find outliers via Algorithm 2.\nLet νm be a clean dataset consisting of 10k MNIST digits and µn be a dataset collected “in the wild” consisting of (different) 8k MNIST digits and 2k Fashion MNIST images. We compute ROBOT(µn, νm) to identify outlier Fashion MNIST images in µn. For each point in µn we obtain a prediction, outlier or clean, which allows us to evaluate accuracy. ROBOT outlier detection is 90% accurate in this experiment. We also comment on λ selection: since we know that νm is clean, we can subsample two datasets from it, compute vanilla OT to obtain transportation plan Π and set λ to be half the maximum distance between matched elements, i.e. 2λ = maxi,j{Cij : Πij > 0}, where C is the cost matrix for the two subsampled datasets. This procedure is essentially estimating maximum distance between matched clean samples. We also present a random sample of outliers identified by our method in Figure 3. All of the sampled outliers are Fashion MNIST images, although 90% accuracy suggests that some of the outliers were not identified. Decreasing λ can help to find more outliers, but may result in some clean samples being mistaken for outliers. We conclude that ROBOT can be used to assist in data collection once an initial set of clean data has been acquired. As we mentioned previously, the Sinkhorn-based UOT POT implementation is too expensive for this experiment due to larger sample size, yielding memory errors on a personal laptop with 16GB RAM.\nFor comparison, we also consider a heuristic distance-based approach for identifying outliers. We estimate diameter τ of the set of clean dataset νm by taking the 99th percentile of the pairwise distance matrix of samples in νm. If outliers and clean data have disjoint support, we can adopt a simple heuristic: for each sample in the potentially contaminated µn compute an average distance to the clean samples in νm and declare a sample as an outlier if this average distance is greater than the diameter τ of the clean data. The accuracy of this procedure is 85.4%, inferior to the ROBOT accuracy of 90%. The disjoint support assumption justifying the distance-based heuristic might be too strong in practice. ROBOT continues to be effective even when the supports of clean and outlier distributions are not easily separable."},{"heading":"5 SUMMARY AND DISCUSSION","text":"We proposed and studied ROBOT, a robust formulation of optimal transport. We showed that although the problem is seemingly asymmetric and challenging to optimize, there is an equivalent formulation based on cost truncation that is symmetric and compatible with modern stochastic optimization methods for OT.\nROBOT closely resembles unbalanced optimal transport (UOT). In our formulation, we added a TV regularizer to the vanilla optimal transport problem. This is motivated by the -contamination model. In UOT, the TV regularizer is typically replaced with a KL divergence. Other choices of the regularizer may lead to new properties and applications. Studying equivalent, simpler formulations of UOT with different divergences may be a fruitful future work direction.\nFrom the practical perspective, in our experiments we observed no degradation of ROBOT GAN in comparison to OT GAN, even when there were no outliers. It is possible that replacing OT with ROBOT may be beneficial for various machine learning applications of OT. Data encountered in practice may not be explicitly contaminated with outliers, but it often has errors and other deficiencies, suggesting that a “no-harm” robustness is desirable."},{"heading":"A PROOF OF THEOREM 3.1","text":""},{"heading":"A.1 PROOF OF DISCRETE VERSION","text":"Proof. Define a matrix Π as:\nΠ(i, j) = { 0, if C(i, j) > 2λ Π∗2(i, j), otherwise\nAlso define s ∈ Rn and t ∈ Rm as:\ns∗1(i) = − m∑ j=1 Π∗2(i, j)1C(i,j)>2λ\nand similarly define:\nt∗1(j) = n∑ i=1 Π∗2(i, j)1C(i,j)>2λ\nThese vectors corresponds to the row sums and the column sums of the elements of the optimal transport plan of Formulation 2, where the cost function exceeds 2λ. Note that, these co-ordinates of the optimal transport plan corresponding to those co-ordinates of cost matrix, where the cost is greater than 2λ and contribute to the objective value via their sum only, hence any different arrangement of these transition probabilities with same sum gives the same objective value.\nNow based on this Π obtained we construct a feasible solution of Formulation 1 following Algorithm 1:\nΠ∗1 = [ 0 Π 0 diag(t∗1) ] The row sums of Π∗1 is:\nΠ∗11 =\n[ µn + s ∗ 1\nt∗1 ] and it is immediate from the construction that the column sums of Π∗1 is νm. Also as:\nn∑ i=1 s∗1(i) = m∑ j=1 t∗1(j) = ∑\n(i,j):Ci,j>2λ\nΠ∗2(i, j)\nand s∗1 0, t∗1 0, we have:\n1>(µn + s ∗ 1 + t ∗ 1) = 1 >p = 1 .\nTherefore, we have (Π∗1, s ∗ 1, t ∗ 1) is a feasible solution of Formulation 1. Now suppose this is not an optimal solution. Pick an optimal solution Π̃, s̃, t̃ of Formulation 1 so that:\n〈Caug, Π̃〉+ λ [ ‖s̃‖1 + ‖t̃‖1 ] < 〈Caug,Π∗1〉+ λ [‖s∗1‖1 + ‖t∗1‖1]\nThe following two lemmas provide some structural properties of any optimal solution of Formulation 1:\nLemma A.1. Suppose Π∗1, s∗1, t∗1 are optimal solution for Formulation 1. Divide Π∗1 into four parts corresponding to augmentation as in algorithm 1:\nΠ∗1 =\n[ Π∗1,11 Π ∗ 1,12\nΠ∗1,21 Π ∗ 1,22 ] Then we have Π∗1,11 = Π ∗ 1,21 = 0 and Π ∗ 1,22 is a diagonal matrix.\nLemma A.2. If Π∗1, s∗1, t∗1 is an optimal solution of Formulation 1 then:\n1. If Ci,j > 2λ then Π∗1(i, j) = 0. 2. If Ci,j < 2λ for some i and for all 1 ≤ j ≤ n, then s∗1(i) = 0. 3. If Ci,j < 2λ for some j and for all 1 ≤ i ≤ m, then t∗1(j) = 0. 4. If Ci,j < 2λ then s∗1(i)t ∗ 1(j) = 0.\nWe provide the proofs in the next subsection. By Lemma A.1 we can assume without loss of generality:\nΠ̃ = [ 0 Π̃12 0 diag(t̃) ] Now based on ( Π̃, s̃, t̃ ) we create a feasible solution namely Π∗2,new of Formulation 2 as follows:\nDefine the set of indices {i1, · · · , ik} and {j1, . . . , jl} as:\ns̃i1 , s̃i2 , . . . , s̃ik > 0 and t̃j1 , t̃j2 , . . . , t̃jl > 0 .\nThen by part (4) of Lemma A.2 we have Ciα,jβ > 2λ for α ∈ {1, . . . , k} and β ∈ {1, . . . , l}. Also by part (2) of Lemma A.2 the value of transport plan at these co-ordinates is 0. Now distribute the mass of slack variables in these co-ordinates such that the marginals of new transport plan becomes exactly µn and νm. This new transport plan is our Π∗2,new. Recall that, ‖s̃‖1 = ‖t̃‖1. Hence, here the regularizer value decreases by 2λ‖s̃‖1 and the cost value increased by exactly 2λ‖s̃‖1 as we are truncating the cost. Hence we have:\n〈Cλ,Π∗2,new〉 = 〈Caug, Π̃〉+ λ [ ‖s̃‖1 + ‖t̃‖1 ] < 〈Caug,Π∗1〉+ λ [‖s∗1‖1 + ‖t∗1‖1] = 〈Cλ,Π∗2〉\nwhich is contradiction as Π∗2 is the optimal solution of Formulation 2. This completes the proof for the discrete part."},{"heading":"A.2 PROOF OF EQUIVALENCE FOR TWO SIDED FORMULATION","text":"Here we prove that our two sided formulation, i.e. Formulation 3 (equation 2.7) is equivalent to Formulation 1 (equation 2.5) for the discrete case. Towards that end, we introduce another auxiliary formulation and show that both Formulation 1 and Formulation 3 are equivalent to the following auxiliary formulation of the problem.\nFormulation 4:\nWR,L,4(p, q) = minΠ∈Rm×n,s1∈Rm,s2∈Rn 〈C,Π〉+ λ [‖s1‖1 + ‖s2‖1] subject to Π1n = p+ s1 ΠT 1m = q + s2\nΠ 0\n(A.1)\nFirst we show that Formulation 1 and Formulation 4 are equivalent in a sense that they have the same optimal objective value.\nTheorem A.3. Suppose C is a cost function such that C(x, x) = 0. Then Formulation 1 and Formulation 4 has same optimal objective value.\nProof. Towards that end, we show that given one optimal variables of one formulation we can get optimal variables of other formulation with the same objective value. Before going into details we need the following lemma whose proof is provided in Appendix B:\nLemma A.4. Suppose Π∗4, s∗4,1, s∗4,2 are the optimal variables of Formulation 4. Then s∗4,1 0 and s∗4,2 0.\nNow we prove that optimal value of Formulation 1 and Formulation 4 are same. Let (Π∗1, s ∗ 1,1, t ∗ 1,1) is an optimal solution of Formulation 1. Then we claim that (Π∗1, s ∗ 1,1, t ∗ 1,1) is also an optimal solution of Formulation 4. Clearly it is feasible solution of Formulation 4. Suppose it is not optimal, i.e. there exists another optimal solution (Π̃4, s̃4,1, s̃4,2) such that:\n〈C, Π̃4〉+ λ(‖s̃4,1‖1 + ‖s̃4,2‖2) < 〈C,Π∗1,12〉+ λ(‖s∗1,1‖1 + ‖t∗1,1‖1)\nNow based on (Π̃4, s̃4,1, s̃4,2) we construct a feasible solution of Formulation 1 as follows:\nΠ̃1 = [ 0 Π̃4 0 −diag(s̃4,2) ] Note that we proved in Lemma A.4 s̃4,2 0, hence we have Π̃1 0. Now as the column sums of Π̃4 is q + s̃4,2, we have column sums of Π̃1 = [0 q>]> and the row sums are [(p+ s̃4,1)> s̃>4,2]\n>. Hence we take s̃1,1 = s̃4,1 and s̃1,2 = s̃4,2. Then it follows:\n〈Caug, Π̃1〉+ λ [‖s̃1,1‖1 + ‖s̃1,2‖1] = 〈C, Π̃4〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] < 〈C,Π∗1,12〉+ λ [ ‖s∗1,1‖1 + ‖t∗1,1‖1 ] = 〈Caug,Π∗1〉+ λ [ ‖s∗1,1‖1 + ‖t∗1,1‖1\n] This is contradiction as we assumed (Π∗1, s ∗ 1,1, t ∗ 1,2) is an optimal solution of Formulation 1. Therefore we conclude (Π∗1, s ∗ 1,1, t ∗ 1,1) is also an optimal solution of Formulation 4 which further concludes Formulation 1 and Formulation 4 have same optimal values. This completes the proof of the theorem.\nTheorem A.5. The optimal objective value of Formulation 3 and Formulation 4 are same.\nProof. Like in the proof of Theorem A.3 we also prove couple of lemmas.\nLemma A.6. Any optimal transport plan Π∗3 of Formulation 3 has the following structure: If we write,\nΠ∗3 =\n[ Π∗3,11 Π ∗ 3,12\nΠ∗3,21 Π ∗ 3,22 ] then Π∗3,11 and Π ∗ 3,22 are diagonal matrices and Π ∗ 3,21 = 0.\nLemma A.7. If s∗3,1, t∗3,1, s∗3,2, t∗3,2 are four optimal slack variables in Formulation 3, then s∗3,1, t ∗ 3,1 0 and s∗3,2, t∗3,2 0.\nProof. The line of argument is same as in proof of Lemma A.4.\nNext we establish equivalence. Suppose (Π∗3, s ∗ 3,1, t ∗ 3,1, s ∗ 3,2, t ∗ 3,2) are optimal values of Formulation 3. We claim that (Π∗3,12, s ∗ 3,1 − s∗3,2, t∗3,1 − t∗3,2) forms an optimal solution of Formulation 4. The objective value will then also be same as s∗3,1 0, s∗3,2 0 (Lemma A.7) implies ‖s∗3,1 − s∗3,2‖1 = ‖s∗3,1‖1 + ‖s∗3,2‖1 and similarly t∗3,1 0, t∗3,2 0 implies ‖t∗3,1 − t∗3,2‖1 = ‖t∗3,1‖1 + ‖t∗3,2‖1. Feasibility is immediate. Now for optimality, we again prove by contradiction. Suppose they are not optimal. Then lets say Π̃4, s̃4,1, s̃4,2 are an optimal triplet of Formulation 4. Now construct another feasible solution of Formulation 3 as follows: Set s̃3,2 = t̃3,2 = 0, s̃3,1 = s̃4,1 and t̃3,1 = s̃4,2. Set the matrix as:\nΠ̃3 = [ 0 Π̃4 0 −diag(s̃4,2) ] Then it follows that ( Π̃3, s̃3,1, s̃3,2, t̃3,1, t̃3,2 ) is a feasible solution of Formulation 3. Finally we\nhave:\n〈Caug, Π̃3〉+ λ [ ‖s̃3,1‖1 + ‖s̃3,2‖1 + ‖t̃3,1‖1 + ‖t̃3,2‖1 ] = 〈Caug, Π̃3〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] = 〈C, Π̃4〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] < 〈C,Π∗3,12〉+ λ [ ‖s∗3,1 − s∗3,2‖1 + ‖t∗3,1 − t∗3,2‖1\n] = 〈Caug,Π∗3〉+ λ [ ‖s∗3,1‖1 + ‖s∗3,2‖1 + ‖t∗3,1‖1 + ‖t∗3,2‖1\n] This contradicts the optimality of (Π∗3, s ∗ 3,1, s ∗ 3,2, t ∗ 3,1, t ∗ 3,2). This completes the proof."},{"heading":"A.3 PROOF OF CONTINUOUS VERSION","text":"Proof. In this proof we denote by F1 the optimization problem of equation equation 2.2 and by F2 the optimization problem equation equation 2.4. Assume that µn and νm denote the respective empirical measures relative to µ, ν. From Villani (2009), we know that µn, νn converge weakly to µ and ν respectively. Therefore, ROBOT2(µn, µ) → 0. Similary for νn and ν. Thus, by triangle inequality,\nlim n→∞\n|F2(µn, νn)− F2(µ, ν)| = 0.\nBut ROBOT2(µn, νn) = ROBOT1(µn, νn). Therefore, our proof is complete if we can show that\nlim n,m→∞\n|F1(µn, νm)− F1(µ, ν)| → 0.\nLet S = {s signed measure : µ+ s is a probability measure in Rd}. For s ∈ S , define\nW (µ+S, ν) = \nminΠ∈F(Rd×Rd) ∫ C(x, y) Π(dx, dy) + λ‖s‖TV\nsubject to\n∫ A Π(dx, dy) ≥ 0 ∀ A ∈ B(Rd × Rd)\nµ(dx) + s(dx)) ≥ 0 ∀ B ∈ B(Rd)∫\nRd×C Π(dx, dy) = ∫ C ν(dy) ∀ C ∈ B(Rd)\nBy Lemma A.8, ∃ s ∈ S such that ROBOT (µ, ν) = W (µ + s, ν) + λ‖s‖TV . Let s = s+ − s−, where s+ and s− are positive measures on Rd. Let ‖s‖TV = γ. Then, ‖s−‖TV = ‖s+‖TV = γ/2. Then consider X1, . . . , Xn ∼ (P − s−)/(1 − γ), Y1, . . . , Yn ∼ s−/γ, Z1, . . . , Zn ∼ s+/γ. Then for any bounded continuous function f ,\nlim n→∞ ∑ i f(Xi)/n = ∫ f(x) (P − s−) (1− γ) (dx)\nlim n→∞ ∑ i f(Zi)/n = ∫ f(x) s+ γ (dx) (A.2)\nTherefore, the distribution given by (P + s)n = (1− γ) n ∑ i δXi + γ n ∑ i δZi satisfies, (P + s)n\nL→ P + s, and therefore from (Villani, 2009), limn→∞WC((P + S)n, νn)→ W (P + S,Q). Here δx is the Dirac mass at x. Moreover, ‖sn‖ = ‖s‖, where sn satisfies sn = γ n ( ∑ i δZi − ∑ i δYi).\nAlso, ROBOT (µn, νn) ≤ W ((P + s)n, νn) + λ‖s‖TV , and therefore ROBOT2(µ, ν)) = lim supn→∞ROBOT (µn, νn) ≤ ROBOT (µ, ν). Now, let s̃n satisfy W1(µn + s̃n, νn) + λ‖s̃n‖TV = ROBOT (µn, νn). Such an s̃n exists by the proof of the discrete part because µn, νn are discrete measures.\nThen, similar to the Step 1 in the proof of Lemma A.8, there exists a probability measure µ⊕ s and a subsequence {nk}k≥1 such that µnk + snk almost surely converges weakly to µ⊕ s. Moreover, similar to Step 2 of Lemma A.8 W1(µnk + snk , µ ⊕ s) → 0 as well as ‖snk‖TV → ‖µ⊕ s−µ‖TV . Thus, W1(µnk + snk , νnk) +λ‖snk‖TV →W1(µ⊕ s, ν) +λ‖µ⊕ s−µ‖TV . But by the proof of the discrete part ROBOT (µnk , νnk) = ROBOT2(µnk , νnk) → ROBOT2(µ, ν). Therefore, with s = µ⊕ s− µ, W1(µ+ s, ν) + λ‖s‖TV = ROBOT2(µ, ν). Therefore, ROBOT2(µ, ν) = lim supn→∞ROBOT (µn, νn) ≥ ROBOT (µ, ν). Thus the equality holds.\nLemma A.8. Assume that µ, ν is such that ∫ ‖x‖dµ, ∫ ‖x‖dν < ∞. Moreover, assume that C(x, y) in equation 2.2 is the l1 norm, i.e., C(x, y) = ‖x− y‖. Then, there exists s with µ+ s being a probability measure such that\nW1(µ+ s, ν) + λ‖s‖TV = ROBOT (µ, ν), (A.3)\nwhere W1 is the Wasserstein-1 norm with the cost function C(·, ·) as mentioned above.\nProof. Let µn, νm be the empirical measures relative to µ, ν respectively. We know that since µn, νm are discrete, there exists sn satisfying W1(µn + sn, νm) = ROBOT (µn, νm). We provide the proof in the following steps.\nStep 1: Almost surely µ× ν, there exists a subsequence {nk}k≥1 such that {µn + sn}n and {νn}n is relatively compact.\nµ and ν are probability measures on Rd and are therefore tight. Let K be such that Pµ(X /∈ K ), Pν(Y /∈ K ) ≤ /4. Consider the empirical distributions νn = ∑ i δYi/n, µn ∑ i δXi/n of ν, µ respectively. Here, Xi ∼ µ and Yi ∼ν . Fix an ω. Then {X1, . . . , Xn, Y1, . . . , Yn} is fixed. Now by the construction for the discrete case, sn has support in {X1, . . . , Xn, Y1, . . . , Yn}. Let Tn be the optimal transport map from µn to νn. Then, for every i ≤ n, there exists a unique j ≤ n, such that Tn(Xi) = Yj . Define τn : {1, . . . , n} → {1, . . . , n} such that τn(i) = j if Tn(Xi) = Yj .Then µn + sn = ∑ i δZi/n, where Zi = Xi or Yτn(i) and δx is the Dirac delta mass at x.\nThen, let Z ∼ µn + sn Pω(Z /∈ K |µn + sn) ≤ ∑ i 1(Xi /∈K )/n+ ∑ i 1(Yi /∈K )/n (A.4)\nTherefore, E(Pω(Z /∈ K |µn+sn)) ≤ /2. Moreover, V ar(Pω(Z /∈ K |µn+sn)) = o(n−1)→ 0. Therefore, limn→∞ Pµn×νn(Pω(Z /∈ K |µn + sn) ≤ ) → 1. Therefore µn + sn is almost surely tight and thus by Prokhorov’s Theorem also relatively compact.\nStep 2: Therefore, for ω almost surely, there exists a subsequence {nk}k≥1 such that µnk + snk converges weakly to a limit (dependent on ω) µ ⊕ s which is a probability measure. Moreover,∫ ‖x‖d(µnk + snk) < ∞ almost surely. By Bolzano-Weierstrass Theorem, there exists a further\nsubsequence {nkl}l such that ∫ ‖x‖d(µnkl + snkl )→ ∫ ‖x‖d(µ+ s) almost surely. For the sake of convenience, without loss of generality, we will replace the sub-subsequence {nkl}l with {nk}k≥1 henceforth.\nThus, by Theorem 6.9 of (Villani, 2009) , W1(µnk + snk , µ ⊕ s) → 0 almost surely. Moreover, W1(µnk , µ)→ 0 almost surely. Therefore ‖snk‖TV → ‖µ⊕ s− µ‖TV almost surely.\nStep 3: Consider an arbitrary S = S+ − S−, such that S+ and S− are positive measures on Rd, and µ+ S is a probability measure. Let ‖S‖TV = γ. Then, ‖S−‖TV = ‖S+‖TV = γ/2. Then consider X1, . . . , Xn ∼ (µ− S−)/(1− γ), Y1, . . . , Yn ∼ S−/γ, Z1, . . . , Zn ∼ S+/γ. Then for any bounded continuous function f ,\nlim n→∞ ∑ i f(Xi)/n = ∫ f(x) (P − s−) (1− γ) (dx)\nlim n→∞ ∑ i f(Zi)/n = ∫ f(x) s+ γ (dx) (A.5)\nTherefore, the distribution given by (µ + S)n(A) = (1 − γ) ∑ i 1Xi∈A + (γ) ∑ i 1Zi∈A satisfies, (µ+S)n L→ µ+S, and therefore from (Villani, 2009), limn→∞W1((µ+S)n, νn)→W1(µ+S, ν). Moreover, ‖Sn‖TV = ‖S‖TV , where Sn satisfies Sn(A) = γ\nn\n∑ i 1Zi∈A − ∑ i 1Yi∈A.\nBut, W1(µnk + snk , νnk) + λ‖snk‖TV ≤ W1((µ + S)nk , νnk) + λ‖Snk‖TV . Therefore, taking limits, W1(µ⊕ s, ν) + λ‖µ⊕ s−µ‖TV ≤W1(µ+S, ν) + λ‖S‖TV , and thus the proof holds with s = µ⊕ s− µ."},{"heading":"B PROOF OF ADDITIONAL LEMMAS","text":""},{"heading":"B.1 PROOF OF LEMMA A.1","text":"Proof. The fact that Π∗1,11 = Π ∗ 1,21 = 0 follows from the fact that Π ∗ 1 0 and Π∗11 = Q. To prove that Π∗1,22 is diagonal, we use the fact that the any diagonal entry the cost matrix is 0. Now suppose Π∗1,22 is not diagonal. Then define a matrix Π̂ as following: set Π̂11 = Π̂21 = 0, Π̂12 = Π ∗ 1,12 and:\nΠ̂22(i, j) =\n{∑m k=1 Π ∗ 1,22(k, i), if j = i\n0, if j 6= i\nAlso define ŝ = s∗1 and t̂ as t̂(i) = Π̂22(i, i). Then clearly (Π̂, ŝ, t̂) is a feasible solution of Formulation 1. Note that:\n‖t̂‖1 = 1>Π̂221 = 1>Π∗1,221 = ‖t∗1‖1\nand by our construction 〈Caug, Π̂〉 < 〈Caug,Π∗1〉. Hence (Π̂, ŝ, t̂) reduces the value of the objective function of Formulation 1 which is a contradiction. This completes the proof."},{"heading":"B.2 PROOF OF LEMMA A.2","text":"Proof. 1. Suppose Π∗1(i, j) > 0. Then dump this mass to s ∗ 1(j) and make it 0. In this way\n〈Caug,Π∗1〉 will decrease by > 2λΠ∗1(i, j) and the regularizer value will increase by atmost 2λΠ∗1(i, j), resulting in overall reduction in the objective value, which leads to a contradiction. 2. Suppose each entry of ith row of C is < 2λ. Then if s∗1(i) > 0, we can distribute this mass in the ith row such that, s∗1(i) = a1 + a2 + · · · + am with the condition that t∗1(j) ≥ aj . Now we reduce t∗1 as:\nt∗1(j)← t∗1(j)− aj Hence the value 〈Caug,Π∗1(i, j)〉 will increase by a value < 2λs∗1(i) but the value of regularizer will decrease by the value of 2λs∗1(i), resulting in overall decrease in the value of objective function. 3. Same as proof of part (2) by interchanging row and column in the argument. 4. Suppose not. Then choose < s∗1(i) ∧ t∗1(j), Add to Π∗1(i, j). Hence the cost function value 〈Caug,Π∗1〉 will increase by < 2λ but the regularizer value will decrease by 2λ , resulting in overall decrease in the objective function."},{"heading":"B.3 PROOF OF LEMMA A.4","text":"Proof. For the notational simplicity, we drop the subscript 4 now as we will only deal with the solution of Formulation 4 and there will be no ambiguity. We prove the Lemma by contradiction. Suppose s∗1,i > 0. Then we show one can come up with another solution (Π̃, s̃1, s̃2) of Formulation 4 such that it has lower objective value. To construct this new solution, make:\ns̃1,j = { s∗1,j , if j 6= i 0, if j = i\nNow to change the optimal transport plan, we will only change ith row of Π∗. We subtract a1, a2, . . . , an ≥ 0 from ith column of Π∗ in such a way, such that none of the elements are negative. Hence the column sum will be change, i.e. the value of s̃2 will be:\ns̃2,j = s ∗ 2,j − aj ∀1 ≤ j ≤ n .\nNow clearly from our construction: 〈C, Π̃〉 ≤ 〈C,Π∗〉\nFor the regularization part, note that, as we only reduced ith element of s∗1, we have ‖s̃1‖1 = ‖s∗1‖1 − s∗1,i. And by simple triangle inequality,\n‖s̃2‖1 ≤ ‖s∗2‖1 + ‖a1‖1 = ‖s∗2‖1 + s∗1,i\nby construction ai’s, as ai ≥ 0 and ∑ i ai = s ∗ 1,i. Hence we have:\n‖s̃1‖1 + ‖s̃2‖1 ≤ ‖s∗1‖1 − s∗1,i + ‖s∗2‖1 + s∗1,i = ‖s∗1‖1 + ‖s∗2‖1 .\nHence the value corresponding to regularizer will also decrease. This completes the proof."},{"heading":"B.4 PROOF OF LEMMA A.6","text":"Proof. We prove this lemma by contradiction. Suppose Π∗3 does not have the structure mentioned in the statement of Lemma. Construct another transport plan for Formulation 3 Π̃3 as follows: Keep Π̃3,12 = Π ∗ 3,12 and set Π̃3,12 = 0. Construct the other parts as:\nΠ̃3,11(i, j) =\n{∑m k=1 Π ∗ 3,11(i, k) + ∑n k=1 Π ∗ 3,21(k, i), if i = j\n0, if i 6= j\nand\nΠ̃3,22(i, j) =\n{∑n k=1 Π ∗ 3,22(k, i), if i = j\n0, if i 6= j It is immediate from the construction that:\n〈Caug, Π̃3〉 ≤ 〈Caug,Π∗3〉\nAs for the regularization term: Note the by our construction s̃4 will be same as s∗4 as column sum of Π̃3,22 is same as Π∗3,22. For the other three:\ns̃3(i) = Π̃3,11(i, i) = m∑ k=1 Π∗3,11(i, k) + n∑ k=1 Π∗3,21(k, i)\ns̃2(i) = Π̃3,22(i, i) = n∑ k=1 Π∗3,22(k, i)\nand hence by construction:\n‖s̃2‖1 = 1>Π∗3,221 = ‖s∗2‖1 − 1>Π∗3,211 .\n‖s̃3‖1 = 1>Π∗3,111 + 1>Π∗3,211 = ‖s∗3‖1 And also by our construction, s̃1 = s∗1 + c where c = (Π ∗ 3,21)\n>1. As a consequence we have ‖c‖1 = 1>Π∗3,211. Then it follows:\n4∑ i=1 ‖s̃i‖1 = ‖s∗1 + c‖+ ‖s∗2‖1 − 1>Π∗3,211 + ‖s∗3‖1 + ‖s∗4‖1\n≤ 4∑ i=1 ‖s∗i ‖1 + ‖c‖1 − 1>Π∗3,211\n= 4∑ i=1 ‖s∗i ‖1\nSo the objective value is overall reduced. This contradicts the optimality of Π∗3 which completes the proof."},{"heading":"C PROOF OF THEOREM 2.1","text":"Proof. The proof is immediate from the Formulation 1. Recall that the Formulation 1 can restructured as:\nROBOT (µ̃, ν) = inf P {OT (P, ν) + λ‖P − µ̃‖TV } .\nwhere the infimum is taking over all measure dominated by some common measure σ (with respect to which µ, µc, ν are dominated). Hence,\nROBOT (µ̃, ν) ≤ OT (P, ν) + λ‖P − µ̃‖TV for any particular choice of P . Taking P = µ we get that\nROBOT (µ̃, ν) ≤ OT (µ, ν) + λ‖µ− µ̃‖TV = OT (µ, ν)) + λ ‖µ− µc‖TV Taking P = ν we get ROBOT (µ̃, ν) ≤ λ‖ν − µ̃‖TV and finally taking P = µ̃ we get ROBOT (µ̃, ν) ≤ OT (µ̃, ν). This completes the proof."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Optimal transport is a fundamental problem in applied mathematics. In its original form (Monge, 1781), the problem entails finding the minimum cost way to transport mass from a prescribed probability distribution µ on X to another prescribed distribution ν on X . Kantorovich (1942) relaxed Monge’s formulation of the optimal transport problem to obtain the Kantorovich formulation:\\nOT(µ, ν) , min Π∈F(µ,ν)\\nE(X1,X2)∼Π [ c(X1, X2) ] , (1.1)\\nwhere F(µ, ν) is the set of couplings between µ and ν (probability distributions on X × X whose marginals are µ and ν) and c is a cost function, where we typically assume c(x, y) ≥ 0 and c(x, x) = 0. Compared to other notions of distance between probability distributions, optimal transport uniquely depends on the geometry of the sample space.\\nRecent advancements in optimization for optimal transport (Cuturi, 2013; Solomon et al., 2015; Genevay et al., 2016; Seguy et al., 2018) enabled its broad adaptation in machine learning applications where geometry of the data is important. See (Peyré & Cuturi, 2018) for a survey. Optimal transport has found applications in natural language processing (Kusner et al., 2015; Huang et al., 2016; Alvarez-Melis & Jaakkola, 2018; Yurochkin et al., 2019), generative modeling (Arjovsky et al., 2017), clustering (Ho et al., 2017), domain adaptation (Courty et al., 2014; 2017), large-scale Bayesian modeling (Srivastava et al., 2018), and many other domains.\\nMany applications use OT as a loss in an optimization problem of the form:\\nθ ∈ arg minθ∈Θ OT(µn, νθ), (1.2) where {νθ}θ∈Θ is a collection of parametric models, µn is the empirical distribution of the samples. Such estimators are called minimum Kantorovich estimators (MKE) (Bassetti et al., 2006). They are popular alternatives to likelihood-based estimators, especially in generative modeling. For example, when OT(·, ·) is the Wasserstein-1 distance and νθ is a generator parameterized by a neural network with weights θ, equation 1.2 corresponds to the Wasserstein GAN (Arjovsky et al., 2017).\\nOne drawback of optimal transport is its sensitivity to outliers. Because all the mass in µ must be transported to ν, a small fraction of outliers can have an outsized impact on the optimal transport problem. For statistics and machine learning applications in which the data is corrupted or noisy, this is a major issue. For example, the poor performance of Wasserstein GANs in the presence of outliers was noted in the recent works on outlier-robust generative learning with f -divergence GANs (Chao et al., 2018; Wu et al., 2020). The problem of outlier-robustness in MKE has not been studied, with the exception of two concurrent works (Staerman et al., 2020; Balaji et al., 2020).\\nIn this paper, we propose a modification of OT to address its sensitivity to outliers. Our formulation can be used as a loss in equation 1.2 so that it is robust to a small fraction of outliers in the data. To keep things simple, we consider the -contamination model (Huber & Ronchetti, 2009). Let νθ0 be a member of a parametric model {νθ : θ ∈ Θ} and let\\nµ = (1− )νθ0 + ν̃,\\nwhere µ is the data-generating distribution, > 0 is the fraction of outliers, and ν̃ is the distribution of the outliers. Although the fraction of outliers is capped at , the value of the outliers is arbitrary, so the outliers may have an arbitrarily large impact on the optimal transport problem. Our goal is to modify the optimal transport problem so that it is more robust to outliers. We have in mind the downstream application of learning θ0 from (samples from) µ in the -contamination model. Our main contributions are as follows:\\n1. We propose a robust OT formulation that is suitable for statistical estimation in the - contamination model using MKE. 2. We show that our formulation is equivalent to the original OT problem with a clipped transport cost. This connection enables us to leverage the voluminous literature on computational optimal transport to develop efficient algorithm to perform MKE robust to outliers. 3. Our formulation enables a new application of optimal transport: outlier detection in data.\"\n },\n {\n \"heading\": \"2 PROBLEM FORMULATION\",\n \"text\": \"\"\n },\n {\n \"heading\": \"2.1 ROBUST OT FOR MKE\",\n \"text\": \"To promote outlier-robustness in MKE, we need to allow the corresponding OT problem to ignore the outliers in the data distribution µ. The -contamination model imposes a cap on the fraction of outliers, so it is not hard to see that ‖µ − νθ0‖TV ≤ , where ‖ · ‖TV is the total-variation norm defined as ‖µ‖TV = ∫ 1 2 |µ(dx)|. This suggests we solve a TV-constrained/regularized version of equation 1.2. The constrained version\\nmin θ∈Θ,µ̃\\nOT(µ̃, νθ)\\nsubject to ‖µ− µ̃‖TV ≤\\nsuffers from identification issues. In particular, it cannot distinguish between “clean” distributions within TV distance of νθ0 . This makes it unsuitable as a loss function for statistical estimation, because it cannot lead to a consistent estimator. However, its regularized counterpart\\nmin θ∈Θ,s\\nOT(µ+ s, νθ) + λ‖s‖TV, (2.1)\\nwhere λ > 0 is a regularization parameter, does not suffer from this issue. In the rest of this paper, we work with the TV-regularized formulation equation 2.1.\\nThe main idea of our formulation is to allow for modifications of µ, while penalizing their magnitude and ensuring that the modified µ is still a probability measure. Below we formulate this intuition in an optimization problem titled ROBOT (ROBust Optimal Transport):\\nFormulation 1:\\nROBOT(µ, ν) = minΠ∈F+(Rd×Rd) s∈F(Rd) ∫ C(x, y) Π(dx,dy) + λ‖s‖TV subject to ∫ B×Rd Π(dx, dy) = ∫ B (µ(dx) + s(dx)) ≥ 0 ∀ B ∈ B(Rd) (Borel σ-algebra)∫ Rd×C Π(dx,dy) = ∫ C ν(dy) ∀ C ∈ B(Rd)∫ s(dx) = 0.\\n(2.2)\\nHere F(Rd) denotes the set of all signed measures with finite total variation on Rd, F+(Rd × Rd) is the set of all measures with finite total variation on Rd × Rd. The first and the last constraints ensure that µ + s is a valid probability measure, while λ‖s‖TV penalizes the amount of modifications in µ. It is worth noting that we can identify exact locations of outliers in µ by inspecting µ+ s, i.e. if µ(x) + s(x) = 0, then x got eliminated and is an outlier.\\nROBOT, unlike classical OT, guarantees that an adversarially picked outliers can not increase the distance arbitrarily. Let µ̃ = (1− )µ+ µc, i.e. µ̃ is µ contaminated with outliers from µc, and let ν be an arbitrary measure (in MKE, µ̃ is the contaminated data and ν is the model we learn). Adversary can arbitrarily increase OT(µ̃, ν) by manipulating the outlier distribution µc. For ROBOT we have the following bound:\\nTheorem 2.1. Let µ̃ = (1− )µ+ µc for some ∈ [0, 1), then\\nROBOT(µ̃, ν) ≤ (OT(µ, ν) + λ ‖µ− µc‖TV) ∧ λ‖µ̃− ν‖TV ∧OT(µ̃, ν). (2.3)\\nThis bound has two key takeaways: since TV norm of any two distributions is bounded by 1, adversary can not increase ROBOT(µ̃, ν) arbitrarily; in the absence of outliers, ROBOT is bounded by classical OT. See Appendix C for the proof.\\nRelated work We note connection between equation 2.2 and unbalanced OT (UOT) (Chizat., 2017; Chizat et al., 2018). UOT is typically formulated by replacing TV norm with KL(µ + s|µ) and adding an analogous term for ν. Chizat et al. (2018) studied entropy regularized UOT with various divergences penalizing marginal violations. Optimization problems similar to equation 2.2 have also been considered outside of the ML literature (Piccoli & Rossi, 2014; Liero et al., 2018). We are unaware of prior applications of UOT to outlier-robustness, but it was studied in the concurrent work of Balaji et al. (2020). Another relevant variation of OT is partial OT (Figalli, 2010; Caffarelli & McCann, 2010). It may also be considered for outlier-robustness, but it has a drawback of forcing mass destruction rather than adjusting marginals to ignore outliers when they are present. A concurrent work by Staerman et al. (2020) took a different path: they replaced the expectation in the Wasserstein-1 dual with a median-of-means to promote robustness. It is unclear what is the corresponding primal, making it hard to interpret as an optimal transport problem.\\nA major challenge with the aforementioned methods, including our Formulation 1, is the difficulty of the optimization problem. This is especially the case for MKEs, where a transport problem has to be solved in every iteration to obtain the gradient of the model parameters. Chizat et al. (2018) proposed a Sinkhorn-like algorithm for entropy regularized UOT, but it is not amenable to stochastic optimization. Balaji et al. (2020) proposed a stochastic optimization algorithm based on the UOT dual, but it requires two additional neural networks (total of four including dual potentials) to parameterize modified marginal distributions (i.e., µ + s and analogous one for ν). Optimizing with a median-of-means in the objective function as in (Staerman et al., 2020) is also challenging. The key contribution of our work is a formulation equivalent to equation 2.2, which is easily compatible with the large body of classical OT optimization techniques (Cuturi, 2013; Solomon et al., 2015; Genevay et al., 2016; Seguy et al., 2018).\\nMore efficient equivalent formulation At a first glance, there are two issues with equation 2.2: it appears asymmetric and it is unclear if it can be optimized efficiently. Below we present an equivalent formulation that is free of these issues:\\nFormulation 2:\\nROBOT(µ, ν) = minΠ∈F+(Rd×Rd) ∫ Cλ(x, y)Π(dx, dy) subject to ∫ B×Rd Π(dx, dy) = ∫ B µ(dx) ∀ B ∈ B(Rd)∫ Rd×C Π(dx, dy) = ∫ C ν(dy) ∀ C ∈ B(Rd), (2.4)\\nwhere Cλ is the truncated cost function defined as Cλ(x, y) = C(x, y) ∧ 2λ. Looking at equation 2.4, it is not apparent that it adds robustness to MKE, but it is symmetric, easy to combine\\nwith entropic regularization by simply truncating the cost, and benefits from stochastic optimization algorithms (Genevay et al., 2016; Seguy et al., 2018). This formulation also has a distant relation to the idea of loss truncation for achieving robustness (Shen & Sanghavi, 2019). Pele & Werman (2009) considered the Earth Mover Distance (discrete OT) with truncated cost to achieve computational improvements; they also mentioned its potential to promote robustness against outlier noise but did not explore this direction.\\nIn Section 3, we establish equivalence between the two ROBOT formulations, equation 2.2 and equation 2.4. This equivalence allows us to obtain an efficient algorithm based on equation 2.4 for robust MKE. We also provide a simple procedure for computing optimal s in equation 2.2 from the solution of equation 2.4, enabling a new OT application: outlier detection. We verify the effectiveness of robust MKE and outlier detection in our experiments in Section 4. Before presenting the equivalence proof, we formulate the discrete analogs of the two ROBOT formulations for their practical value.\"\n },\n {\n \"heading\": \"2.2 DISCRETE ROBOT FORMULATIONS\",\n \"text\": \"In practice we typically encounter samples from the distributions, rather then the distributions themselves. Sampling is also built into stochastic optimization. In this subsection, we present the discrete versions of the ROBOT formulations. The key detail is that, in equation 2.2, µ, ν and s are all supported on Rd, while in the discrete case the empirical measures µn ∈ ∆n−1 and νm ∈ ∆m−1 are supported on a set of points (∆r is the unit probability simplex in Rr). As a result, to formulate a discrete version of equation 2.2, we need to augment µn and νm with each others’ supports. To be precise, let supp(µn) = {X1, . . . , Xn} and supp(νm) = {Y1, . . . , Ym}. Define C = {Z1, Z2, . . . , Zm+n} = {X1, . . . , Xn, Y1, . . . , Ym}. Then discrete analog of equation 2.2 is Formulation 1 (discrete):\\nROBOT(µn, νm) = minΠ∈R(m+n)×(m+n) s∈Rm+n 〈Caug,Π〉+ λ [‖s1‖1 + ‖t1‖1] subject to Π1m+n = [ µn + s1 t1 ] , Π>1m+n = [ 0 νm ] Π 0, 1>m+ns = 0, (2.5)\\nwhere Caug ∈ R(m+n)×(m+n) is the augmented cost function Caug,i,j = c(Zi, Zj) (c is the ground cost, e.g., squared Euclidean distance), s = (s1, t1) and 1r is the vector all ones in Rr. The TV norm got replaced with its discrete analog, the L1 norm. Similarly to its continuous counterpart, the optimization problem is harder than the typical OT due to additional constraint optimization variable s and increased cost matrix size.\\nThe discrete analog of equation 2.4 is straightforward:\\nFormulation 2 (discrete):\\nROBOT(µn, νm) = { minΠ∈Rn×m 〈Cλ,Π〉 subject to Π1n = µn, Π >1m = νm, Π 0, (2.6)\\nwhereCλ,i,j = c(Xi, Yj)∧2λ. As in the continuous case, it is easy to adapt modern (regularized) OT solvers without any computational overhead. As in the continuous case, formulations of equation 2.5 and equation 2.6 are equivalent. It is also possible to recover s of equation 2.5 from the solution of equation 2.6 to perform outlier detection.\\nTwo-sided formulation So far we have assumed that one of the input distributions does not have outliers, which is the setting of MKE, where the clean distribution corresponds to the model we learn. In some applications, both distributions may be corrupted. To address this case, we provide an equivalent two-sided formulation, analogous to UOT with TV norm:\\nFormulation 3 (two-sided):\\nROBOT(µn, νm) = min Π∈R(m+n)×(m+n) s1∈Rm+n, s2∈Rm+n 〈Caug,Π〉+ λ [‖s1‖1 + ‖t1‖1 + ‖s2‖1 + ‖t2‖1] subject to Π1m+n = [ µn + s1 t1 ] , Π>1m+n = [ s2 νm + t2 ] Π 0, 1>m+ns1 = 0, 1>m+ns2 = 0. (2.7) where s1 = (s>1 , t > 1 ) > and s2 = (s>2 , t > 2 ) >.\"\n },\n {\n \"heading\": \"3 EQUIVALENCE OF THE ROBOT FORMULATIONS\",\n \"text\": \"In this section we present our main theorem, which demonstrates the equivalence between two formulations of the robust optimal transport:\\nTheorem 3.1. For any two measures µ and ν, ROBOT(µ, ν) has same value for both the formulations, i.e., Formulation 1 is equivalent to Formulation 2 both for continuous and discrete case. Moreover, we can recover optimal coupling of one formulation from the other.\\nBelow we sketch the proof of this theorem and highlight some important techniques used in the proof. We focus on the discrete case as it is more intuitive and has concrete practical implications in our experiments. A complete proof can be found in Appendix A. Please also see Appendix A.2 for the proof of equivalence between Formulations 1, 2 and 3 in the discrete case.\"\n },\n {\n \"heading\": \"3.1 PROOF SKETCH\",\n \"text\": \"In the remainder of this section we consider the discrete case, i.e., equation 2.5 for Formulation 1 (F1) and equation 2.6 for Formulation 2 (F2). Suppose Π∗2 is an optimal solution of F2. Then we construct a feasible solution Π∗1, s ∗ 1 = (s ∗ 1, t ∗ 1) of F1 based on Π ∗ 2 with the same value of the objective function as F2 and claim that (Π∗1, s ∗ 1) is an optimal solution. We prove the claim by contradiction: if (Π∗1, s ∗ 1) is not optimal, then there exists another pair (Π̃1, s̃1) which is optimal for F1 with strictly less objective value. We then construct another feasible solution Π∗2,new of Formulation 2 which has the same objective value as of (Π̃1, s̃1) for F1. This implies Π∗2,new has strictly less objective value for F2 than Π∗2, which is a contradiction.\\nThe two main pillars of this proof are (1) to construct a feasible solution of F1 starting from a feasible solution of F2 and (2) to show that the solution constructed is indeed optimal for F1. Hence step (1) gives a recipe to construct an optimal solution of F1 starting from an optimal solution of F2. We elaborate the first point in the next subsection, which has practical implications for outlier detection. The other point is more technical; interested readers may go through the proof in Appendix A.1.\\nAlgorithm 1 Generating optimal solution of F1 from F2\\n1: Start with Π∗2 ∈ Rn×m, an optimal solution of Formulation 2. 2: Create an augmented matrix Π ∈ Rm+n×m+n with all 0. Divide Π into four blocks:\\nΠ = Π11︸︷︷︸ n×n Π12︸︷︷︸ n×m\\nΠ21︸︷︷︸ m×n Π22︸︷︷︸ m×m 3: Set Π12 ← Π∗2 and collect all the indices I = {(i, j) : Ci,j > 2λ}. 4: Set Π12(i, j)← 0 for (i, j) ∈ I. 5: Set Π22(j, j)← ∑n i=1 Π ∗ 2(i, j)1(i,j)∈I for all 1 ≤ j ≤ m and set Π∗1 ← Π.\\n6: Set s∗1(i) ≤ ∑m j=1 Π ∗ 2(i, j)1(i,j)∈I for all 1 ≤ i ≤ n. 7: Set t∗1(j) = Π22(j, j) for all 1 ≤ j ≤ m. 8: return Π∗1, s ∗ 1, t ∗ 1.\"\n },\n {\n \"heading\": \"3.2 GOING FROM FORMULATION 2 TO FORMULATION 1\",\n \"text\": \"Let Π∗2 (respectively Π ∗ 1) be an optimal solution of F2 (respectively F1). Recall that Π ∗ 1 has dimension (m + n) × (m + n). From the column sum constraint in F1, we need to take the first n columns of Π∗1 to be exactly 0, whereas the last m columns must sum up to νm. For any matrix A, we denote by A[(a : b)× (c : d)] the submatrix consisting of rows from a to b and columns from c to d. Our main idea is to put a modified version of Π∗2 in Π ∗ 1[(1 : n) × (n + 1 : m + n)] and make Π∗1[(n+1 : m+n)×(n+1 : m+n)] diagonal. First we describe how to modify Π∗2. Observe that, if for some (i, j) Ci,j > 2λ, we expectXi ∈ supp(µn) to be an outlier resulting in high transportation cost, which is why we truncate the cost in F2. Therefore, to get an optimal solution of F1, we make the corresponding value of optimal plan 0 and dump the mass into the corresponding slack variable t∗1 in the diagonal of the bottom right submatrix. This changes the row sum, which is taken care of by s∗1. But, as we are not moving this mass outside the corresponding column, the column sum of Π∗1[(1 : (m + n)) : ((n + 1) : (m + n))] remains same as column sum of Π ∗ 2, which is νn. We summarize this procedure in Algorithm 1.\\nExample. In Figure 1, we provide an example to visualize the construction. On the left, we have Π∗2, an optimal solution of Formulation 2. The blue triangles denote the positions where the corresponding cost value is ≤ 2λ, and light-green squares denote the positions where the corresponding value of the cost matrix is> 2λ. To construct an optimal solution Π∗1 of Formulation 1 from this Π ∗ 2, we first create an augmented matrix of size 6× 6. We keep all the entries of of left 6× 3 sub-matrix as 0 (in this picture blank elements indicate 0). On the right submatrix, we put Π∗2 into the top-right block, but remove the masses from light-green squares, i.e. where cost value is > 2λ, and put it in the diagonal entries of the bottom right block as shown in Figure 1. This mass contributes to the slack variables s1 and t1, and this augmented matrix along with s1, t1 give us an optimal solution of Formulation 1.\"\n },\n {\n \"heading\": \"3.3 OUTLIER DETECTION WITH ROBOT\",\n \"text\": \"Our construction algorithm has practical consequences for outlier detection. Suppose we have two datasets, a clean dataset νm (i.e., has no outliers) and an outlier-contaminated dataset µn. We can detect the outliers in µn without directly solving costly Formulation 1 by following Algorithm 2. In this algorithm, λ is a regularization parameter that can be chosen via cross-validation or heuristically (see Section 4.2 for an example). In Section 4.2, we use this algorithm to perform outlier detection on image data.\\nAlgorithm 2 Outlier detection in contaminated data 1: Start with µn (contaminted data) and νm (clean data). 2: Solve Formulation 2 and obtain Π∗2 using a suitable value of λ. 3: Use Algorithm 1 to obtain Π∗1, s ∗ 1, t ∗ 1 from Π ∗ 2.\\n4: Find I, the set of all the indices where µn + s∗1 = 0. 5: Return I as the indices of outliers in µn.\\nTable 1: Robust mean estimation with GANs using different distribution divergences. True mean is η0 = 05; sample size n = 1000; contamination proportion = 0.2. We report results over 30 experiment restarts.\\nContamination JS Loss SH Loss RKL Loss ROBOT UOT\\nN (0.1 · 15, I5) 0.09 ± 0.03 0.11 ± 0.03 0.115 ± 0.03 0.1 ± 0.03 0.1 ± 0.04 N (0.5 · 15, I5) 0.23 ± 0.04 0.24 ± 0.05 0.24 ± 0.05 0.117 ± 0.03 0.2 ± 0.04 N (1 · 15, I5) 0.43 ± 0.05 0.43 ± 0.06 0.43 ± 0.06 0.261 ± 0.06 0.25 ± 0.05 N (2 · 15, I5) 0.67 ± 0.07 0.67 ± 0.08 0.67 ± 0.08 0.106 ± 0.03 0.1 ± 0.03\\n(a) Varying proportion of contamination (b) Varying outlier distribution mean\\nFigure 2: Empirical study of regularization hyperparameter λ sensitivity\"\n },\n {\n \"heading\": \"4 EMPIRICAL STUDIES\",\n \"text\": \"To evaluate effectiveness of ROBOT, we consider the task of robust mean estimation under the Huber contamination model. The data is generated from (1 − )N (η0, Id) + N (η1, Id) and the goal is to estimate η0. Prior work has advocated for using f -divergence GANs (Chao et al., 2018; Wu et al., 2020) for this problem and pointed out inefficiencies of Wasserstein GAN in the presence of outliers. We show that our robust OT formulation allows us to estimate the uncontaminated mean η0 comparably or better than a variety of f -divergence GANs. We also use this simulated setup to study sensitivity to the regularization hyperparameter λ.\\nIn our second experiment, we present a new application of optimal transport enabled by ROBOT. Suppose we have collected a curated dataset νm (i.e., we know that it has no outliers)—such data collection is expensive, and we want to benefit from it to automate subsequent data collection. Let µn be a second dataset collected “in the wild,” i.e., it may or may not have outliers. We demonstrate how ROBOT can be used to identify outliers in µn using the curated dataset νm.\"\n },\n {\n \"heading\": \"4.1 ROBUST MEAN ESTIMATION\",\n \"text\": \"Following Wu et al. (2020), we consider a simple generator of the form gθ(x) = x + θ, x ∼ N (0, Id), d is the data dimension. The basic idea of robust mean estimation with GANs is to minimize various distributional divergences between samples from gθ and observed data simulated from (1 − )N (η0, Id) + N (η1, Id). The goal is to estimate η0 with θ. To efficiently implement ROBOT GAN, we use a standard min-max optimization approach: solve the inner max (ROBOT) and use gradient descent for the outer min parameter. To solve ROBOT, it is straightforward to adopt any of the prior stochastic regularized OT solvers: the only modification is the truncation of the cost entries as in equation 2.6. We use the stochastic algorithm for semi-discrete regularized OT from\\n(Genevay et al., 2016, Algorithm 2). We summarize ROBOT GAN in Algorithm 3. Line 5 - Line 10 perform the inner optimization where we solve entropy regularized OT dual with truncated cost and Line 11 - Line 12 perform gradient update of θ.\\nAlgorithm 3 ROBOT GAN\\n1: Input: robustness regularizion λ, entropic regularization α, data distribution µn ∈ ∆n−1, supp(µn) = X = [X1, . . . , Xn], steps sizes τ and γ 2: Initialize: Initialize θ = θinit, set number of iterations M and L, i = 0, v = ṽ = 0. 3: for j = 1, . . . ,M do 4: Generate z̃ ∼ N (0, Id) and set z = z̃ + θ. 5: Set the cost vector c ∈ Rn as c(k) = c(Xk, z) ∧ 2λ for k = 1, . . . , n. 6: for i = 1, . . . , L do . solve entropy regularized OT dual 7: Set h← ṽ−cα and do the normalized exponential transformation u← eh\\n〈1,eh〉 . 8: Calculate the gradient∇ṽ← µn − u. 9: Update ṽ← ṽ + γ∇ṽ and v← (1/(j + i))ṽ + (j + i− 1/(j + i))v.\\n10: Do the same transformation of v as in Step 7, i.e. set h← v−cα and set Π← eh\\n〈1,eh〉 . 11: Set Π(k) = 0 for k such that C(Xk, z) > 2λ for k = 1, . . . , n. 12: Calculate gradient with respect to θ as ∇θ = 2 [ z ∑ k Π(k)−X>Π ] 13: Update θ ← θ − τ∇θ. 14: Ouput: θ\\nFor the f -divergence GANs (Nowozin et al., 2016) we use the code of Wu et al. (2020) for GANs with Jensen-Shannon (JS) loss, squared Hellinger (SH) loss and Reverse Kullback-Leibler (RKL) loss. For the exact expression of these divergences see Table 1 of Wu et al. (2020). We report estimation error measured by the Euclidean distance between true uncontaminated mean η0 and estimated mean θ for various contamination distributions in Table 1. ROBOT GAN performs well across all considered contamination distributions. As the difference between true mean η0 and contamination mean η1 increases, the estimation error of all methods tends to increase. However, when it becomes easier to distinguish outliers from clean samples, i.e., η1 = 2 · 15, performance of ROBOT noticeably improves.\\nWe also compared to the Sinkhorn-based UOT algorithm (Chizat et al., 2018) available in the Python Optimal Transport (POT) library (Flamary & Courty, 2017); to obtain a UOT GAN, we modified steps 5-11 of Algorithm 3 for computing Π. Unsurprisingly, both ROBOT and UOT perform similarly: recall equivalence to Formulation 3, which is similar to UOT with TV norm. The key insight of our work is the equivalence to classical OT with truncated cost, that greatly simplifies optimization and allows to use existing stochastic OT algorithms. In this experiment, the sample size n = 1000 is sufficiently small for the Sinkhorn-based UOT POT implementation to be effective, but it breaks in the experiment we present in Section 4.2. We also tried the code of Balaji et al. (2020) based on CVXPY (Diamond & Boyd, 2016), but it is too slow even for the n = 1000 sample size.\\nIn the previous experiment, we set λ = 0.5. Now we demonstrate empirically that there is a broad range of λ values performing well. In Figure 2a, we study sensitivity of λ under various contamination proportions holding η0 = 15 and η1 = 5 · 15 fixed. Horizontal lines correspond to λ = ∞, i.e., vanilla OT. The key observations are: there is a wide range of λ efficient at all contamination proportions, and ROBOT is always at least as good as vanilla OT (even when there is no contamination = 0). In Figure 2b, we present a similar study varying the mean of the contamination distribution and holding = 0.2 fixed. We see that as the contamination distribution gets closer to the true distribution, it becomes harder to pick a good λ, but the performance is always at least as good as the vanilla OT (horizontal lines).\"\n },\n {\n \"heading\": \"4.2 OUTLIER DETECTION FOR DATA COLLECTION\",\n \"text\": \"Our robust OT formulation equation 2.5 enables outlier identification. Let νm be a clean dataset and µn potentially contaminated with outliers. Recall that ROBOT allows modification of one of the input distributions to eliminate potential outliers. We can identify outliers in µn as follows: if µn(i)+s ∗ 1(i) = 0, thenXi, the ith point in µn, is an outlier. Instead of directly solving equation 2.5,\\nwhich may be inefficient, we use our equivalence results and solve an easier optimization problem equation 2.6, followed by recovering s to find outliers via Algorithm 2.\\nLet νm be a clean dataset consisting of 10k MNIST digits and µn be a dataset collected “in the wild” consisting of (different) 8k MNIST digits and 2k Fashion MNIST images. We compute ROBOT(µn, νm) to identify outlier Fashion MNIST images in µn. For each point in µn we obtain a prediction, outlier or clean, which allows us to evaluate accuracy. ROBOT outlier detection is 90% accurate in this experiment. We also comment on λ selection: since we know that νm is clean, we can subsample two datasets from it, compute vanilla OT to obtain transportation plan Π and set λ to be half the maximum distance between matched elements, i.e. 2λ = maxi,j{Cij : Πij > 0}, where C is the cost matrix for the two subsampled datasets. This procedure is essentially estimating maximum distance between matched clean samples. We also present a random sample of outliers identified by our method in Figure 3. All of the sampled outliers are Fashion MNIST images, although 90% accuracy suggests that some of the outliers were not identified. Decreasing λ can help to find more outliers, but may result in some clean samples being mistaken for outliers. We conclude that ROBOT can be used to assist in data collection once an initial set of clean data has been acquired. As we mentioned previously, the Sinkhorn-based UOT POT implementation is too expensive for this experiment due to larger sample size, yielding memory errors on a personal laptop with 16GB RAM.\\nFor comparison, we also consider a heuristic distance-based approach for identifying outliers. We estimate diameter τ of the set of clean dataset νm by taking the 99th percentile of the pairwise distance matrix of samples in νm. If outliers and clean data have disjoint support, we can adopt a simple heuristic: for each sample in the potentially contaminated µn compute an average distance to the clean samples in νm and declare a sample as an outlier if this average distance is greater than the diameter τ of the clean data. The accuracy of this procedure is 85.4%, inferior to the ROBOT accuracy of 90%. The disjoint support assumption justifying the distance-based heuristic might be too strong in practice. ROBOT continues to be effective even when the supports of clean and outlier distributions are not easily separable.\"\n },\n {\n \"heading\": \"5 SUMMARY AND DISCUSSION\",\n \"text\": \"We proposed and studied ROBOT, a robust formulation of optimal transport. We showed that although the problem is seemingly asymmetric and challenging to optimize, there is an equivalent formulation based on cost truncation that is symmetric and compatible with modern stochastic optimization methods for OT.\\nROBOT closely resembles unbalanced optimal transport (UOT). In our formulation, we added a TV regularizer to the vanilla optimal transport problem. This is motivated by the -contamination model. In UOT, the TV regularizer is typically replaced with a KL divergence. Other choices of the regularizer may lead to new properties and applications. Studying equivalent, simpler formulations of UOT with different divergences may be a fruitful future work direction.\\nFrom the practical perspective, in our experiments we observed no degradation of ROBOT GAN in comparison to OT GAN, even when there were no outliers. It is possible that replacing OT with ROBOT may be beneficial for various machine learning applications of OT. Data encountered in practice may not be explicitly contaminated with outliers, but it often has errors and other deficiencies, suggesting that a “no-harm” robustness is desirable.\"\n },\n {\n \"heading\": \"A PROOF OF THEOREM 3.1\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A.1 PROOF OF DISCRETE VERSION\",\n \"text\": \"Proof. Define a matrix Π as:\\nΠ(i, j) = { 0, if C(i, j) > 2λ Π∗2(i, j), otherwise\\nAlso define s ∈ Rn and t ∈ Rm as:\\ns∗1(i) = − m∑ j=1 Π∗2(i, j)1C(i,j)>2λ\\nand similarly define:\\nt∗1(j) = n∑ i=1 Π∗2(i, j)1C(i,j)>2λ\\nThese vectors corresponds to the row sums and the column sums of the elements of the optimal transport plan of Formulation 2, where the cost function exceeds 2λ. Note that, these co-ordinates of the optimal transport plan corresponding to those co-ordinates of cost matrix, where the cost is greater than 2λ and contribute to the objective value via their sum only, hence any different arrangement of these transition probabilities with same sum gives the same objective value.\\nNow based on this Π obtained we construct a feasible solution of Formulation 1 following Algorithm 1:\\nΠ∗1 = [ 0 Π 0 diag(t∗1) ] The row sums of Π∗1 is:\\nΠ∗11 =\\n[ µn + s ∗ 1\\nt∗1 ] and it is immediate from the construction that the column sums of Π∗1 is νm. Also as:\\nn∑ i=1 s∗1(i) = m∑ j=1 t∗1(j) = ∑\\n(i,j):Ci,j>2λ\\nΠ∗2(i, j)\\nand s∗1 0, t∗1 0, we have:\\n1>(µn + s ∗ 1 + t ∗ 1) = 1 >p = 1 .\\nTherefore, we have (Π∗1, s ∗ 1, t ∗ 1) is a feasible solution of Formulation 1. Now suppose this is not an optimal solution. Pick an optimal solution Π̃, s̃, t̃ of Formulation 1 so that:\\n〈Caug, Π̃〉+ λ [ ‖s̃‖1 + ‖t̃‖1 ] < 〈Caug,Π∗1〉+ λ [‖s∗1‖1 + ‖t∗1‖1]\\nThe following two lemmas provide some structural properties of any optimal solution of Formulation 1:\\nLemma A.1. Suppose Π∗1, s∗1, t∗1 are optimal solution for Formulation 1. Divide Π∗1 into four parts corresponding to augmentation as in algorithm 1:\\nΠ∗1 =\\n[ Π∗1,11 Π ∗ 1,12\\nΠ∗1,21 Π ∗ 1,22 ] Then we have Π∗1,11 = Π ∗ 1,21 = 0 and Π ∗ 1,22 is a diagonal matrix.\\nLemma A.2. If Π∗1, s∗1, t∗1 is an optimal solution of Formulation 1 then:\\n1. If Ci,j > 2λ then Π∗1(i, j) = 0. 2. If Ci,j < 2λ for some i and for all 1 ≤ j ≤ n, then s∗1(i) = 0. 3. If Ci,j < 2λ for some j and for all 1 ≤ i ≤ m, then t∗1(j) = 0. 4. If Ci,j < 2λ then s∗1(i)t ∗ 1(j) = 0.\\nWe provide the proofs in the next subsection. By Lemma A.1 we can assume without loss of generality:\\nΠ̃ = [ 0 Π̃12 0 diag(t̃) ] Now based on ( Π̃, s̃, t̃ ) we create a feasible solution namely Π∗2,new of Formulation 2 as follows:\\nDefine the set of indices {i1, · · · , ik} and {j1, . . . , jl} as:\\ns̃i1 , s̃i2 , . . . , s̃ik > 0 and t̃j1 , t̃j2 , . . . , t̃jl > 0 .\\nThen by part (4) of Lemma A.2 we have Ciα,jβ > 2λ for α ∈ {1, . . . , k} and β ∈ {1, . . . , l}. Also by part (2) of Lemma A.2 the value of transport plan at these co-ordinates is 0. Now distribute the mass of slack variables in these co-ordinates such that the marginals of new transport plan becomes exactly µn and νm. This new transport plan is our Π∗2,new. Recall that, ‖s̃‖1 = ‖t̃‖1. Hence, here the regularizer value decreases by 2λ‖s̃‖1 and the cost value increased by exactly 2λ‖s̃‖1 as we are truncating the cost. Hence we have:\\n〈Cλ,Π∗2,new〉 = 〈Caug, Π̃〉+ λ [ ‖s̃‖1 + ‖t̃‖1 ] < 〈Caug,Π∗1〉+ λ [‖s∗1‖1 + ‖t∗1‖1] = 〈Cλ,Π∗2〉\\nwhich is contradiction as Π∗2 is the optimal solution of Formulation 2. This completes the proof for the discrete part.\"\n },\n {\n \"heading\": \"A.2 PROOF OF EQUIVALENCE FOR TWO SIDED FORMULATION\",\n \"text\": \"Here we prove that our two sided formulation, i.e. Formulation 3 (equation 2.7) is equivalent to Formulation 1 (equation 2.5) for the discrete case. Towards that end, we introduce another auxiliary formulation and show that both Formulation 1 and Formulation 3 are equivalent to the following auxiliary formulation of the problem.\\nFormulation 4:\\nWR,L,4(p, q) = minΠ∈Rm×n,s1∈Rm,s2∈Rn 〈C,Π〉+ λ [‖s1‖1 + ‖s2‖1] subject to Π1n = p+ s1 ΠT 1m = q + s2\\nΠ 0\\n(A.1)\\nFirst we show that Formulation 1 and Formulation 4 are equivalent in a sense that they have the same optimal objective value.\\nTheorem A.3. Suppose C is a cost function such that C(x, x) = 0. Then Formulation 1 and Formulation 4 has same optimal objective value.\\nProof. Towards that end, we show that given one optimal variables of one formulation we can get optimal variables of other formulation with the same objective value. Before going into details we need the following lemma whose proof is provided in Appendix B:\\nLemma A.4. Suppose Π∗4, s∗4,1, s∗4,2 are the optimal variables of Formulation 4. Then s∗4,1 0 and s∗4,2 0.\\nNow we prove that optimal value of Formulation 1 and Formulation 4 are same. Let (Π∗1, s ∗ 1,1, t ∗ 1,1) is an optimal solution of Formulation 1. Then we claim that (Π∗1, s ∗ 1,1, t ∗ 1,1) is also an optimal solution of Formulation 4. Clearly it is feasible solution of Formulation 4. Suppose it is not optimal, i.e. there exists another optimal solution (Π̃4, s̃4,1, s̃4,2) such that:\\n〈C, Π̃4〉+ λ(‖s̃4,1‖1 + ‖s̃4,2‖2) < 〈C,Π∗1,12〉+ λ(‖s∗1,1‖1 + ‖t∗1,1‖1)\\nNow based on (Π̃4, s̃4,1, s̃4,2) we construct a feasible solution of Formulation 1 as follows:\\nΠ̃1 = [ 0 Π̃4 0 −diag(s̃4,2) ] Note that we proved in Lemma A.4 s̃4,2 0, hence we have Π̃1 0. Now as the column sums of Π̃4 is q + s̃4,2, we have column sums of Π̃1 = [0 q>]> and the row sums are [(p+ s̃4,1)> s̃>4,2]\\n>. Hence we take s̃1,1 = s̃4,1 and s̃1,2 = s̃4,2. Then it follows:\\n〈Caug, Π̃1〉+ λ [‖s̃1,1‖1 + ‖s̃1,2‖1] = 〈C, Π̃4〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] < 〈C,Π∗1,12〉+ λ [ ‖s∗1,1‖1 + ‖t∗1,1‖1 ] = 〈Caug,Π∗1〉+ λ [ ‖s∗1,1‖1 + ‖t∗1,1‖1\\n] This is contradiction as we assumed (Π∗1, s ∗ 1,1, t ∗ 1,2) is an optimal solution of Formulation 1. Therefore we conclude (Π∗1, s ∗ 1,1, t ∗ 1,1) is also an optimal solution of Formulation 4 which further concludes Formulation 1 and Formulation 4 have same optimal values. This completes the proof of the theorem.\\nTheorem A.5. The optimal objective value of Formulation 3 and Formulation 4 are same.\\nProof. Like in the proof of Theorem A.3 we also prove couple of lemmas.\\nLemma A.6. Any optimal transport plan Π∗3 of Formulation 3 has the following structure: If we write,\\nΠ∗3 =\\n[ Π∗3,11 Π ∗ 3,12\\nΠ∗3,21 Π ∗ 3,22 ] then Π∗3,11 and Π ∗ 3,22 are diagonal matrices and Π ∗ 3,21 = 0.\\nLemma A.7. If s∗3,1, t∗3,1, s∗3,2, t∗3,2 are four optimal slack variables in Formulation 3, then s∗3,1, t ∗ 3,1 0 and s∗3,2, t∗3,2 0.\\nProof. The line of argument is same as in proof of Lemma A.4.\\nNext we establish equivalence. Suppose (Π∗3, s ∗ 3,1, t ∗ 3,1, s ∗ 3,2, t ∗ 3,2) are optimal values of Formulation 3. We claim that (Π∗3,12, s ∗ 3,1 − s∗3,2, t∗3,1 − t∗3,2) forms an optimal solution of Formulation 4. The objective value will then also be same as s∗3,1 0, s∗3,2 0 (Lemma A.7) implies ‖s∗3,1 − s∗3,2‖1 = ‖s∗3,1‖1 + ‖s∗3,2‖1 and similarly t∗3,1 0, t∗3,2 0 implies ‖t∗3,1 − t∗3,2‖1 = ‖t∗3,1‖1 + ‖t∗3,2‖1. Feasibility is immediate. Now for optimality, we again prove by contradiction. Suppose they are not optimal. Then lets say Π̃4, s̃4,1, s̃4,2 are an optimal triplet of Formulation 4. Now construct another feasible solution of Formulation 3 as follows: Set s̃3,2 = t̃3,2 = 0, s̃3,1 = s̃4,1 and t̃3,1 = s̃4,2. Set the matrix as:\\nΠ̃3 = [ 0 Π̃4 0 −diag(s̃4,2) ] Then it follows that ( Π̃3, s̃3,1, s̃3,2, t̃3,1, t̃3,2 ) is a feasible solution of Formulation 3. Finally we\\nhave:\\n〈Caug, Π̃3〉+ λ [ ‖s̃3,1‖1 + ‖s̃3,2‖1 + ‖t̃3,1‖1 + ‖t̃3,2‖1 ] = 〈Caug, Π̃3〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] = 〈C, Π̃4〉+ λ [‖s̃4,1‖1 + ‖s̃4,2‖1] < 〈C,Π∗3,12〉+ λ [ ‖s∗3,1 − s∗3,2‖1 + ‖t∗3,1 − t∗3,2‖1\\n] = 〈Caug,Π∗3〉+ λ [ ‖s∗3,1‖1 + ‖s∗3,2‖1 + ‖t∗3,1‖1 + ‖t∗3,2‖1\\n] This contradicts the optimality of (Π∗3, s ∗ 3,1, s ∗ 3,2, t ∗ 3,1, t ∗ 3,2). This completes the proof.\"\n },\n {\n \"heading\": \"A.3 PROOF OF CONTINUOUS VERSION\",\n \"text\": \"Proof. In this proof we denote by F1 the optimization problem of equation equation 2.2 and by F2 the optimization problem equation equation 2.4. Assume that µn and νm denote the respective empirical measures relative to µ, ν. From Villani (2009), we know that µn, νn converge weakly to µ and ν respectively. Therefore, ROBOT2(µn, µ) → 0. Similary for νn and ν. Thus, by triangle inequality,\\nlim n→∞\\n|F2(µn, νn)− F2(µ, ν)| = 0.\\nBut ROBOT2(µn, νn) = ROBOT1(µn, νn). Therefore, our proof is complete if we can show that\\nlim n,m→∞\\n|F1(µn, νm)− F1(µ, ν)| → 0.\\nLet S = {s signed measure : µ+ s is a probability measure in Rd}. For s ∈ S , define\\nW (µ+S, ν) = \\nminΠ∈F(Rd×Rd) ∫ C(x, y) Π(dx, dy) + λ‖s‖TV\\nsubject to\\n∫ A Π(dx, dy) ≥ 0 ∀ A ∈ B(Rd × Rd)\\nµ(dx) + s(dx)) ≥ 0 ∀ B ∈ B(Rd)∫\\nRd×C Π(dx, dy) = ∫ C ν(dy) ∀ C ∈ B(Rd)\\nBy Lemma A.8, ∃ s ∈ S such that ROBOT (µ, ν) = W (µ + s, ν) + λ‖s‖TV . Let s = s+ − s−, where s+ and s− are positive measures on Rd. Let ‖s‖TV = γ. Then, ‖s−‖TV = ‖s+‖TV = γ/2. Then consider X1, . . . , Xn ∼ (P − s−)/(1 − γ), Y1, . . . , Yn ∼ s−/γ, Z1, . . . , Zn ∼ s+/γ. Then for any bounded continuous function f ,\\nlim n→∞ ∑ i f(Xi)/n = ∫ f(x) (P − s−) (1− γ) (dx)\\nlim n→∞ ∑ i f(Zi)/n = ∫ f(x) s+ γ (dx) (A.2)\\nTherefore, the distribution given by (P + s)n = (1− γ) n ∑ i δXi + γ n ∑ i δZi satisfies, (P + s)n\\nL→ P + s, and therefore from (Villani, 2009), limn→∞WC((P + S)n, νn)→ W (P + S,Q). Here δx is the Dirac mass at x. Moreover, ‖sn‖ = ‖s‖, where sn satisfies sn = γ n ( ∑ i δZi − ∑ i δYi).\\nAlso, ROBOT (µn, νn) ≤ W ((P + s)n, νn) + λ‖s‖TV , and therefore ROBOT2(µ, ν)) = lim supn→∞ROBOT (µn, νn) ≤ ROBOT (µ, ν). Now, let s̃n satisfy W1(µn + s̃n, νn) + λ‖s̃n‖TV = ROBOT (µn, νn). Such an s̃n exists by the proof of the discrete part because µn, νn are discrete measures.\\nThen, similar to the Step 1 in the proof of Lemma A.8, there exists a probability measure µ⊕ s and a subsequence {nk}k≥1 such that µnk + snk almost surely converges weakly to µ⊕ s. Moreover, similar to Step 2 of Lemma A.8 W1(µnk + snk , µ ⊕ s) → 0 as well as ‖snk‖TV → ‖µ⊕ s−µ‖TV . Thus, W1(µnk + snk , νnk) +λ‖snk‖TV →W1(µ⊕ s, ν) +λ‖µ⊕ s−µ‖TV . But by the proof of the discrete part ROBOT (µnk , νnk) = ROBOT2(µnk , νnk) → ROBOT2(µ, ν). Therefore, with s = µ⊕ s− µ, W1(µ+ s, ν) + λ‖s‖TV = ROBOT2(µ, ν). Therefore, ROBOT2(µ, ν) = lim supn→∞ROBOT (µn, νn) ≥ ROBOT (µ, ν). Thus the equality holds.\\nLemma A.8. Assume that µ, ν is such that ∫ ‖x‖dµ, ∫ ‖x‖dν < ∞. Moreover, assume that C(x, y) in equation 2.2 is the l1 norm, i.e., C(x, y) = ‖x− y‖. Then, there exists s with µ+ s being a probability measure such that\\nW1(µ+ s, ν) + λ‖s‖TV = ROBOT (µ, ν), (A.3)\\nwhere W1 is the Wasserstein-1 norm with the cost function C(·, ·) as mentioned above.\\nProof. Let µn, νm be the empirical measures relative to µ, ν respectively. We know that since µn, νm are discrete, there exists sn satisfying W1(µn + sn, νm) = ROBOT (µn, νm). We provide the proof in the following steps.\\nStep 1: Almost surely µ× ν, there exists a subsequence {nk}k≥1 such that {µn + sn}n and {νn}n is relatively compact.\\nµ and ν are probability measures on Rd and are therefore tight. Let K be such that Pµ(X /∈ K ), Pν(Y /∈ K ) ≤ /4. Consider the empirical distributions νn = ∑ i δYi/n, µn ∑ i δXi/n of ν, µ respectively. Here, Xi ∼ µ and Yi ∼ν . Fix an ω. Then {X1, . . . , Xn, Y1, . . . , Yn} is fixed. Now by the construction for the discrete case, sn has support in {X1, . . . , Xn, Y1, . . . , Yn}. Let Tn be the optimal transport map from µn to νn. Then, for every i ≤ n, there exists a unique j ≤ n, such that Tn(Xi) = Yj . Define τn : {1, . . . , n} → {1, . . . , n} such that τn(i) = j if Tn(Xi) = Yj .Then µn + sn = ∑ i δZi/n, where Zi = Xi or Yτn(i) and δx is the Dirac delta mass at x.\\nThen, let Z ∼ µn + sn Pω(Z /∈ K |µn + sn) ≤ ∑ i 1(Xi /∈K )/n+ ∑ i 1(Yi /∈K )/n (A.4)\\nTherefore, E(Pω(Z /∈ K |µn+sn)) ≤ /2. Moreover, V ar(Pω(Z /∈ K |µn+sn)) = o(n−1)→ 0. Therefore, limn→∞ Pµn×νn(Pω(Z /∈ K |µn + sn) ≤ ) → 1. Therefore µn + sn is almost surely tight and thus by Prokhorov’s Theorem also relatively compact.\\nStep 2: Therefore, for ω almost surely, there exists a subsequence {nk}k≥1 such that µnk + snk converges weakly to a limit (dependent on ω) µ ⊕ s which is a probability measure. Moreover,∫ ‖x‖d(µnk + snk) < ∞ almost surely. By Bolzano-Weierstrass Theorem, there exists a further\\nsubsequence {nkl}l such that ∫ ‖x‖d(µnkl + snkl )→ ∫ ‖x‖d(µ+ s) almost surely. For the sake of convenience, without loss of generality, we will replace the sub-subsequence {nkl}l with {nk}k≥1 henceforth.\\nThus, by Theorem 6.9 of (Villani, 2009) , W1(µnk + snk , µ ⊕ s) → 0 almost surely. Moreover, W1(µnk , µ)→ 0 almost surely. Therefore ‖snk‖TV → ‖µ⊕ s− µ‖TV almost surely.\\nStep 3: Consider an arbitrary S = S+ − S−, such that S+ and S− are positive measures on Rd, and µ+ S is a probability measure. Let ‖S‖TV = γ. Then, ‖S−‖TV = ‖S+‖TV = γ/2. Then consider X1, . . . , Xn ∼ (µ− S−)/(1− γ), Y1, . . . , Yn ∼ S−/γ, Z1, . . . , Zn ∼ S+/γ. Then for any bounded continuous function f ,\\nlim n→∞ ∑ i f(Xi)/n = ∫ f(x) (P − s−) (1− γ) (dx)\\nlim n→∞ ∑ i f(Zi)/n = ∫ f(x) s+ γ (dx) (A.5)\\nTherefore, the distribution given by (µ + S)n(A) = (1 − γ) ∑ i 1Xi∈A + (γ) ∑ i 1Zi∈A satisfies, (µ+S)n L→ µ+S, and therefore from (Villani, 2009), limn→∞W1((µ+S)n, νn)→W1(µ+S, ν). Moreover, ‖Sn‖TV = ‖S‖TV , where Sn satisfies Sn(A) = γ\\nn\\n∑ i 1Zi∈A − ∑ i 1Yi∈A.\\nBut, W1(µnk + snk , νnk) + λ‖snk‖TV ≤ W1((µ + S)nk , νnk) + λ‖Snk‖TV . Therefore, taking limits, W1(µ⊕ s, ν) + λ‖µ⊕ s−µ‖TV ≤W1(µ+S, ν) + λ‖S‖TV , and thus the proof holds with s = µ⊕ s− µ.\"\n },\n {\n \"heading\": \"B PROOF OF ADDITIONAL LEMMAS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"B.1 PROOF OF LEMMA A.1\",\n \"text\": \"Proof. The fact that Π∗1,11 = Π ∗ 1,21 = 0 follows from the fact that Π ∗ 1 0 and Π∗11 = Q. To prove that Π∗1,22 is diagonal, we use the fact that the any diagonal entry the cost matrix is 0. Now suppose Π∗1,22 is not diagonal. Then define a matrix Π̂ as following: set Π̂11 = Π̂21 = 0, Π̂12 = Π ∗ 1,12 and:\\nΠ̂22(i, j) =\\n{∑m k=1 Π ∗ 1,22(k, i), if j = i\\n0, if j 6= i\\nAlso define ŝ = s∗1 and t̂ as t̂(i) = Π̂22(i, i). Then clearly (Π̂, ŝ, t̂) is a feasible solution of Formulation 1. Note that:\\n‖t̂‖1 = 1>Π̂221 = 1>Π∗1,221 = ‖t∗1‖1\\nand by our construction 〈Caug, Π̂〉 < 〈Caug,Π∗1〉. Hence (Π̂, ŝ, t̂) reduces the value of the objective function of Formulation 1 which is a contradiction. This completes the proof.\"\n },\n {\n \"heading\": \"B.2 PROOF OF LEMMA A.2\",\n \"text\": \"Proof. 1. Suppose Π∗1(i, j) > 0. Then dump this mass to s ∗ 1(j) and make it 0. In this way\\n〈Caug,Π∗1〉 will decrease by > 2λΠ∗1(i, j) and the regularizer value will increase by atmost 2λΠ∗1(i, j), resulting in overall reduction in the objective value, which leads to a contradiction. 2. Suppose each entry of ith row of C is < 2λ. Then if s∗1(i) > 0, we can distribute this mass in the ith row such that, s∗1(i) = a1 + a2 + · · · + am with the condition that t∗1(j) ≥ aj . Now we reduce t∗1 as:\\nt∗1(j)← t∗1(j)− aj Hence the value 〈Caug,Π∗1(i, j)〉 will increase by a value < 2λs∗1(i) but the value of regularizer will decrease by the value of 2λs∗1(i), resulting in overall decrease in the value of objective function. 3. Same as proof of part (2) by interchanging row and column in the argument. 4. Suppose not. Then choose < s∗1(i) ∧ t∗1(j), Add to Π∗1(i, j). Hence the cost function value 〈Caug,Π∗1〉 will increase by < 2λ but the regularizer value will decrease by 2λ , resulting in overall decrease in the objective function.\"\n },\n {\n \"heading\": \"B.3 PROOF OF LEMMA A.4\",\n \"text\": \"Proof. For the notational simplicity, we drop the subscript 4 now as we will only deal with the solution of Formulation 4 and there will be no ambiguity. We prove the Lemma by contradiction. Suppose s∗1,i > 0. Then we show one can come up with another solution (Π̃, s̃1, s̃2) of Formulation 4 such that it has lower objective value. To construct this new solution, make:\\ns̃1,j = { s∗1,j , if j 6= i 0, if j = i\\nNow to change the optimal transport plan, we will only change ith row of Π∗. We subtract a1, a2, . . . , an ≥ 0 from ith column of Π∗ in such a way, such that none of the elements are negative. Hence the column sum will be change, i.e. the value of s̃2 will be:\\ns̃2,j = s ∗ 2,j − aj ∀1 ≤ j ≤ n .\\nNow clearly from our construction: 〈C, Π̃〉 ≤ 〈C,Π∗〉\\nFor the regularization part, note that, as we only reduced ith element of s∗1, we have ‖s̃1‖1 = ‖s∗1‖1 − s∗1,i. And by simple triangle inequality,\\n‖s̃2‖1 ≤ ‖s∗2‖1 + ‖a1‖1 = ‖s∗2‖1 + s∗1,i\\nby construction ai’s, as ai ≥ 0 and ∑ i ai = s ∗ 1,i. Hence we have:\\n‖s̃1‖1 + ‖s̃2‖1 ≤ ‖s∗1‖1 − s∗1,i + ‖s∗2‖1 + s∗1,i = ‖s∗1‖1 + ‖s∗2‖1 .\\nHence the value corresponding to regularizer will also decrease. This completes the proof.\"\n },\n {\n \"heading\": \"B.4 PROOF OF LEMMA A.6\",\n \"text\": \"Proof. We prove this lemma by contradiction. Suppose Π∗3 does not have the structure mentioned in the statement of Lemma. Construct another transport plan for Formulation 3 Π̃3 as follows: Keep Π̃3,12 = Π ∗ 3,12 and set Π̃3,12 = 0. Construct the other parts as:\\nΠ̃3,11(i, j) =\\n{∑m k=1 Π ∗ 3,11(i, k) + ∑n k=1 Π ∗ 3,21(k, i), if i = j\\n0, if i 6= j\\nand\\nΠ̃3,22(i, j) =\\n{∑n k=1 Π ∗ 3,22(k, i), if i = j\\n0, if i 6= j It is immediate from the construction that:\\n〈Caug, Π̃3〉 ≤ 〈Caug,Π∗3〉\\nAs for the regularization term: Note the by our construction s̃4 will be same as s∗4 as column sum of Π̃3,22 is same as Π∗3,22. For the other three:\\ns̃3(i) = Π̃3,11(i, i) = m∑ k=1 Π∗3,11(i, k) + n∑ k=1 Π∗3,21(k, i)\\ns̃2(i) = Π̃3,22(i, i) = n∑ k=1 Π∗3,22(k, i)\\nand hence by construction:\\n‖s̃2‖1 = 1>Π∗3,221 = ‖s∗2‖1 − 1>Π∗3,211 .\\n‖s̃3‖1 = 1>Π∗3,111 + 1>Π∗3,211 = ‖s∗3‖1 And also by our construction, s̃1 = s∗1 + c where c = (Π ∗ 3,21)\\n>1. As a consequence we have ‖c‖1 = 1>Π∗3,211. Then it follows:\\n4∑ i=1 ‖s̃i‖1 = ‖s∗1 + c‖+ ‖s∗2‖1 − 1>Π∗3,211 + ‖s∗3‖1 + ‖s∗4‖1\\n≤ 4∑ i=1 ‖s∗i ‖1 + ‖c‖1 − 1>Π∗3,211\\n= 4∑ i=1 ‖s∗i ‖1\\nSo the objective value is overall reduced. This contradicts the optimality of Π∗3 which completes the proof.\"\n },\n {\n \"heading\": \"C PROOF OF THEOREM 2.1\",\n \"text\": \"Proof. The proof is immediate from the Formulation 1. Recall that the Formulation 1 can restructured as:\\nROBOT (µ̃, ν) = inf P {OT (P, ν) + λ‖P − µ̃‖TV } .\\nwhere the infimum is taking over all measure dominated by some common measure σ (with respect to which µ, µc, ν are dominated). Hence,\\nROBOT (µ̃, ν) ≤ OT (P, ν) + λ‖P − µ̃‖TV for any particular choice of P . Taking P = µ we get that\\nROBOT (µ̃, ν) ≤ OT (µ, ν) + λ‖µ− µ̃‖TV = OT (µ, ν)) + λ ‖µ− µc‖TV Taking P = ν we get ROBOT (µ̃, ν) ≤ λ‖ν − µ̃‖TV and finally taking P = µ̃ we get ROBOT (µ̃, ν) ≤ OT (µ̃, ν). This completes the proof.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"string","value":"OUTLIER-ROBUST OPTIMAL TRANSPORT"}}},{"rowIdx":90,"cells":{"paper_id":{"kind":"string","value":"SP:198d7f650c930a1423f7f30688cd2f73d2719920"},"summaries":{"kind":"list like","value":["This paper aims to extend the continuous optimization approach to causal discovery to handle interventional data as well as observational data. It describes a method for learning the causal structure over a set of categorical variables and reports strong empirical performance. However, no theoretical guarantee or analysis is provided, which is a significant weakness in my view. It also makes no comment on or comparison to a paper that has essentially the same goal, https://arxiv.org/pdf/2007.01754.pdf. The latter paper seems to me more principled and convincing. "],"string":"[\n \"This paper aims to extend the continuous optimization approach to causal discovery to handle interventional data as well as observational data. It describes a method for learning the causal structure over a set of categorical variables and reports strong empirical performance. However, no theoretical guarantee or analysis is provided, which is a significant weakness in my view. It also makes no comment on or comparison to a paper that has essentially the same goal, https://arxiv.org/pdf/2007.01754.pdf. The latter paper seems to me more principled and convincing. \"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Promising results have driven a recent surge of interest in continuous optimization methods for Bayesian network structure learning from observational data. However, there are theoretical limitations on the identifiability of underlying structures obtained from observational data alone. Interventional data provides much richer information about the underlying data-generating process. However, the extension and application of methods designed for observational data to include interventions is not straightforward and remains an open problem. In this paper we provide a general framework based on continuous optimization and neural networks to create models for the combination of observational and interventional data. The proposed method is applicable even in the challenging and realistic case that the identity of the intervened upon variable is unknown. We examine the proposed method in the setting of graph recovery both de novo and from a partially-known edge set. We establish strong benchmark results on several structure learning tasks, including structure recovery of both synthetic graphs as well as standard graphs from the Bayesian Network Repository."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Bruce Abramson","John Brown","Ward Edwards","Allan Murphy","Robert L Winkler"],"title":"Hailfinder: A bayesian system for forecasting severe weather","venue":"International Journal of Forecasting,","year":1996},{"authors":["Ingo A Beinlich","Henri Jacques Suermondt","R Martin Chavez","Gregory F Cooper"],"title":"The alarm monitoring system: A case study with two probabilistic inference techniques for belief networks","venue":"In AIME","year":1989},{"authors":["Yoshua Bengio","Tristan Deleu","Nasim Rahaman","Rosemary Ke","Sébastien Lachapelle","Olexa Bilaniuk","Anirudh Goyal","Christopher Pal"],"title":"A meta-transfer objective for learning to disentangle causal mechanisms","venue":null,"year":1901},{"authors":["David Maxwell Chickering"],"title":"Optimal structure identification with greedy search","venue":"Journal of machine learning research,","year":2002},{"authors":["Gregory F. 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Cooper\",\n \"Changwon Yoo\"\n ],\n \"title\": \"Causal Discovery from a Mixture of Experimental and Observational Data\",\n \"venue\": \"In Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence,\",\n \"year\": 1999\n },\n {\n \"authors\": [\n \"Laura Douglas\",\n \"Iliyan Zarov\",\n \"Konstantinos Gourgoulias\",\n \"Chris Lucas\",\n \"Chris Hart\",\n \"Adam Baker\",\n \"Maneesh Sahani\",\n \"Yura Perov\",\n \"Saurabh Johri\"\n ],\n \"title\": \"A universal marginalizer for amortized inference in generative models\",\n \"venue\": \"arXiv preprint arXiv:1711.00695,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Daniel Eaton\",\n \"Kevin Murphy\"\n ],\n \"title\": \"Belief net structure learning from uncertain interventions\",\n \"venue\": \"J Mach Learn Res,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Daniel Eaton\",\n \"Kevin Murphy\"\n ],\n \"title\": \"Exact bayesian structure learning from uncertain interventions\",\n \"venue\": \"In Artificial Intelligence and Statistics, pp\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Daniel Eaton\",\n \"Kevin Murphy\"\n ],\n \"title\": \"Bayesian structure learning using dynamic programming and MCMC\",\n \"venue\": \"In Uncertainty in Artificial Intelligence, pp\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Frederick Eberhardt\",\n \"Clark Glymour\",\n \"Richard Scheines\"\n ],\n \"title\": \"On the number of experiments sufficient and in the worst case necessary to identify all causal relations among n variables\",\n \"venue\": \"arXiv preprint arXiv:1207.1389,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"AmirEmad Ghassami\",\n \"Saber Salehkaleybar\",\n \"Negar Kiyavash\",\n \"Kun Zhang\"\n ],\n \"title\": \"Learning causal structures using regression invariance\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Ian Goodfellow\",\n \"Jean Pouget-Abadie\",\n \"Mehdi Mirza\",\n \"Bing Xu\",\n \"David Warde-Farley\",\n \"Sherjil Ozair\",\n \"Aaron Courville\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Generative adversarial nets\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Olivier Goudet\",\n \"Diviyan Kalainathan\",\n \"Philippe Caillou\",\n \"Isabelle Guyon\",\n \"David Lopez-Paz\",\n \"Michèle Sebag\"\n ],\n \"title\": \"Causal generative neural networks\",\n \"venue\": \"arXiv preprint arXiv:1711.08936,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Olivier Goudet\",\n \"Diviyan Kalainathan\",\n \"Philippe Caillou\",\n \"Isabelle Guyon\",\n \"David Lopez-Paz\",\n \"Michele Sebag\"\n ],\n \"title\": \"Learning functional causal models with generative neural networks\",\n \"venue\": \"In Explainable and Interpretable Models in Computer Vision and Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Isabelle Guyon\"\n ],\n \"title\": \"Cause-effect pairs kaggle competition\",\n \"venue\": \"URL https://www. kaggle. com/c/causeeffect-pairs,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Isabelle Guyon\"\n ],\n \"title\": \"Chalearn fast causation coefficient challenge\",\n \"venue\": \"URL https://www. codalab. org/competitions/1381,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Alain Hauser\",\n \"Peter Bühlmann\"\n ],\n \"title\": \"Characterization and greedy learning of interventional markov equivalence classes of directed acyclic graphs\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"David Heckerman\",\n \"Dan Geiger\",\n \"David M Chickering\"\n ],\n \"title\": \"Learning bayesian networks: The combination of knowledge and statistical data\",\n \"venue\": \"Machine learning,\",\n \"year\": 1995\n },\n {\n \"authors\": [\n \"Christina Heinze-Deml\",\n \"Marloes H Maathuis\",\n \"Nicolai Meinshausen\"\n ],\n \"title\": \"Causal structure learning\",\n \"venue\": \"Annual Review of Statistics and Its Application,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Christina Heinze-Deml\",\n \"Jonas Peters\",\n \"Nicolai Meinshausen\"\n ],\n \"title\": \"Invariant causal prediction for nonlinear models\",\n \"venue\": \"Journal of Causal Inference,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Steven M Hill\",\n \"Laura M Heiser\",\n \"Thomas Cokelaer\",\n \"Michael Unger\",\n \"Nicole K Nesser\",\n \"Daniel E Carlin\",\n \"Yang Zhang\",\n \"Artem Sokolov\",\n \"Evan O Paull\",\n \"Chris K Wong\"\n ],\n \"title\": \"Inferring causal molecular networks: empirical assessment through a community-based effort\",\n \"venue\": \"Nature methods,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Biwei Huang\",\n \"Kun Zhang\",\n \"Jiji Zhang\",\n \"Joseph Ramsey\",\n \"Ruben Sanchez-Romero\",\n \"Clark Glymour\",\n \"Bernhard Schölkopf\"\n ],\n \"title\": \"Causal discovery from heterogeneous/nonstationary data\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Oleg Ivanov\",\n \"Michael Figurnov\",\n \"Dmitry Vetrov\"\n ],\n \"title\": \"Variational autoencoder with arbitrary conditioning\",\n \"venue\": \"arXiv preprint arXiv:1806.02382,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Amin Jaber\",\n \"Murat Kocaoglu\",\n \"Karthikeyan Shanmugam\",\n \"Elias Bareinboim\"\n ],\n \"title\": \"Causal discovery from soft interventions with unknown targets: Characterization and learning\",\n \"venue\": \"Advances in Neural Information Processing Systems,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Diviyan Kalainathan\",\n \"Olivier Goudet\",\n \"Isabelle Guyon\",\n \"David Lopez-Paz\",\n \"Michèle Sebag\"\n ],\n \"title\": \"Sam: Structural agnostic model, causal discovery and penalized adversarial learning\",\n \"venue\": \"arXiv preprint arXiv:1803.04929,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Diederik P Kingma\",\n \"Jimmy Ba\"\n ],\n \"title\": \"Adam: A method for stochastic optimization\",\n 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\"title\": \"Gradient-based neural dag learning\",\n \"venue\": \"arXiv preprint arXiv:1906.02226,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Steffen L Lauritzen\",\n \"David J Spiegelhalter\"\n ],\n \"title\": \"Local computations with probabilities on graphical structures and their application to expert systems\",\n \"venue\": \"Journal of the Royal Statistical Society: Series B (Methodological),\",\n \"year\": 1988\n },\n {\n \"authors\": [\n \"Yang Li\",\n \"Shoaib Akbar\",\n \"Junier B Oliva\"\n ],\n \"title\": \"Flow models for arbitrary conditional likelihoods\",\n \"venue\": \"arXiv preprint arXiv=1909.06319,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"David Lopez-Paz\",\n \"Krikamol Muandet\",\n \"Bernhard Schölkopf\",\n \"Iliya Tolstikhin\"\n ],\n \"title\": \"Towards a learning theory of cause-effect inference\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Ricardo Pio Monti\",\n \"Kun 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\"Journal of the Royal Statistical Society: Series B (Statistical Methodology),\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Jonas Peters\",\n \"Dominik Janzing\",\n \"Bernhard Schölkopf\"\n ],\n \"title\": \"Elements of causal inference: foundations and learning algorithms\",\n \"venue\": \"MIT press,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Mateo Rojas-Carulla\",\n \"Bernhard Schölkopf\",\n \"Richard Turner\",\n \"Jonas Peters\"\n ],\n \"title\": \"Invariant models for causal transfer learning\",\n \"venue\": \"The Journal of Machine Learning Research,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Dominik Rothenhäusler\",\n \"Christina Heinze\",\n \"Jonas Peters\",\n \"Nicolai Meinshausen\"\n ],\n \"title\": \"Backshift: Learning causal cyclic graphs from unknown shift interventions\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Karen Sachs\",\n \"Omar Perez\",\n \"Dana Pe’er\",\n \"Douglas A Lauffenburger\",\n \"Garry P Nolan\"\n ],\n \"title\": \"Causal proteinsignaling networks derived from multiparameter single-cell data\",\n \"venue\": null,\n \"year\": 2005\n },\n {\n \"authors\": [\n \"Bernhard Schölkopf\",\n \"Dominik Janzing\",\n \"Jonas Peters\",\n \"Eleni Sgouritsa\",\n \"Kun Zhang\",\n \"Joris Mooij\"\n ],\n \"title\": \"On causal and anticausal learning\",\n \"venue\": \"Proceedings of the 29th International Conference on Machine Learning (ICML),\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"Shohei Shimizu\",\n \"Patrik O Hoyer\",\n \"Aapo Hyvärinen\",\n \"Antti Kerminen\"\n ],\n \"title\": \"A linear non-gaussian acyclic model for causal discovery\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2003\n },\n {\n \"authors\": [\n \"Chandler Squires\",\n \"Yuhao Wang\",\n \"Caroline Uhler\"\n ],\n \"title\": \"Permutation-based causal structure learning with unknown intervention targets\",\n \"venue\": \"In Conference on Uncertainty in Artificial Intelligence,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Xiaohai Sun\",\n \"Dominik Janzing\",\n \"Bernhard Schölkopf\",\n \"Kenji Fukumizu\"\n ],\n \"title\": \"A kernel-based causal learning algorithm\",\n \"venue\": \"In Proceedings of the 24th international conference on Machine learning,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Ioannis Tsamardinos\",\n \"Laura E Brown\",\n \"Constantin F Aliferis\"\n ],\n \"title\": \"The max-min hill-climbing bayesian network structure learning algorithm\",\n \"venue\": \"Machine learning,\",\n \"year\": 2006\n },\n {\n \"authors\": [\n \"Jinsung Yoon\",\n \"James Jordon\",\n \"Mihaela Van Der Schaar\"\n ],\n \"title\": \"Gain: Missing data imputation using generative adversarial nets\",\n \"venue\": \"arXiv preprint arXiv:1806.02920,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Yue Yu\",\n \"Jie Chen\",\n \"Tian Gao\",\n \"Mo Yu\"\n ],\n \"title\": \"Dag-gnn: Dag structure learning with graph neural networks\",\n \"venue\": \"arXiv preprint arXiv:1904.10098,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Kun Zhang\",\n \"Jonas Peters\",\n \"Dominik Janzing\",\n \"Bernhard Schölkopf\"\n ],\n \"title\": \"Kernel-based conditional independence test and application in causal discovery\",\n \"venue\": \"arXiv preprint arXiv:1202.3775,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"Xun Zheng\",\n \"Bryon Aragam\",\n \"Pradeep K Ravikumar\",\n \"Eric P Xing\"\n ],\n \"title\": \"DAGs with NO TEARS: Continuous optimization for structure learning\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Shengyu Zhu\",\n \"Zhitang Chen\"\n ],\n \"title\": \"Causal discovery with reinforcement learning\",\n \"venue\": \"arXiv preprint arXiv:1906.04477,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Peters\"\n ],\n \"title\": \"Adam(γt, g) graph is always identifiable in principle (Eberhardt et al., 2012; Heinze-Deml et al., 2018a). We also consider here situations where a single variable is randomly selected and intervened upon with a soft or imprecise intervention, its identity is unknown and must be inferred\",\n \"venue\": null,\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Structure learning concerns itself with the recovery of the graph structure of Bayesian networks (BNs) from data samples. A natural application of Bayesian networks is to describe cause-effect relationships between variables. In that context, one may speak of causal structure learning. Causal structure learning is challenging because purely observational data may be satisfactorily explained by multiple Bayesian networks (a Markov equivalence class), but only one is the most robust to distributional shifts: The one with the correct graph. A more powerful tool than BNs is thus needed to model causal relationships.\nStructural Causal Models (SCMs) are that tool. An SCM over a set of random variables is a collection of assignments to these variables and a directed acyclic graph of dependencies between them (Peters et al., 2017, §6.2). Each assignment is a function of only the direct causes of a variable, plus an independent noise source. An SCM entails precisely one (observational) data distribution. Interventions on an SCM’s assignments, such as setting a random variable to a fixed value (a hard intervention), entail new interventional data distributions (Peters et al., 2017, §6.3).\nSCMs can be used to answer higher-order questions of cause-and-effect, up the ladder of causation (Pearl & Mackenzie, 2018). Causal structure learning using SCMs has been attempted in several disciplines including biology (Sachs et al., 2005; Hill et al., 2016), weather forecasting (Abramson et al., 1996) and medicine (Lauritzen & Spiegelhalter, 1988; Korb & Nicholson, 2010).\nCausal structure is most frequently learned from data drawn from observational distributions. Structure learning methods generally cannot do more than identify the causal graph up to a Markov equivalence class (Spirtes et al., 2000). In order to fully identify the true causal graph, a method must either make restrictive assumptions about the underlying data-generating process, such as linear but non-Gaussian data (Shimizu et al., 2006), or must access enough data from outside the observational distribution (i.e., from interventions).\nUnder certain assumptions about the number, diversity, and nature of the interventions, the true underlying causal graph is always identifiable, given that the method knows the intervention performed (Heckerman et al., 1995). In much of the prior work on causal model induction it is assumed that\nthere is an experimenter and this experimenter performs interventions. However, in the real world, interventions can also be performed by other agents, which could lead to unknown interventions (interventions with unknown target variables). A few works have attempted to learn structures from unknown-intervention data (Eaton & Murphy, 2007a; Squires et al., 2020; Huang et al., 2020). A notable such work, (Mooij et al., 2016), has been extended in (Kocaoglu et al., 2019; Jaber et al., 2020). Although there is no theoretical guarantee that the true causal graph can be identified in that setting, evidence so far points to that still being the case.\nAnother common setting is when the graph structure is partially provided, but must be completed. An example is protein structure learning in biology, where we may have definitive knowledge of some causal edges in the protein-protein interactome, but the remaining causal edges must be discovered. We will call this setting “partial graph completion”. This is an easier task compared to learning the entire graph, since it limits the number of edges that have to be learned.\nRecently, a flurry of work on structure learning using continuous optimization methods has appeared (Zheng et al., 2018; Yu et al., 2019). These methods operate on observational data and are competitive with other methods. Because of the theoretical limitations on identification from purely observational data cited above, it would be interesting to extend these methods to interventional data. However, it is not straightforward to apply continuous optimization methods to structure learning from interventional data. Our key contributions are to answer the following questions experimentally:\n1. Can the proposed model recover true causal structure? Yes, see Figure §4. 2. How does the proposed model compare against state of the art causal methods on real-world\ndatasets? Favourably; see §5.4 and Table §1. 3. Does a proposed model generalize well to unseen interventions? Yes, see §5.5. 4. How does the proposed model perform on partial graph recovery? It scales to∼ 50 variables\nwhile the other baselines can’t. see §5.7."},{"heading":"2 PRELIMINARIES","text":"Causal modeling. A Structural Causal Model (SCM) (Peters et al., 2017) over a finite number M of random variables Xi is a set of structural assignments\nXi := fi(Xpa(i,C), Ni) , ∀i ∈ {0, . . . ,M − 1} (1)\nIdentifiability. In a purely-observational setting, it is known that causal graphs can be distinguished only up to a Markov equivalence class. In order to identify the true causal graph structure, interventional data is needed (Eberhardt et al., 2012).\nInterventions. There are several types of common interventions which may be available (Eaton & Murphy, 2007b). These are: No intervention: only observational data is obtained from the ground truth model. Hard/perfect: the value of a single or several variables is fixed and then ancestral sampling is performed on the other variables. Soft/imperfect: the conditional distribution of the variable on which the intervention is performed is changed. Uncertain: the learner is not sure of which variable exactly the intervention affected directly. Here we make use of soft intervention because they include hard intervention as a limiting case and hence are more general.\nStructure discovery using continuous optimization. Structure discovery is a super-exponential search problem that searches though all possible directed acyclic graphs (DAGs). Previous continuousoptimization structure learning works (Zheng et al., 2018; Yu et al., 2019; Lachapelle et al., 2019) mitigate the problem of searching in the super-exponential set of graph structures by considering the degree to which a hypothesis graph violates “DAG-ness” as an additional penalty to be optimized. If there are M such variables, the strategy of considering all the possible structural graphs as separate hypotheses is not feasible because it would require maintaining O(2M 2\n) models of the data.2"},{"heading":"3 RELATED WORK","text":"The recovery of the underlying structural causal graph from observational and interventional data is a fundamental problem (Pearl, 1995; 2009; Spirtes et al., 2000). Different approaches have been studied: score-based, constraint-based, asymmetry-based and continuous optimization methods. Score-based methods search through the space of all possible directed acyclic graphs (DAGs) representing the causal structure based on some form of scoring function for network structures (Heckerman et al., 1995; Chickering, 2002; Tsamardinos et al., 2006; Hauser & Bühlmann, 2012; Goudet et al., 2017; Cooper & Yoo, 1999; Zhu & Chen, 2019). Constraint-based methods (Spirtes et al., 2000; Sun et al., 2007; Zhang et al., 2012; Monti et al., 2019; Zhu & Chen, 2019) infer the DAG by analyzing conditional independences in the data. Eaton & Murphy (2007c) use dynamic programming techniques to accelerate Markov Chain Monte Carlo (MCMC) sampling in a Bayesian approach to structure learning for discrete variable DAGs. Peters et al. (2016); Ghassami et al. (2017); Rojas-Carulla et al. (2018) exploit invariance across environments to infer causal structure, which faces difficulty scaling due to the iteration over the super-exponential set of possible graphs. Recently, (Zheng et al., 2018; Yu et al., 2019; Lachapelle et al., 2019) framed the structure search as a continuous optimization problem, however, the methods only uses observational data and are non-trivial to extend to interventional data. In our paper, we present a method that uses continuous optimization methods that works on both observational and interventional data.\nFor interventional data, it is often assumed that the models have access to full intervention information, which is rare in the real world. Rothenhäusler et al. (2015) have investigated the case of additive shift interventions, while Eaton & Murphy (2007b) have examined the situation where the targets of experimental interventions are imperfect or uncertain. This is different from our setting where the intervention is unknown to start with and is assumed to arise from other agents and the environment.\nLearning based methods have been proposed (Guyon, 2013; 2014; Lopez-Paz et al., 2015) and there also exist recent approaches using the generalization ability of neural networks to learn causal signals from purely observational data (Kalainathan et al., 2018; Goudet et al., 2018). Neural network methods equipped with learned masks, such as (Ivanov et al., 2018; Li et al., 2019; Yoon et al., 2018; Douglas et al., 2017), exist in the literature, but only a few (Kalainathan et al., 2018) have been adapted to causal inference. This last work is, however, tailored for causal inference on continuous variables and from observations only. Adapting it to a discrete-variable setting is made difficult by its use of a Generative Adversarial Network (GAN) Goodfellow et al. (2014) framework."},{"heading":"4 STRUCTURE DISCOVERY FROM INTERVENTIONS METHOD","text":"Scope of Applicability and Objective. The proposed method, like any structure learning algorithm, assumes the availability of a data-generating process based on ancestral sampling of a ground-truth SCM of M variables, which can be queried for samples. The SCM supports applying and retracting known or unknown interventions. The method can support infinite- or finite-data as well as infiniteor finite-intervention regimes.\nThe objective is, then, to learn the SCM’s structure from the insights that each intervention gives about cause-effect relationships between variables in the SCM."},{"heading":"4.1 PROBLEM SETTING AND ASSUMPTIONS","text":"In this paper, we restrict the problem setting to specific, but still broad classes of SCMs and interventions. In particular, we assume that:\nData is discrete-valued. The SCM’s random variables are all categorical.\nCausal sufficiency. For every data sample, the value of all variables are available; There are no latent confounders.\nInterventions are localized. They affect only a single variable (but which one may not be known).\nInterventions are soft. An intervention does not necessarily pin its target random variable to a fixed value (though it may, as a special case). It changes the relationship of a variable with its parents.\nInterventions do not stack. Before a new intervention is made, the previous one is fully retracted. This stops the SCM from wandering away from its initial, observational configuration after a long series of interventions.\nNo control over interventions. The structure learning algorithm has control neither of the target, nor the nature of the next intervention on the SCM.\nFor a detailed description of the interventions, refer to §A.2."},{"heading":"4.2 VARIATIONS AND PRIOR KNOWLEDGE","text":"In the problem setting above, the ground-truth SCM is completely opaque to us. However, we consider two interesting relaxations of this formulation:\nComplete or partial graph recovery. We may already know the existence of certain cause-effect edges and non-edges within the ground-truth SCM. If such prior information is available, it turns a complete graph recovery problem into one of partial graph recovery. Larger SCMs can be tackled if only parts of the graph need to be recovered.\nKnown or unknown interventions: The interventions can either be known or unknown to the learned model.\nWe demonstrate that the proposed method can naturally incorporate this prior information to improve its performance.\n4.3 METHOD OVERVIEW\nThe proposed method is a score-based, iterative, continuousoptimization method consisting of three phases that flow into one other (See Figure 2). During the three-phase procedure, a structural representation of a DAG and a functional representation of a set of independent causal mechanisms are trained jointly until convergence. Because the structural and functional parameters are not independent and do influence each other, we train them in alternating phases, a form of block coordinate descent optimization."},{"heading":"4.3.1 PARAMETRIZATION","text":"We distinguish two sets of parameters: The structural parameters γ and the functional parameters θ. Given a graph of M variables, we parametrize the structure γ as a matrix RM×M such that σ(γij) is our belief in random variable Xj being a direct cause of Xi, where\nσ(x) = 1/(1 + exp(−x)) is the sigmoid function. The matrix σ(γ) is thus a soft adjacency matrix. The set of functional parameters θi parametrizes the conditional probability distribution of Xi given its parent set Xpa(i,C), with C ∼ Ber(σ(γ)) a hypothesized configuration of the SCM’s DAG."},{"heading":"4.3.2 PHASE 1: GRAPH FITTING ON OBSERVATIONAL DATA","text":"During Phase 1, the functional parameters θ are trained to maximize the likelihood of randomly drawn observational data under graphs randomly drawn from our current beliefs about the edge structure. We draw graph configurations Cij ∼ Ber(σ(γij)) and batches of observational data from the unintervened ground-truth SCM, then maximize the log-likelihood of the batch under that configuration using SGD. The use of graph configurations sampling from Bernoulli distributions is analogous to dropout on the inputs of the functional models (in our implementation, MLPs), giving us an ensemble of neural networks that can model the observational data."},{"heading":"4.3.3 PHASE 2: GRAPH SCORING ON INTERVENTIONAL DATA","text":"During Phase 2, a number of graph configurations are sampled from the current edge beliefs parametrized by γ, and scored on data samples drawn from the intervention SCM.\nIntervention applied: At the beginning of Phase 2, an intervention is applied to the ground-truth SCM. This intervention is not under the control of the method. In our implementation, and unbeknownst to the model, the target variable is chosen uniformly randomly from all M variables throughout the optimization process.\nIntervention predicted: If the target of the intervention is not known, it is predicted using a simple heuristic. A small number of interventional data samples are drawn from the SCM and more graphs are sampled from our current edge beliefs. The average log-likelihood of each individual variable Xi across the samples is then computed using the functional model parameters θ fine-tuned on observational data in Phase 1. The variable Xi showing the greatest deterioration in log-likelihood is assumed to be the target because the observational distribution most poorly predicts that variable.\nIf the target of the intervention is known, then this is taken as ground-truth knowledge for the purpose of subsequent steps, and no prediction needs to be done.\nGraphs Sampled and Scored: A new set of interventional data samples and graph configurations are now drawn from the intervention SCM and edge beliefs respectively. The log-likelihood of the data batches under the hypothesized configurations is computed, with one modification: The contribution to the total log-likelihood of a sample X coming from the target (or predicted-target) intervention variable Xi is masked. Because Xi was intervened upon (in the manner of a Pearl do-operation, soft or hard), the values one gets for that variable should be taken as givens, not as contributors to the total log-likelihood of the sample. As well, no gradient should be allowed to propagate into the variable’s learned functional parametrization θi, because it was not actually responsible for the outcome.\nIntervention retracted: After Phase 2, the intervention is retracted, per our modelling assumptions."},{"heading":"4.3.4 PHASE 3: CREDIT ASSIGNMENT TO STRUCTURAL PARAMETERS","text":"During Phase 3, the scores of the interventional data batches over various graph configurations are aggregated into a gradient for the structural parameters γ. Because a discrete Bernoulli random sampling process was used to sample graph configurations under which the log-likelihoods were computed, we require a gradient estimator to propagate gradient through to the γ structural parameters. Several alternatives exist, but we adopt for this purpose the REINFORCE-like gradient estimator gij proposed by Bengio et al. (2019):\ngij =\n∑ k(σ(γij)− c\n(k) ij )LC(k),i (X)∑\nk LC(k),i (X) , ∀i, j ∈ {0, . . . ,M−1} (2)\nwhere the (k) superscript indicates the values obtained for the k-th draw of C under the current edge beliefs parametrized by γ. Therefore, L(k)C,i(X) can be read as the log-likelihood of variable Xi in the data sample X under the k’th configuration, C(k), drawn from our edge beliefs. Using the estimated gradient, we then update γ with SGD, and return to Phase 1 of the continuous optimization process.\nThe gradient estimator gij minimizes an implicit empirical risk objective with respect to γij . When the functional and structural parameters θ and γ are “sufficiently close” to their minima, the estimator gij empirically converges quickly towards that minimum γ∗ as shown in Figure 16 of Appendix A.13.\nAcyclic Constraint: We include a regularization term JDAG(γ) that penalizes length-2 cycles in the learned adjacency matrix σ(γ), with a tunable strength λDAG. The regularization term is JDAG(γ) =∑ i 6=j cosh(σ(γij)σ(γji)), ∀i, j ∈ {0, . . . ,M−1} and is derived from Zheng et al. (2018). The details of the derivation are in the Appendix. We explore several different values of λDAG and their effects in our experimental setup. Suppression of longer-length cycles was not found to be worthwhile for the increased computational expense."},{"heading":"5 EXPERIMENTAL SETUP AND RESULTS","text":"We first evaluate the proposed method on a synthetic dataset where we have control over the number of variables and causal edges in the ground-truth SCM. This allows us to analyze the performance of the proposed method under various conditions. We then evaluate the proposed method on real-world datasets from the BnLearn dataset repository. We also consider the two variations of §4.2: Recovering only part of the graph (when the rest is known), and exploiting knowledge of the intervention target.\nThe summary of our findings is: 1) We show strong results for graph recovery for all synthetic graphs in comparisons with other baselines, measured by Hamming distance. 2) The proposed method achieves high accuracy on partial graph recovery for large, real-world graphs. 3) The proposed method’s intervention target prediction heuristic closes the gap between the known- and unknowntarget intervention scenarios. 4) The proposed method generalizes well to unseen interventions. 5) The proposed method’s time-to-solution scaling appears to be driven by the number of edges in the groundtruth graph moreso than the number of variables.\n5.1 MODEL DESCRIPTION\nLearner model. Without loss of generality, we let θi = {W0i,B0i,W1i,B1i} define a stack of M one-hidden-layer MLPs, one for each random variable Xi. A more appropriate model, such as a CNN, can be chosen using domainspecific knowledge; the primary advantage of using MLPs is that the hypothesized DAG configurations cij can be readily used to mask the inputs of MLP i, as shown in Figure 3.\nTo force the structural equation fi corresponding to Xi to rely exclusively on its direct ancestor set pa(i, C) under hypothesis adjacency matrix C (See Eqn. 1), the one-hot input vector Xj for variable Xi’s MLP is masked by the Boolean element cij . An example of the multi-MLP architecture with M=4 categorical variables of N=3 categories is shown in Figure 3. For more details, refer to Appendix A.4.\nGround-truth model. Ground-truth SCM models are parametrized either as CPTs with parameters from BnLearn (in the case of real-world graphs), or as a second stack of MLPs similar to the learner model, with randomly-initialized functional parameters θGT and the desired adjacency matrix γGT.\nInterventions. In all experiments, at most one (soft) intervention is concurrently performed. To simulate a soft intervention on variable Xi, we reinitialize its ground-truth conditional distribution’s MLP parameters or CPT table randomly, while leaving the other variables untouched. For more details about the interventions, please refer to Appendix A.2.\n5.2 SYNTHETIC DATASETS EXPERIMENTS\nWe first evaluate the model’s performance on several randomlyinitialized SCMs with specific, representative graph structures. Since the number of possible DAGs grows super-exponentially with the number of variables, for M=4 up to 13 a selection of\nrepresentative and edge-case DAGs are chosen. chainM and fullM (M=3-13) are the minimallyand maximally-connected M -variable DAGs, while treeM and jungleM are tree-like intermediate graphs. colliderM is the (M−1)→ 1 collider graph. The details of the setup is in Appendix A.6.\nResults. The model can recover most synthetic DAGs with high accuracy, as measured by Structural Hamming Distance (SHD) between learned and ground-truth DAGs. Table 1 shows our proposed method outperforming all other baseline methods, and learns all graphs perfectly for 3 to 13 variables (excepting full). For DAGs ranging from 3 to 8 variables, the AUROCs all eventually reach 1.0 (indicating perfect classification into edge/non-edge; Refer to Figure 4). For both large (M > 10)\nand dense DAGs (e.g. full13) the model begins encountering difficulties, as shown in Table 1 and Appendix §A.6.1.\nSmall graphs (M < 10) are less sensitive than larger ones to our hyperparameters, notably the sparsity and acyclic regularization (§4.3.4) terms. In §A.5, we perform an analysis of these hyperparameters."},{"heading":"5.3 REAL-WORLD DATASETS: BNLEARN","text":"The Bayesian Network Repository is a collection of commonly-used causal Bayesian networks from the literature, suitable for Bayesian and causal learning benchmarks. We evaluate the proposed method on the Earthquake (Korb & Nicholson, 2010), Cancer (Korb & Nicholson, 2010), Asia (Lauritzen & Spiegelhalter, 1988) and Sachs (Sachs et al., 2005) datasets (M =5, 5, 8 and 11-variables respectively, maximum in-degree 3) in the BnLearn dataset repository.\nResults. As shown in Table 1, the proposed method perfectly recovers the DAG of Asia, while making a small number of errors (SHD=6) for Sachs (11-variables). It thus significantly outperforms all other baselines models. Figures 8 & 9 visualize what the model has learned at several stages of learning. Results for Cancer and Asia can be found in the appendices, Figure 17 and 18."},{"heading":"5.4 COMPARISONS WITH OTHER METHODS","text":"As shown in Table 1, we compared the proposed SDI method to ICP ((Peters et al., 2016)), non-linear ICP ((Heinze-Deml et al., 2018b)), and (Eaton & Murphy, 2007b; Zheng et al., 2018; Yu et al., 2019) on Asia (Lauritzen & Spiegelhalter, 1988), Sachs (Sachs et al., 2005) and representative synthetic graphs. Eaton & Murphy (2007b) handles uncertain interventions and Peters et al. (2016), Heinze-Deml et al. (2018b) handles unknown interventions. However, neither attempts to predict the intervention. As shown in Table 1, we significantly outperform ICP, non-linear ICP, and the methods in (Yu et al., 2019) and (Zheng et al., 2018). Furthermore, Eaton & Murphy (2007b) runs out of memory for graphs larger than M = 10 because modelling of uncertain interventions is done using “shadow” random variables (as suggested by the authors), and thus recovering the DAG internally requires solving a d = 2M -variable problem. Their method’s extremely poor time- and space-scaling of O(d2d) makes it unusable beyond d > 20.\nFor SDIs, we threshold our edge beliefs at σ(γ) = 0.5 to derive a graph, but the continued decrease of the cross-entropy loss (Figure 4) hints at SDI’s convergence onto the correct causal model. Please refer to Appendix §A.8 for full details and results."},{"heading":"5.5 GENERALIZATION TO PREVIOUSLY UNSEEN INTERVENTIONS","text":"It is often argued that machine learning approaches based purely on capturing joint distributions do not necessarily yield models that generalize to unseen experiments, since they do not explicitly model changes through interventions. By way of contrast,\ncausal models use the concept of interventions to explicitly model changing environments and thus hold the promise of robustness under distributional shifts (Pearl, 2009; Schölkopf et al., 2012; Peters et al., 2017). To test the robustness of causal modelling to previously unseen interventions (new values for an intervened variable), we evaluate a well-trained causal model against a variant, non-causal model trained with cij = 1, i 6= j. An intervention is performed on the ground-truth SCM, fresh interventional data is drawn from it, and the models, with knowledge of the intervention target, are asked to predict the other variables given their parents. The average log-likelihoods of the data under both models are computed and contrasted. The intervention variable’s contribution to the loglikelihood is masked. For all 3-variable graphs (chain3, fork3, collider3, confounder3), the causal model attributes higher log-likelihood to the intervention distribution’s samples than the non-causal variant, thereby demonstrating causal models’ superior generalization ability in transfer tasks. Table 2 collects these results."},{"heading":"5.6 VARIANT: PREDICTING INTERVENTIONS","text":"In Phase 2 (§4.3.3), we use a simple heuristic to predict the intervention target variable. Experiments show that this heuristic functions well in practice, yielding correct predictions far more often than by chance alone (Table 3). Guessing the intervention variable\nrandomly, or not guessing it at all, leads to a significant drop in the model performance, even for 3-variable graphs (Fig. 11 Left). Training SDI with intervention prediction closely tracks training with leaked knowledge of the ground-truth intervention on larger, 7-variable graphs (Fig. 11 Right)."},{"heading":"5.7 VARIANT: PARTIAL GRAPH RECOVERY","text":"Instead of learning causal structures de novo, we may have partial information about the ground-truth SCM and may only need to fill in missing information (§4.2). An example is protein structure discovery in biology, where some causal relationships have been definitely established and others remain open hypotheses. This is an easier task compared to full graph recovery, since the model only has to search for missing edges.\nTable 4: Partial Graph Recovery on Alarm (Beinlich et al., 1989) and Barley (Kristensen & Rasmussen, 2002). The model is asked to predict 50 edges in Barley and 40 edges in Alarm. The accuracy is measured in Structural Hamming Distance (SHD). SDI achieved over 90% accuracy on both graphs.\nGraph Alarm Barley Number of variables 37 48 Total Edges 46 84\nEdges to recover 40 50 Recovered Edges 37 45 Errors (in SHD) 3 5\nWe evaluate the proposed method on Barley (Kristensen & Rasmussen, 2002) (M = 48) and Alarm (Beinlich et al., 1989) (M = 37) from the BnLearn repository. The model is asked to predict 50 edges from Barley and 40 edges from Alarm. The model reached ≥ 90% accuracy on both datasets, as shown in Table 4."},{"heading":"5.8 ABLATION AND ANALYSIS","text":"As shown in Figure 12, larger graphs (such as M > 6) and denser graphs (such as full8) are progressively more difficult to learn. For denser graphs, the learned models have higher sample complexity, higher variance and slightly worse results. Refer to Appendix §A.9 for complete results on all graphs. Hyperparameters. Hyperparameters for all experiments were kept identical unless otherwise stated. We study the effect of DAG and sparsity penalties in the following paragraph. For more details, please refer to Appendix §A.5 .\nImportance of regularization. Valid configurations C for a causal model are expected to be a) sparse and b) acyclic. To promote such solutions, we use DAG and sparsity regularization with tunable hyperparameters. We set the DAG penalty to 0.5 and sparsity penalty to 0.1. We run ablation studies on different values of the regularizers and study their effect. We find that smaller graphs are less sensitive to different values of regularizer than larger graphs. For details, refer to Appendix §A.12.\nImportance of dropout. To train functional parameter for an observational distribution, sampling adjacency matrices is required. We \"drop out\" each edge (with a probability of σ(γ)) in our experiments during functional parameter training of the conditional distributions of the SCM. Please refer to Appendix §A.14 for a more detailed analysis."},{"heading":"6 CONCLUSION","text":"In this work, we introduced an experimentally successful method (SDI) for causal structure discovery using continuous optimization, combining information from both observational and interventional data. We show in experiments that it can recover true causal structure, that it generalizes well to unseen interventions, that it compares very well against start-of-the-art causal discovery methods on real world datasets, and that it scales even better on problems where only part of the graph is known."},{"heading":"Appendix","text":""},{"heading":"Table of Contents","text":""},{"heading":"A Annexes 13","text":"A.1 Training Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.4 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.5 Hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.6 Synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.7 BnLearn data repository . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A.8 Comparisons to other methods . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.9 Sparsity of Ground-Truth Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.10 Predicting interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.11 Sample complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.12 Effect of regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A.13 Near-Optimum Performance of Gradient Estimator . . . . . . . . . . . . . . . . 20 A.14 Importance of dropout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21"},{"heading":"A ANNEXES","text":""},{"heading":"A.1 TRAINING ALGORITHM","text":"Algorithm 1 shows the pseudocode of the method described in §4. Typical values for the loop trip counts are found in §A.11."},{"heading":"A.2 PRELIMINARIES","text":"Interventions. In a purely-observational setting, it is known that causal graphs can be distinguished only up to a Markov equivalence class. In order to identify the true causal graph intervention data is needed (Eberhardt et al., 2012). Several types of common interventions may be available (Eaton & Murphy, 2007b). These are: No intervention: only observational data is obtained from the ground truth causal model. Hard/perfect: the value of a single or several variables is fixed and then ancestral sampling is performed on the other variables. Soft/imperfect: the conditional distribution of the variable on which the intervention is performed is changed. Uncertain: the learner is not sure of which variable exactly the intervention affected directly. Here we make use of soft interventions for several reasons: First, they include hard interventions as a limiting case and hence are more general. Second, in many real-world scenarios, it is more difficult to perform a hard intervention compared to a soft one. We also deal with a special case of uncertain interventions, where the variable selected for intervention is random and unknown. We call these unidentified or unknown interventions.\nIntervention setup. For our experiments, the groundtruth models of the synthetic datasets are modeled by neural networks as described in section A.6. Each neural network models the relationship of the causal parents and a variable. We perform our intervention by first randomly selecting which variable to intervene on, then soft-intervening on it. The selected variable is sampled from a uniform distribution. The soft intervention is a reinitialization of its neural network’s parameters.\nCausal sufficiency. The inability to distinguish which causal graph, within a Markov equivalence class, is the correct one in the purely-observational setting is called the identifiability problem. In our setting, all variables are observed (there are no latent confounders) and all interventions are random and independent. Hence, within our setting, if the interventions are known, then the true causal\nAlgorithm 1 Training Algorithm 1: procedure TRAINING(SCM Ground-Truth Entailed Distribution D, with M nodes and N categories) 2: Let i an index from 0 to M − 1\n3: for I iterations, or until convergence, do 4: if I % reinitialization_period == 0 then 5: D ← reinitialize(D)\n6: for F functional parameter training steps do . Phase 1 7: X ∼ D 8: C ∼ Ber(σ(γ)) 9: L = − logP (X|C ; θ)\n10: θt+1 ← Adam(θt,∇θL)\n11: for Q interventions do . Phase 2 12: I_N← randint(0, M − 1) . Uniform selection of target 13: Dint :=D with intervention on node I_N . Apply intervention\n14: if predicting intervention then . Phase 2 Prediction 15: Li ← 0 ∀i 16: for NP prediction steps do 17: X ∼ Dint 18: for CP configurations do 19: C ∼ Ber(σ(γ)) 20: Li ← Li − logPi(X|Ci; θslow) ∀i 21: I_N← argmax(Li)\n22: gammagrads, logregrets = [], [] . Phase 2 Scoring 23: for NS scoring steps do 24: X ∼ Dint 25: gammagrad, logregret = 0, 0 26: for CS configurations do 27: C ∼ Ber(σ(γ)) 28: Li = − logPi(X|Ci; θslow) ∀i 29: gammagrad += σ(γ)− C . Collect σ(γ)− C for Equation 2 30: logregret +=\n∑ i6=I_N Li . Collect LC(k),i (X) for Equation 2\n31: gammagrads.append(gammagrad) 32: logregrets.append(logregret)\n. Phase 3\n33: gij =\n∑ k(σ(γij)− c (k) ij )LC\n(k) ,i (X)∑\nk LC (k) ,i (X)\n. Gradient Estimator, Equation 2\n34: g ← g +∇γ (λsparse Lsparse(γ) + λDAG LDAG(γ)) . Regularizers 35: γt+1 ← Adam(γt, g)\ngraph is always identifiable in principle (Eberhardt et al., 2012; Heinze-Deml et al., 2018a). We also consider here situations where a single variable is randomly selected and intervened upon with a soft or imprecise intervention, its identity is unknown and must be inferred. In this case, there is no theoretical guarantee that the causal graph is identifiable. However, there is existing work Peters et al. (2016) that handles this scenario and the proposed method is also proven to work empirically.\nFaithfulness. It is possible for causally-related variables to be probabilistically independent purely by happenstance, such as when causal effects along multiple paths cancel out. This is called unfaithfulness. We assume that faithfulness holds, since the γ gradient estimate is extracted from shifts in probability distributions. However, because of the “soft” nature of our interventions and their infinite variety, it would be exceedingly unlikely for cancellation-related unfaithfulness to persist throughout the causal-learning procedure."},{"heading":"A.3 EXPERIMENTAL SETUP","text":"For all datasets, the weight parameters for the learned model is initialized randomly. In order to not bias the structural parameters, all σ(γ) are initialized to 0.5 in the beginning of training. Details of hyperparameters of the learner model are described in Section A.5. The experimental setup for the groundtruth model for the synthetic data can be found in Section A.6 and the details for the real world data are described in Section A.7."},{"heading":"A.4 MODEL SETUP","text":"As discussed in section 4, we model the M variables in the graph using M independent MLPs, each possesses an input layer of M × N neurons (for M one-hot vectors of length N each), a single hidden layer chosen arbitrarily to have max(4M, 4N) neurons with a LeakyReLU activation of slope 0.1, and a linear output layer of N neurons representing the unnormalized log-probabilities of each category (a softmax then recovers the conditional probabilities from these logits). To force fi to rely exclusively on the direct ancestor set pa(i, C) under adjacency matrix C (See Eqn. 2), the one-hot input vector Xj for variable Xi’s MLP is masked by the Boolean element cij . The functional parameters of the MLP are the set θ = {W0ihjn,B0ih,W1inh,B1in}.An example of the multi-MLP architecture with M=3 categorical variables of N=2 categories is shown in Figure 3."},{"heading":"A.5 HYPERPARAMETERS","text":"Learner model. All experiments on the synthetic graphs of size 3-8 use the same hyperparameters. Both the functional and structural parameters are optimized using the Adam optimizer Kingma & Ba (2014). We use a learning rate of 5e− 2 with alpha of 0.9 for the functional parameters, and we use a learning rate of 5e− 3 with alpha of 0.1 for the structural parameters. We perform 5 runs of each experiment with random seeds 1− 5 and error bars are plotted for various graphs from size 3 to 8 in Figure 4. We use a batch size of 256. The L1 norm regularizer is set to 0.1 and the DAG regularizer is set to 0.5 for all experiments. For each γ update step, we sample 25 structural configurations from the current γ. In all experiments, we use 100 batches from the interventional distribution to predict the intervened node."},{"heading":"A.6 SYNTHETIC DATA","text":"Synthetic datasets. The synthetic datasets in the paper are modeled by neural networks. All neural networks are 2 layered feed forward neural networks (MLPs) with Leaky ReLU activations between layers. The parameters of the neural network are initialized orthogonally within the range of (−2.5, 2.5). This range was selected such that they output a non-trivial distribution. The biases are initialized uniformly between (−1.1, 1.1). SCM with n variables are modeled by n feedforward neural networks (MLPs) as described in §5.1. We assume an acyclic causal graph so that we may easily sample from them. Hence, given any pair of random variables A and B, either A −→ B, B −→ A or A and B are independent. The MLP representing the ground-truth SCM has its weights θ initialized use orthogonal initialization with gain 2.5 and the biases are initialized using a uniform initialization between−1.1 and 1.1, which was empirically found to yield \"interesting\" yet learnable random SCMs.\nWe study a variety of SCMs with different ground-truth edge structures γ. Our selection of synthetic graphs explores various extremes in the space of DAGs, stress-testing SDI. The chain graphs are the sparsest connected graphs possible, and are relatively easy to learn. The bidiag graphs are extensions of chain where there are 2-hops as well as single hops between nodes, doubling the number of edges and creating a meshed chain of forks and colliders. The jungle graphs are binary-tree-like graphs, but with each node connected directly to its grandparent in the tree as well. Half the nodes in a jungle graph are leaves, and the out-degree is up to 6. The collider graphs deliberately collide independent M − 1 ancestors into the last node; They stress maximum in-degree. Lastly, the full graphs are the maximally dense DAGs. All nodes are direct parents of all nodes below them in the topological order. The maximum in- and out-degree are both M − 1. These graphs are depicted in Figure 6."},{"heading":"A.6.1 SYNTHETIC DATA RESULTS","text":"The model can recover correctly all synthetic graphs with 10 variables or less, as shown in Figure 10 and Table 1. For graphs larger than 10 variables, the model found it more challenging to recover the denser graphs (e.g. fullM), as shown in Table 1. Plots of the training curves showing average cross entropy (CE) and Area-Under-Curve(AUC/AUCROC) for edge probabilities of the learned graph against the ground-truth graph for synthetic SCMs with 3-13 variables are available in Figure 10."},{"heading":"A.7 BNLEARN DATA REPOSITORY","text":"The repo contains many datasets with various sizes and structures modeling different variables. We evaluate the proposed method on 3 of the datasets in the repo, namely the Earthquake (Korb & Nicholson, 2010), Cancer (Korb & Nicholson, 2010) and Asia (Lauritzen & Spiegelhalter, 1988) datasets. The ground-truth model structure for the Cancer (Korb & Nicholson, 2010) and Earthquake (Korb & Nicholson, 2010) datasets are shown in Figure 7. Note that even though the structure for the two datasets seems to be the same, the conditional probability tables (CPTs) for these datasets are very different and hence results in different structured causal models (SCMs) for each."},{"heading":"A.8 COMPARISONS TO OTHER METHODS","text":"As described in section 5.4, we compare to 5 other methods. The full comparison between SDIs and other methods on various graphs can be found in Table 1.\nOne of these methods, DAG-GNN Yu et al. (2019), outputs 3 graphs based on different criteria: best mean square error (MSE), best negative loglikelihood (NLL) and best evidence lower bound (ELBO). We report performance of all outputs of DAG-GNN Yu et al. (2019) in Table 6, and the best one is selected for Table 1."},{"heading":"A.9 SPARSITY OF GROUND-TRUTH GRAPH","text":"We evaluated the performance of SDI on graphs of various size and sparsity to better understand the performance of the model. We evaluated the proposed model on 4 representative types of graphs in increasing order of density. They are the chain, jungle, bidiag and full graphs. As shown in the results in figure 12, for graphs of size 5 or smaller, there is almost no difference in the final results in terms of variance and sample complexity. However, as the graphs gets larger (than 6), the denser graphs (full graphs) gets progressively more difficult to learn compared to the sparser graphs (chain, jungle and bidiag). The models learned for denser graphs have higher complexity, higher variance and slightly worse results."},{"heading":"A.10 PREDICTING INTERVENTIONS","text":"In Phase 2, we score graph configurations based on how well they fit the interventional data. We find that it is necessary to avoid disturbing the learned parameters of intervened variables, and to ignore its contribution to the total negative log-likelihood of the sample. Intuitively, this is because, having been intervened upon, that variable should be taken as a given. It should especially not be interpreted as a poorly-learned variable requiring a tuning of its functional parameters, because those functional parameters were not responsible for the value of that variable; The extrinsic intervention was.\nSince an intervened variable is likely to be unusually poorly predicted, we heuristically determine that the most poorly predicted variable is the intervention variable. We then zero out its contribution to the log-likelihood of the sample and block gradient into its functional parameters.\nFigure 11 illustrates the necessity of this process. When using the prediction heuristic, the training curve closely tracks training with ground-truth knowledge of the identity of the intervention. If no prediction is made, or a random prediction is made, training proceeds much more slowly, or fails entirely."},{"heading":"A.11 SAMPLE COMPLEXITY","text":"Our method is heavily reliant on sampling of configurations and data in Phases 1 and 2. We present here the breakdown of the sample complexity. Let\n• I be the number of iterations of the method, (typical: 500-2000) • B the number of samples per batch, (typical: 256) • F the number of functional parameter training iterations in Phase 1, (typical: 10000) • Q the number of interventions performed in Phase 2, (typical: 100) • NP the number of data batches for prediction, (typical: 100)\n• CP the number of graph configurations drawn per prediction data batch, (typical: 10) • NS the number of data batches for scoring, (typical: 10) • CS the number of graph configurations drawn per scoring data batch. (typical: 20-30)\nThen the total number of interventions performed, and configurations and samples drawn, over an entire run are:\nInterventions = IQ = γ updates (3) Samples = I( F︸︷︷︸\nPhase 1 +Q(NP +NS)︸ ︷︷ ︸ Phase 2 )B (4)\nConfigurations = I( F︸︷︷︸ Phase 1 +Q(CPNP + CSNS)︸ ︷︷ ︸ Phase 2 ) (5)\nBecause of the multiplicative effect of these factors, the number of data samples required can quickly spiral out of control. For typical values, as many as 500 × 10000 × 256 = 1.28e9 observational and 500 × 100 × (100 + 10) × 256 = 1.408e9 interventional samples are required. To alleviate this problem slightly, we limit the number of samples generated for each intervention; This limit is usually 500-2000."},{"heading":"A.12 EFFECT OF REGULARIZATION","text":"Importance of sparsity regularizer. We use a L1 regularizer on the structure parameters γ to encourage a sparse representation of edges in the causal graph. In order to better understand the effect of the L1 regularizer, we conducted ablation studies on the L1 regularizer. It seems that the regularizer has an small effect on rate of converges and that the model converges faster with the regularizer, This is shown in Figure 13. However, this does not seem to affect the final value the model converges to, as is shown in Table 7.\nImportance of DAG regularizer. We use an acyclic regularizer to discourage length-2 cycles in the learned model. We found that for small models (≤ 5 variables), the acyclic regularizer helps with faster convergence, without improving significantly the final cross-entropy. This is illustrated for the 3-variable graphs in Figure 14. However, for graphs larger than 5 variables, the acyclic regularizer starts playing an important role in encouraging the model to learn the correct structure. This is shown in the ablation study in Table 7."},{"heading":"A.13 NEAR-OPTIMUM PERFORMANCE OF GRADIENT ESTIMATOR","text":"The gradient estimator gij we use to minimize the empirical risk w.r.t. the structural parameters γ, defined in Eq. 2 is adapted from Bengio et al. (2019). We verify that the estimator samples the correct gradient by an experiment that tests convergence near the optimum.\nTo do this, we pre-initialize the structural and functional parameters near the global minimum, and verify that γ converges. Specifically, the ground-truth functional parameters θ are copied and disturbed by a small Gaussian noise, while the ground-truth structural parameters γ are copied, but the confidences in an edge or non-edge are set to 88% and 12% rather than 100% and 0%. The experiment is then expected to quickly converge to the global minimum.\nAs shown in Figure 16, the gradient estimator correctly enables Stochastic Gradient Descent towards the minimum, for the chain and jungle graphs of size 15, 20 and 25. The average cross-entropy rapidly approaches its floor of 0.01, a consequence of our clamping of all γij to the range ±5 (equivalently, clamping σ(γij) to the range [0.0067, 0.9933]).\nA.14 IMPORTANCE OF DROPOUT\nTo train the functional parameters on an observational distribution, one would need sampling adjacency matrices. One may be tempted to make these “complete directed graph” (all-ones except for a zero diagonal), to give the MLP maximum freedom to learn any potential causal relations itself. We\ndemonstrate that functional parameter training cannot be carried out this way, and that it is necessary to “drop out” each edge (with probability of the current γ value in our experiments) during pretraining of the conditional distributions of the SCM. We attempt to recover the previously-recoverable graphs chain3, fork3 and confounder3 without dropout, but fail to do so, as shown in Figure 15.\nFigure 17: Cross-entropy for edge probability between learned and ground-truth SCM for Cancer at varying temperatures.\nFigure 18: Cross-entropy for edge probability between learned and ground-truth SCM. Left: The Earthquake dataset with 6 variables. Right: The Asia dataset with 8 variables"}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Structure learning concerns itself with the recovery of the graph structure of Bayesian networks (BNs) from data samples. A natural application of Bayesian networks is to describe cause-effect relationships between variables. In that context, one may speak of causal structure learning. Causal structure learning is challenging because purely observational data may be satisfactorily explained by multiple Bayesian networks (a Markov equivalence class), but only one is the most robust to distributional shifts: The one with the correct graph. A more powerful tool than BNs is thus needed to model causal relationships.\\nStructural Causal Models (SCMs) are that tool. An SCM over a set of random variables is a collection of assignments to these variables and a directed acyclic graph of dependencies between them (Peters et al., 2017, §6.2). Each assignment is a function of only the direct causes of a variable, plus an independent noise source. An SCM entails precisely one (observational) data distribution. Interventions on an SCM’s assignments, such as setting a random variable to a fixed value (a hard intervention), entail new interventional data distributions (Peters et al., 2017, §6.3).\\nSCMs can be used to answer higher-order questions of cause-and-effect, up the ladder of causation (Pearl & Mackenzie, 2018). Causal structure learning using SCMs has been attempted in several disciplines including biology (Sachs et al., 2005; Hill et al., 2016), weather forecasting (Abramson et al., 1996) and medicine (Lauritzen & Spiegelhalter, 1988; Korb & Nicholson, 2010).\\nCausal structure is most frequently learned from data drawn from observational distributions. Structure learning methods generally cannot do more than identify the causal graph up to a Markov equivalence class (Spirtes et al., 2000). In order to fully identify the true causal graph, a method must either make restrictive assumptions about the underlying data-generating process, such as linear but non-Gaussian data (Shimizu et al., 2006), or must access enough data from outside the observational distribution (i.e., from interventions).\\nUnder certain assumptions about the number, diversity, and nature of the interventions, the true underlying causal graph is always identifiable, given that the method knows the intervention performed (Heckerman et al., 1995). In much of the prior work on causal model induction it is assumed that\\nthere is an experimenter and this experimenter performs interventions. However, in the real world, interventions can also be performed by other agents, which could lead to unknown interventions (interventions with unknown target variables). A few works have attempted to learn structures from unknown-intervention data (Eaton & Murphy, 2007a; Squires et al., 2020; Huang et al., 2020). A notable such work, (Mooij et al., 2016), has been extended in (Kocaoglu et al., 2019; Jaber et al., 2020). Although there is no theoretical guarantee that the true causal graph can be identified in that setting, evidence so far points to that still being the case.\\nAnother common setting is when the graph structure is partially provided, but must be completed. An example is protein structure learning in biology, where we may have definitive knowledge of some causal edges in the protein-protein interactome, but the remaining causal edges must be discovered. We will call this setting “partial graph completion”. This is an easier task compared to learning the entire graph, since it limits the number of edges that have to be learned.\\nRecently, a flurry of work on structure learning using continuous optimization methods has appeared (Zheng et al., 2018; Yu et al., 2019). These methods operate on observational data and are competitive with other methods. Because of the theoretical limitations on identification from purely observational data cited above, it would be interesting to extend these methods to interventional data. However, it is not straightforward to apply continuous optimization methods to structure learning from interventional data. Our key contributions are to answer the following questions experimentally:\\n1. Can the proposed model recover true causal structure? Yes, see Figure §4. 2. How does the proposed model compare against state of the art causal methods on real-world\\ndatasets? Favourably; see §5.4 and Table §1. 3. Does a proposed model generalize well to unseen interventions? Yes, see §5.5. 4. How does the proposed model perform on partial graph recovery? It scales to∼ 50 variables\\nwhile the other baselines can’t. see §5.7.\"\n },\n {\n \"heading\": \"2 PRELIMINARIES\",\n \"text\": \"Causal modeling. A Structural Causal Model (SCM) (Peters et al., 2017) over a finite number M of random variables Xi is a set of structural assignments\\nXi := fi(Xpa(i,C), Ni) , ∀i ∈ {0, . . . ,M − 1} (1)\\nIdentifiability. In a purely-observational setting, it is known that causal graphs can be distinguished only up to a Markov equivalence class. In order to identify the true causal graph structure, interventional data is needed (Eberhardt et al., 2012).\\nInterventions. There are several types of common interventions which may be available (Eaton & Murphy, 2007b). These are: No intervention: only observational data is obtained from the ground truth model. Hard/perfect: the value of a single or several variables is fixed and then ancestral sampling is performed on the other variables. Soft/imperfect: the conditional distribution of the variable on which the intervention is performed is changed. Uncertain: the learner is not sure of which variable exactly the intervention affected directly. Here we make use of soft intervention because they include hard intervention as a limiting case and hence are more general.\\nStructure discovery using continuous optimization. Structure discovery is a super-exponential search problem that searches though all possible directed acyclic graphs (DAGs). Previous continuousoptimization structure learning works (Zheng et al., 2018; Yu et al., 2019; Lachapelle et al., 2019) mitigate the problem of searching in the super-exponential set of graph structures by considering the degree to which a hypothesis graph violates “DAG-ness” as an additional penalty to be optimized. If there are M such variables, the strategy of considering all the possible structural graphs as separate hypotheses is not feasible because it would require maintaining O(2M 2\\n) models of the data.2\"\n },\n {\n \"heading\": \"3 RELATED WORK\",\n \"text\": \"The recovery of the underlying structural causal graph from observational and interventional data is a fundamental problem (Pearl, 1995; 2009; Spirtes et al., 2000). Different approaches have been studied: score-based, constraint-based, asymmetry-based and continuous optimization methods. Score-based methods search through the space of all possible directed acyclic graphs (DAGs) representing the causal structure based on some form of scoring function for network structures (Heckerman et al., 1995; Chickering, 2002; Tsamardinos et al., 2006; Hauser & Bühlmann, 2012; Goudet et al., 2017; Cooper & Yoo, 1999; Zhu & Chen, 2019). Constraint-based methods (Spirtes et al., 2000; Sun et al., 2007; Zhang et al., 2012; Monti et al., 2019; Zhu & Chen, 2019) infer the DAG by analyzing conditional independences in the data. Eaton & Murphy (2007c) use dynamic programming techniques to accelerate Markov Chain Monte Carlo (MCMC) sampling in a Bayesian approach to structure learning for discrete variable DAGs. Peters et al. (2016); Ghassami et al. (2017); Rojas-Carulla et al. (2018) exploit invariance across environments to infer causal structure, which faces difficulty scaling due to the iteration over the super-exponential set of possible graphs. Recently, (Zheng et al., 2018; Yu et al., 2019; Lachapelle et al., 2019) framed the structure search as a continuous optimization problem, however, the methods only uses observational data and are non-trivial to extend to interventional data. In our paper, we present a method that uses continuous optimization methods that works on both observational and interventional data.\\nFor interventional data, it is often assumed that the models have access to full intervention information, which is rare in the real world. Rothenhäusler et al. (2015) have investigated the case of additive shift interventions, while Eaton & Murphy (2007b) have examined the situation where the targets of experimental interventions are imperfect or uncertain. This is different from our setting where the intervention is unknown to start with and is assumed to arise from other agents and the environment.\\nLearning based methods have been proposed (Guyon, 2013; 2014; Lopez-Paz et al., 2015) and there also exist recent approaches using the generalization ability of neural networks to learn causal signals from purely observational data (Kalainathan et al., 2018; Goudet et al., 2018). Neural network methods equipped with learned masks, such as (Ivanov et al., 2018; Li et al., 2019; Yoon et al., 2018; Douglas et al., 2017), exist in the literature, but only a few (Kalainathan et al., 2018) have been adapted to causal inference. This last work is, however, tailored for causal inference on continuous variables and from observations only. Adapting it to a discrete-variable setting is made difficult by its use of a Generative Adversarial Network (GAN) Goodfellow et al. (2014) framework.\"\n },\n {\n \"heading\": \"4 STRUCTURE DISCOVERY FROM INTERVENTIONS METHOD\",\n \"text\": \"Scope of Applicability and Objective. The proposed method, like any structure learning algorithm, assumes the availability of a data-generating process based on ancestral sampling of a ground-truth SCM of M variables, which can be queried for samples. The SCM supports applying and retracting known or unknown interventions. The method can support infinite- or finite-data as well as infiniteor finite-intervention regimes.\\nThe objective is, then, to learn the SCM’s structure from the insights that each intervention gives about cause-effect relationships between variables in the SCM.\"\n },\n {\n \"heading\": \"4.1 PROBLEM SETTING AND ASSUMPTIONS\",\n \"text\": \"In this paper, we restrict the problem setting to specific, but still broad classes of SCMs and interventions. In particular, we assume that:\\nData is discrete-valued. The SCM’s random variables are all categorical.\\nCausal sufficiency. For every data sample, the value of all variables are available; There are no latent confounders.\\nInterventions are localized. They affect only a single variable (but which one may not be known).\\nInterventions are soft. An intervention does not necessarily pin its target random variable to a fixed value (though it may, as a special case). It changes the relationship of a variable with its parents.\\nInterventions do not stack. Before a new intervention is made, the previous one is fully retracted. This stops the SCM from wandering away from its initial, observational configuration after a long series of interventions.\\nNo control over interventions. The structure learning algorithm has control neither of the target, nor the nature of the next intervention on the SCM.\\nFor a detailed description of the interventions, refer to §A.2.\"\n },\n {\n \"heading\": \"4.2 VARIATIONS AND PRIOR KNOWLEDGE\",\n \"text\": \"In the problem setting above, the ground-truth SCM is completely opaque to us. However, we consider two interesting relaxations of this formulation:\\nComplete or partial graph recovery. We may already know the existence of certain cause-effect edges and non-edges within the ground-truth SCM. If such prior information is available, it turns a complete graph recovery problem into one of partial graph recovery. Larger SCMs can be tackled if only parts of the graph need to be recovered.\\nKnown or unknown interventions: The interventions can either be known or unknown to the learned model.\\nWe demonstrate that the proposed method can naturally incorporate this prior information to improve its performance.\\n4.3 METHOD OVERVIEW\\nThe proposed method is a score-based, iterative, continuousoptimization method consisting of three phases that flow into one other (See Figure 2). During the three-phase procedure, a structural representation of a DAG and a functional representation of a set of independent causal mechanisms are trained jointly until convergence. Because the structural and functional parameters are not independent and do influence each other, we train them in alternating phases, a form of block coordinate descent optimization.\"\n },\n {\n \"heading\": \"4.3.1 PARAMETRIZATION\",\n \"text\": \"We distinguish two sets of parameters: The structural parameters γ and the functional parameters θ. Given a graph of M variables, we parametrize the structure γ as a matrix RM×M such that σ(γij) is our belief in random variable Xj being a direct cause of Xi, where\\nσ(x) = 1/(1 + exp(−x)) is the sigmoid function. The matrix σ(γ) is thus a soft adjacency matrix. The set of functional parameters θi parametrizes the conditional probability distribution of Xi given its parent set Xpa(i,C), with C ∼ Ber(σ(γ)) a hypothesized configuration of the SCM’s DAG.\"\n },\n {\n \"heading\": \"4.3.2 PHASE 1: GRAPH FITTING ON OBSERVATIONAL DATA\",\n \"text\": \"During Phase 1, the functional parameters θ are trained to maximize the likelihood of randomly drawn observational data under graphs randomly drawn from our current beliefs about the edge structure. We draw graph configurations Cij ∼ Ber(σ(γij)) and batches of observational data from the unintervened ground-truth SCM, then maximize the log-likelihood of the batch under that configuration using SGD. The use of graph configurations sampling from Bernoulli distributions is analogous to dropout on the inputs of the functional models (in our implementation, MLPs), giving us an ensemble of neural networks that can model the observational data.\"\n },\n {\n \"heading\": \"4.3.3 PHASE 2: GRAPH SCORING ON INTERVENTIONAL DATA\",\n \"text\": \"During Phase 2, a number of graph configurations are sampled from the current edge beliefs parametrized by γ, and scored on data samples drawn from the intervention SCM.\\nIntervention applied: At the beginning of Phase 2, an intervention is applied to the ground-truth SCM. This intervention is not under the control of the method. In our implementation, and unbeknownst to the model, the target variable is chosen uniformly randomly from all M variables throughout the optimization process.\\nIntervention predicted: If the target of the intervention is not known, it is predicted using a simple heuristic. A small number of interventional data samples are drawn from the SCM and more graphs are sampled from our current edge beliefs. The average log-likelihood of each individual variable Xi across the samples is then computed using the functional model parameters θ fine-tuned on observational data in Phase 1. The variable Xi showing the greatest deterioration in log-likelihood is assumed to be the target because the observational distribution most poorly predicts that variable.\\nIf the target of the intervention is known, then this is taken as ground-truth knowledge for the purpose of subsequent steps, and no prediction needs to be done.\\nGraphs Sampled and Scored: A new set of interventional data samples and graph configurations are now drawn from the intervention SCM and edge beliefs respectively. The log-likelihood of the data batches under the hypothesized configurations is computed, with one modification: The contribution to the total log-likelihood of a sample X coming from the target (or predicted-target) intervention variable Xi is masked. Because Xi was intervened upon (in the manner of a Pearl do-operation, soft or hard), the values one gets for that variable should be taken as givens, not as contributors to the total log-likelihood of the sample. As well, no gradient should be allowed to propagate into the variable’s learned functional parametrization θi, because it was not actually responsible for the outcome.\\nIntervention retracted: After Phase 2, the intervention is retracted, per our modelling assumptions.\"\n },\n {\n \"heading\": \"4.3.4 PHASE 3: CREDIT ASSIGNMENT TO STRUCTURAL PARAMETERS\",\n \"text\": \"During Phase 3, the scores of the interventional data batches over various graph configurations are aggregated into a gradient for the structural parameters γ. Because a discrete Bernoulli random sampling process was used to sample graph configurations under which the log-likelihoods were computed, we require a gradient estimator to propagate gradient through to the γ structural parameters. Several alternatives exist, but we adopt for this purpose the REINFORCE-like gradient estimator gij proposed by Bengio et al. (2019):\\ngij =\\n∑ k(σ(γij)− c\\n(k) ij )LC(k),i (X)∑\\nk LC(k),i (X) , ∀i, j ∈ {0, . . . ,M−1} (2)\\nwhere the (k) superscript indicates the values obtained for the k-th draw of C under the current edge beliefs parametrized by γ. Therefore, L(k)C,i(X) can be read as the log-likelihood of variable Xi in the data sample X under the k’th configuration, C(k), drawn from our edge beliefs. Using the estimated gradient, we then update γ with SGD, and return to Phase 1 of the continuous optimization process.\\nThe gradient estimator gij minimizes an implicit empirical risk objective with respect to γij . When the functional and structural parameters θ and γ are “sufficiently close” to their minima, the estimator gij empirically converges quickly towards that minimum γ∗ as shown in Figure 16 of Appendix A.13.\\nAcyclic Constraint: We include a regularization term JDAG(γ) that penalizes length-2 cycles in the learned adjacency matrix σ(γ), with a tunable strength λDAG. The regularization term is JDAG(γ) =∑ i 6=j cosh(σ(γij)σ(γji)), ∀i, j ∈ {0, . . . ,M−1} and is derived from Zheng et al. (2018). The details of the derivation are in the Appendix. We explore several different values of λDAG and their effects in our experimental setup. Suppression of longer-length cycles was not found to be worthwhile for the increased computational expense.\"\n },\n {\n \"heading\": \"5 EXPERIMENTAL SETUP AND RESULTS\",\n \"text\": \"We first evaluate the proposed method on a synthetic dataset where we have control over the number of variables and causal edges in the ground-truth SCM. This allows us to analyze the performance of the proposed method under various conditions. We then evaluate the proposed method on real-world datasets from the BnLearn dataset repository. We also consider the two variations of §4.2: Recovering only part of the graph (when the rest is known), and exploiting knowledge of the intervention target.\\nThe summary of our findings is: 1) We show strong results for graph recovery for all synthetic graphs in comparisons with other baselines, measured by Hamming distance. 2) The proposed method achieves high accuracy on partial graph recovery for large, real-world graphs. 3) The proposed method’s intervention target prediction heuristic closes the gap between the known- and unknowntarget intervention scenarios. 4) The proposed method generalizes well to unseen interventions. 5) The proposed method’s time-to-solution scaling appears to be driven by the number of edges in the groundtruth graph moreso than the number of variables.\\n5.1 MODEL DESCRIPTION\\nLearner model. Without loss of generality, we let θi = {W0i,B0i,W1i,B1i} define a stack of M one-hidden-layer MLPs, one for each random variable Xi. A more appropriate model, such as a CNN, can be chosen using domainspecific knowledge; the primary advantage of using MLPs is that the hypothesized DAG configurations cij can be readily used to mask the inputs of MLP i, as shown in Figure 3.\\nTo force the structural equation fi corresponding to Xi to rely exclusively on its direct ancestor set pa(i, C) under hypothesis adjacency matrix C (See Eqn. 1), the one-hot input vector Xj for variable Xi’s MLP is masked by the Boolean element cij . An example of the multi-MLP architecture with M=4 categorical variables of N=3 categories is shown in Figure 3. For more details, refer to Appendix A.4.\\nGround-truth model. Ground-truth SCM models are parametrized either as CPTs with parameters from BnLearn (in the case of real-world graphs), or as a second stack of MLPs similar to the learner model, with randomly-initialized functional parameters θGT and the desired adjacency matrix γGT.\\nInterventions. In all experiments, at most one (soft) intervention is concurrently performed. To simulate a soft intervention on variable Xi, we reinitialize its ground-truth conditional distribution’s MLP parameters or CPT table randomly, while leaving the other variables untouched. For more details about the interventions, please refer to Appendix A.2.\\n5.2 SYNTHETIC DATASETS EXPERIMENTS\\nWe first evaluate the model’s performance on several randomlyinitialized SCMs with specific, representative graph structures. Since the number of possible DAGs grows super-exponentially with the number of variables, for M=4 up to 13 a selection of\\nrepresentative and edge-case DAGs are chosen. chainM and fullM (M=3-13) are the minimallyand maximally-connected M -variable DAGs, while treeM and jungleM are tree-like intermediate graphs. colliderM is the (M−1)→ 1 collider graph. The details of the setup is in Appendix A.6.\\nResults. The model can recover most synthetic DAGs with high accuracy, as measured by Structural Hamming Distance (SHD) between learned and ground-truth DAGs. Table 1 shows our proposed method outperforming all other baseline methods, and learns all graphs perfectly for 3 to 13 variables (excepting full). For DAGs ranging from 3 to 8 variables, the AUROCs all eventually reach 1.0 (indicating perfect classification into edge/non-edge; Refer to Figure 4). For both large (M > 10)\\nand dense DAGs (e.g. full13) the model begins encountering difficulties, as shown in Table 1 and Appendix §A.6.1.\\nSmall graphs (M < 10) are less sensitive than larger ones to our hyperparameters, notably the sparsity and acyclic regularization (§4.3.4) terms. In §A.5, we perform an analysis of these hyperparameters.\"\n },\n {\n \"heading\": \"5.3 REAL-WORLD DATASETS: BNLEARN\",\n \"text\": \"The Bayesian Network Repository is a collection of commonly-used causal Bayesian networks from the literature, suitable for Bayesian and causal learning benchmarks. We evaluate the proposed method on the Earthquake (Korb & Nicholson, 2010), Cancer (Korb & Nicholson, 2010), Asia (Lauritzen & Spiegelhalter, 1988) and Sachs (Sachs et al., 2005) datasets (M =5, 5, 8 and 11-variables respectively, maximum in-degree 3) in the BnLearn dataset repository.\\nResults. As shown in Table 1, the proposed method perfectly recovers the DAG of Asia, while making a small number of errors (SHD=6) for Sachs (11-variables). It thus significantly outperforms all other baselines models. Figures 8 & 9 visualize what the model has learned at several stages of learning. Results for Cancer and Asia can be found in the appendices, Figure 17 and 18.\"\n },\n {\n \"heading\": \"5.4 COMPARISONS WITH OTHER METHODS\",\n \"text\": \"As shown in Table 1, we compared the proposed SDI method to ICP ((Peters et al., 2016)), non-linear ICP ((Heinze-Deml et al., 2018b)), and (Eaton & Murphy, 2007b; Zheng et al., 2018; Yu et al., 2019) on Asia (Lauritzen & Spiegelhalter, 1988), Sachs (Sachs et al., 2005) and representative synthetic graphs. Eaton & Murphy (2007b) handles uncertain interventions and Peters et al. (2016), Heinze-Deml et al. (2018b) handles unknown interventions. However, neither attempts to predict the intervention. As shown in Table 1, we significantly outperform ICP, non-linear ICP, and the methods in (Yu et al., 2019) and (Zheng et al., 2018). Furthermore, Eaton & Murphy (2007b) runs out of memory for graphs larger than M = 10 because modelling of uncertain interventions is done using “shadow” random variables (as suggested by the authors), and thus recovering the DAG internally requires solving a d = 2M -variable problem. Their method’s extremely poor time- and space-scaling of O(d2d) makes it unusable beyond d > 20.\\nFor SDIs, we threshold our edge beliefs at σ(γ) = 0.5 to derive a graph, but the continued decrease of the cross-entropy loss (Figure 4) hints at SDI’s convergence onto the correct causal model. Please refer to Appendix §A.8 for full details and results.\"\n },\n {\n \"heading\": \"5.5 GENERALIZATION TO PREVIOUSLY UNSEEN INTERVENTIONS\",\n \"text\": \"It is often argued that machine learning approaches based purely on capturing joint distributions do not necessarily yield models that generalize to unseen experiments, since they do not explicitly model changes through interventions. By way of contrast,\\ncausal models use the concept of interventions to explicitly model changing environments and thus hold the promise of robustness under distributional shifts (Pearl, 2009; Schölkopf et al., 2012; Peters et al., 2017). To test the robustness of causal modelling to previously unseen interventions (new values for an intervened variable), we evaluate a well-trained causal model against a variant, non-causal model trained with cij = 1, i 6= j. An intervention is performed on the ground-truth SCM, fresh interventional data is drawn from it, and the models, with knowledge of the intervention target, are asked to predict the other variables given their parents. The average log-likelihoods of the data under both models are computed and contrasted. The intervention variable’s contribution to the loglikelihood is masked. For all 3-variable graphs (chain3, fork3, collider3, confounder3), the causal model attributes higher log-likelihood to the intervention distribution’s samples than the non-causal variant, thereby demonstrating causal models’ superior generalization ability in transfer tasks. Table 2 collects these results.\"\n },\n {\n \"heading\": \"5.6 VARIANT: PREDICTING INTERVENTIONS\",\n \"text\": \"In Phase 2 (§4.3.3), we use a simple heuristic to predict the intervention target variable. Experiments show that this heuristic functions well in practice, yielding correct predictions far more often than by chance alone (Table 3). Guessing the intervention variable\\nrandomly, or not guessing it at all, leads to a significant drop in the model performance, even for 3-variable graphs (Fig. 11 Left). Training SDI with intervention prediction closely tracks training with leaked knowledge of the ground-truth intervention on larger, 7-variable graphs (Fig. 11 Right).\"\n },\n {\n \"heading\": \"5.7 VARIANT: PARTIAL GRAPH RECOVERY\",\n \"text\": \"Instead of learning causal structures de novo, we may have partial information about the ground-truth SCM and may only need to fill in missing information (§4.2). An example is protein structure discovery in biology, where some causal relationships have been definitely established and others remain open hypotheses. This is an easier task compared to full graph recovery, since the model only has to search for missing edges.\\nTable 4: Partial Graph Recovery on Alarm (Beinlich et al., 1989) and Barley (Kristensen & Rasmussen, 2002). The model is asked to predict 50 edges in Barley and 40 edges in Alarm. The accuracy is measured in Structural Hamming Distance (SHD). SDI achieved over 90% accuracy on both graphs.\\nGraph Alarm Barley Number of variables 37 48 Total Edges 46 84\\nEdges to recover 40 50 Recovered Edges 37 45 Errors (in SHD) 3 5\\nWe evaluate the proposed method on Barley (Kristensen & Rasmussen, 2002) (M = 48) and Alarm (Beinlich et al., 1989) (M = 37) from the BnLearn repository. The model is asked to predict 50 edges from Barley and 40 edges from Alarm. The model reached ≥ 90% accuracy on both datasets, as shown in Table 4.\"\n },\n {\n \"heading\": \"5.8 ABLATION AND ANALYSIS\",\n \"text\": \"As shown in Figure 12, larger graphs (such as M > 6) and denser graphs (such as full8) are progressively more difficult to learn. For denser graphs, the learned models have higher sample complexity, higher variance and slightly worse results. Refer to Appendix §A.9 for complete results on all graphs. Hyperparameters. Hyperparameters for all experiments were kept identical unless otherwise stated. We study the effect of DAG and sparsity penalties in the following paragraph. For more details, please refer to Appendix §A.5 .\\nImportance of regularization. Valid configurations C for a causal model are expected to be a) sparse and b) acyclic. To promote such solutions, we use DAG and sparsity regularization with tunable hyperparameters. We set the DAG penalty to 0.5 and sparsity penalty to 0.1. We run ablation studies on different values of the regularizers and study their effect. We find that smaller graphs are less sensitive to different values of regularizer than larger graphs. For details, refer to Appendix §A.12.\\nImportance of dropout. To train functional parameter for an observational distribution, sampling adjacency matrices is required. We \\\"drop out\\\" each edge (with a probability of σ(γ)) in our experiments during functional parameter training of the conditional distributions of the SCM. Please refer to Appendix §A.14 for a more detailed analysis.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"In this work, we introduced an experimentally successful method (SDI) for causal structure discovery using continuous optimization, combining information from both observational and interventional data. We show in experiments that it can recover true causal structure, that it generalizes well to unseen interventions, that it compares very well against start-of-the-art causal discovery methods on real world datasets, and that it scales even better on problems where only part of the graph is known.\"\n },\n {\n \"heading\": \"Appendix\",\n \"text\": \"\"\n },\n {\n \"heading\": \"Table of Contents\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A Annexes 13\",\n \"text\": \"A.1 Training Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.4 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.5 Hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.6 Synthetic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 A.7 BnLearn data repository . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A.8 Comparisons to other methods . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.9 Sparsity of Ground-Truth Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 17 A.10 Predicting interventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.11 Sample complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 A.12 Effect of regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 A.13 Near-Optimum Performance of Gradient Estimator . . . . . . . . . . . . . . . . 20 A.14 Importance of dropout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\"\n },\n {\n \"heading\": \"A ANNEXES\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A.1 TRAINING ALGORITHM\",\n \"text\": \"Algorithm 1 shows the pseudocode of the method described in §4. Typical values for the loop trip counts are found in §A.11.\"\n },\n {\n \"heading\": \"A.2 PRELIMINARIES\",\n \"text\": \"Interventions. In a purely-observational setting, it is known that causal graphs can be distinguished only up to a Markov equivalence class. In order to identify the true causal graph intervention data is needed (Eberhardt et al., 2012). Several types of common interventions may be available (Eaton & Murphy, 2007b). These are: No intervention: only observational data is obtained from the ground truth causal model. Hard/perfect: the value of a single or several variables is fixed and then ancestral sampling is performed on the other variables. Soft/imperfect: the conditional distribution of the variable on which the intervention is performed is changed. Uncertain: the learner is not sure of which variable exactly the intervention affected directly. Here we make use of soft interventions for several reasons: First, they include hard interventions as a limiting case and hence are more general. Second, in many real-world scenarios, it is more difficult to perform a hard intervention compared to a soft one. We also deal with a special case of uncertain interventions, where the variable selected for intervention is random and unknown. We call these unidentified or unknown interventions.\\nIntervention setup. For our experiments, the groundtruth models of the synthetic datasets are modeled by neural networks as described in section A.6. Each neural network models the relationship of the causal parents and a variable. We perform our intervention by first randomly selecting which variable to intervene on, then soft-intervening on it. The selected variable is sampled from a uniform distribution. The soft intervention is a reinitialization of its neural network’s parameters.\\nCausal sufficiency. The inability to distinguish which causal graph, within a Markov equivalence class, is the correct one in the purely-observational setting is called the identifiability problem. In our setting, all variables are observed (there are no latent confounders) and all interventions are random and independent. Hence, within our setting, if the interventions are known, then the true causal\\nAlgorithm 1 Training Algorithm 1: procedure TRAINING(SCM Ground-Truth Entailed Distribution D, with M nodes and N categories) 2: Let i an index from 0 to M − 1\\n3: for I iterations, or until convergence, do 4: if I % reinitialization_period == 0 then 5: D ← reinitialize(D)\\n6: for F functional parameter training steps do . Phase 1 7: X ∼ D 8: C ∼ Ber(σ(γ)) 9: L = − logP (X|C ; θ)\\n10: θt+1 ← Adam(θt,∇θL)\\n11: for Q interventions do . Phase 2 12: I_N← randint(0, M − 1) . Uniform selection of target 13: Dint :=D with intervention on node I_N . Apply intervention\\n14: if predicting intervention then . Phase 2 Prediction 15: Li ← 0 ∀i 16: for NP prediction steps do 17: X ∼ Dint 18: for CP configurations do 19: C ∼ Ber(σ(γ)) 20: Li ← Li − logPi(X|Ci; θslow) ∀i 21: I_N← argmax(Li)\\n22: gammagrads, logregrets = [], [] . Phase 2 Scoring 23: for NS scoring steps do 24: X ∼ Dint 25: gammagrad, logregret = 0, 0 26: for CS configurations do 27: C ∼ Ber(σ(γ)) 28: Li = − logPi(X|Ci; θslow) ∀i 29: gammagrad += σ(γ)− C . Collect σ(γ)− C for Equation 2 30: logregret +=\\n∑ i6=I_N Li . Collect LC(k),i (X) for Equation 2\\n31: gammagrads.append(gammagrad) 32: logregrets.append(logregret)\\n. Phase 3\\n33: gij =\\n∑ k(σ(γij)− c (k) ij )LC\\n(k) ,i (X)∑\\nk LC (k) ,i (X)\\n. Gradient Estimator, Equation 2\\n34: g ← g +∇γ (λsparse Lsparse(γ) + λDAG LDAG(γ)) . Regularizers 35: γt+1 ← Adam(γt, g)\\ngraph is always identifiable in principle (Eberhardt et al., 2012; Heinze-Deml et al., 2018a). We also consider here situations where a single variable is randomly selected and intervened upon with a soft or imprecise intervention, its identity is unknown and must be inferred. In this case, there is no theoretical guarantee that the causal graph is identifiable. However, there is existing work Peters et al. (2016) that handles this scenario and the proposed method is also proven to work empirically.\\nFaithfulness. It is possible for causally-related variables to be probabilistically independent purely by happenstance, such as when causal effects along multiple paths cancel out. This is called unfaithfulness. We assume that faithfulness holds, since the γ gradient estimate is extracted from shifts in probability distributions. However, because of the “soft” nature of our interventions and their infinite variety, it would be exceedingly unlikely for cancellation-related unfaithfulness to persist throughout the causal-learning procedure.\"\n },\n {\n \"heading\": \"A.3 EXPERIMENTAL SETUP\",\n \"text\": \"For all datasets, the weight parameters for the learned model is initialized randomly. In order to not bias the structural parameters, all σ(γ) are initialized to 0.5 in the beginning of training. Details of hyperparameters of the learner model are described in Section A.5. The experimental setup for the groundtruth model for the synthetic data can be found in Section A.6 and the details for the real world data are described in Section A.7.\"\n },\n {\n \"heading\": \"A.4 MODEL SETUP\",\n \"text\": \"As discussed in section 4, we model the M variables in the graph using M independent MLPs, each possesses an input layer of M × N neurons (for M one-hot vectors of length N each), a single hidden layer chosen arbitrarily to have max(4M, 4N) neurons with a LeakyReLU activation of slope 0.1, and a linear output layer of N neurons representing the unnormalized log-probabilities of each category (a softmax then recovers the conditional probabilities from these logits). To force fi to rely exclusively on the direct ancestor set pa(i, C) under adjacency matrix C (See Eqn. 2), the one-hot input vector Xj for variable Xi’s MLP is masked by the Boolean element cij . The functional parameters of the MLP are the set θ = {W0ihjn,B0ih,W1inh,B1in}.An example of the multi-MLP architecture with M=3 categorical variables of N=2 categories is shown in Figure 3.\"\n },\n {\n \"heading\": \"A.5 HYPERPARAMETERS\",\n \"text\": \"Learner model. All experiments on the synthetic graphs of size 3-8 use the same hyperparameters. Both the functional and structural parameters are optimized using the Adam optimizer Kingma & Ba (2014). We use a learning rate of 5e− 2 with alpha of 0.9 for the functional parameters, and we use a learning rate of 5e− 3 with alpha of 0.1 for the structural parameters. We perform 5 runs of each experiment with random seeds 1− 5 and error bars are plotted for various graphs from size 3 to 8 in Figure 4. We use a batch size of 256. The L1 norm regularizer is set to 0.1 and the DAG regularizer is set to 0.5 for all experiments. For each γ update step, we sample 25 structural configurations from the current γ. In all experiments, we use 100 batches from the interventional distribution to predict the intervened node.\"\n },\n {\n \"heading\": \"A.6 SYNTHETIC DATA\",\n \"text\": \"Synthetic datasets. The synthetic datasets in the paper are modeled by neural networks. All neural networks are 2 layered feed forward neural networks (MLPs) with Leaky ReLU activations between layers. The parameters of the neural network are initialized orthogonally within the range of (−2.5, 2.5). This range was selected such that they output a non-trivial distribution. The biases are initialized uniformly between (−1.1, 1.1). SCM with n variables are modeled by n feedforward neural networks (MLPs) as described in §5.1. We assume an acyclic causal graph so that we may easily sample from them. Hence, given any pair of random variables A and B, either A −→ B, B −→ A or A and B are independent. The MLP representing the ground-truth SCM has its weights θ initialized use orthogonal initialization with gain 2.5 and the biases are initialized using a uniform initialization between−1.1 and 1.1, which was empirically found to yield \\\"interesting\\\" yet learnable random SCMs.\\nWe study a variety of SCMs with different ground-truth edge structures γ. Our selection of synthetic graphs explores various extremes in the space of DAGs, stress-testing SDI. The chain graphs are the sparsest connected graphs possible, and are relatively easy to learn. The bidiag graphs are extensions of chain where there are 2-hops as well as single hops between nodes, doubling the number of edges and creating a meshed chain of forks and colliders. The jungle graphs are binary-tree-like graphs, but with each node connected directly to its grandparent in the tree as well. Half the nodes in a jungle graph are leaves, and the out-degree is up to 6. The collider graphs deliberately collide independent M − 1 ancestors into the last node; They stress maximum in-degree. Lastly, the full graphs are the maximally dense DAGs. All nodes are direct parents of all nodes below them in the topological order. The maximum in- and out-degree are both M − 1. These graphs are depicted in Figure 6.\"\n },\n {\n \"heading\": \"A.6.1 SYNTHETIC DATA RESULTS\",\n \"text\": \"The model can recover correctly all synthetic graphs with 10 variables or less, as shown in Figure 10 and Table 1. For graphs larger than 10 variables, the model found it more challenging to recover the denser graphs (e.g. fullM), as shown in Table 1. Plots of the training curves showing average cross entropy (CE) and Area-Under-Curve(AUC/AUCROC) for edge probabilities of the learned graph against the ground-truth graph for synthetic SCMs with 3-13 variables are available in Figure 10.\"\n },\n {\n \"heading\": \"A.7 BNLEARN DATA REPOSITORY\",\n \"text\": \"The repo contains many datasets with various sizes and structures modeling different variables. We evaluate the proposed method on 3 of the datasets in the repo, namely the Earthquake (Korb & Nicholson, 2010), Cancer (Korb & Nicholson, 2010) and Asia (Lauritzen & Spiegelhalter, 1988) datasets. The ground-truth model structure for the Cancer (Korb & Nicholson, 2010) and Earthquake (Korb & Nicholson, 2010) datasets are shown in Figure 7. Note that even though the structure for the two datasets seems to be the same, the conditional probability tables (CPTs) for these datasets are very different and hence results in different structured causal models (SCMs) for each.\"\n },\n {\n \"heading\": \"A.8 COMPARISONS TO OTHER METHODS\",\n \"text\": \"As described in section 5.4, we compare to 5 other methods. The full comparison between SDIs and other methods on various graphs can be found in Table 1.\\nOne of these methods, DAG-GNN Yu et al. (2019), outputs 3 graphs based on different criteria: best mean square error (MSE), best negative loglikelihood (NLL) and best evidence lower bound (ELBO). We report performance of all outputs of DAG-GNN Yu et al. (2019) in Table 6, and the best one is selected for Table 1.\"\n },\n {\n \"heading\": \"A.9 SPARSITY OF GROUND-TRUTH GRAPH\",\n \"text\": \"We evaluated the performance of SDI on graphs of various size and sparsity to better understand the performance of the model. We evaluated the proposed model on 4 representative types of graphs in increasing order of density. They are the chain, jungle, bidiag and full graphs. As shown in the results in figure 12, for graphs of size 5 or smaller, there is almost no difference in the final results in terms of variance and sample complexity. However, as the graphs gets larger (than 6), the denser graphs (full graphs) gets progressively more difficult to learn compared to the sparser graphs (chain, jungle and bidiag). The models learned for denser graphs have higher complexity, higher variance and slightly worse results.\"\n },\n {\n \"heading\": \"A.10 PREDICTING INTERVENTIONS\",\n \"text\": \"In Phase 2, we score graph configurations based on how well they fit the interventional data. We find that it is necessary to avoid disturbing the learned parameters of intervened variables, and to ignore its contribution to the total negative log-likelihood of the sample. Intuitively, this is because, having been intervened upon, that variable should be taken as a given. It should especially not be interpreted as a poorly-learned variable requiring a tuning of its functional parameters, because those functional parameters were not responsible for the value of that variable; The extrinsic intervention was.\\nSince an intervened variable is likely to be unusually poorly predicted, we heuristically determine that the most poorly predicted variable is the intervention variable. We then zero out its contribution to the log-likelihood of the sample and block gradient into its functional parameters.\\nFigure 11 illustrates the necessity of this process. When using the prediction heuristic, the training curve closely tracks training with ground-truth knowledge of the identity of the intervention. If no prediction is made, or a random prediction is made, training proceeds much more slowly, or fails entirely.\"\n },\n {\n \"heading\": \"A.11 SAMPLE COMPLEXITY\",\n \"text\": \"Our method is heavily reliant on sampling of configurations and data in Phases 1 and 2. We present here the breakdown of the sample complexity. Let\\n• I be the number of iterations of the method, (typical: 500-2000) • B the number of samples per batch, (typical: 256) • F the number of functional parameter training iterations in Phase 1, (typical: 10000) • Q the number of interventions performed in Phase 2, (typical: 100) • NP the number of data batches for prediction, (typical: 100)\\n• CP the number of graph configurations drawn per prediction data batch, (typical: 10) • NS the number of data batches for scoring, (typical: 10) • CS the number of graph configurations drawn per scoring data batch. (typical: 20-30)\\nThen the total number of interventions performed, and configurations and samples drawn, over an entire run are:\\nInterventions = IQ = γ updates (3) Samples = I( F︸︷︷︸\\nPhase 1 +Q(NP +NS)︸ ︷︷ ︸ Phase 2 )B (4)\\nConfigurations = I( F︸︷︷︸ Phase 1 +Q(CPNP + CSNS)︸ ︷︷ ︸ Phase 2 ) (5)\\nBecause of the multiplicative effect of these factors, the number of data samples required can quickly spiral out of control. For typical values, as many as 500 × 10000 × 256 = 1.28e9 observational and 500 × 100 × (100 + 10) × 256 = 1.408e9 interventional samples are required. To alleviate this problem slightly, we limit the number of samples generated for each intervention; This limit is usually 500-2000.\"\n },\n {\n \"heading\": \"A.12 EFFECT OF REGULARIZATION\",\n \"text\": \"Importance of sparsity regularizer. We use a L1 regularizer on the structure parameters γ to encourage a sparse representation of edges in the causal graph. In order to better understand the effect of the L1 regularizer, we conducted ablation studies on the L1 regularizer. It seems that the regularizer has an small effect on rate of converges and that the model converges faster with the regularizer, This is shown in Figure 13. However, this does not seem to affect the final value the model converges to, as is shown in Table 7.\\nImportance of DAG regularizer. We use an acyclic regularizer to discourage length-2 cycles in the learned model. We found that for small models (≤ 5 variables), the acyclic regularizer helps with faster convergence, without improving significantly the final cross-entropy. This is illustrated for the 3-variable graphs in Figure 14. However, for graphs larger than 5 variables, the acyclic regularizer starts playing an important role in encouraging the model to learn the correct structure. This is shown in the ablation study in Table 7.\"\n },\n {\n \"heading\": \"A.13 NEAR-OPTIMUM PERFORMANCE OF GRADIENT ESTIMATOR\",\n \"text\": \"The gradient estimator gij we use to minimize the empirical risk w.r.t. the structural parameters γ, defined in Eq. 2 is adapted from Bengio et al. (2019). We verify that the estimator samples the correct gradient by an experiment that tests convergence near the optimum.\\nTo do this, we pre-initialize the structural and functional parameters near the global minimum, and verify that γ converges. Specifically, the ground-truth functional parameters θ are copied and disturbed by a small Gaussian noise, while the ground-truth structural parameters γ are copied, but the confidences in an edge or non-edge are set to 88% and 12% rather than 100% and 0%. The experiment is then expected to quickly converge to the global minimum.\\nAs shown in Figure 16, the gradient estimator correctly enables Stochastic Gradient Descent towards the minimum, for the chain and jungle graphs of size 15, 20 and 25. The average cross-entropy rapidly approaches its floor of 0.01, a consequence of our clamping of all γij to the range ±5 (equivalently, clamping σ(γij) to the range [0.0067, 0.9933]).\\nA.14 IMPORTANCE OF DROPOUT\\nTo train the functional parameters on an observational distribution, one would need sampling adjacency matrices. One may be tempted to make these “complete directed graph” (all-ones except for a zero diagonal), to give the MLP maximum freedom to learn any potential causal relations itself. We\\ndemonstrate that functional parameter training cannot be carried out this way, and that it is necessary to “drop out” each edge (with probability of the current γ value in our experiments) during pretraining of the conditional distributions of the SCM. We attempt to recover the previously-recoverable graphs chain3, fork3 and confounder3 without dropout, but fail to do so, as shown in Figure 15.\\nFigure 17: Cross-entropy for edge probability between learned and ground-truth SCM for Cancer at varying temperatures.\\nFigure 18: Cross-entropy for edge probability between learned and ground-truth SCM. Left: The Earthquake dataset with 6 variables. Right: The Asia dataset with 8 variables\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":91,"cells":{"paper_id":{"kind":"string","value":"SP:d8c4980cf2187b549f2f2a4fbb2fba4101337459"},"summaries":{"kind":"list like","value":[".** Autoregressive models have demonstrate their potential utility for modeling images and other types of complex data with high flexibility (particularly in density estimation). However, its sampling ability is not that good as explained in the paper. Authors show that one of the main weaknesses of autoregressive models comes from the propagation of mistakes due to the mismatch of conditionals. Inspired in the promising results of randomized smoothing in adversarial models (Cohen et al. 2019), authors propose a similar strategy. The addition of Gaussian noise and posterior modeling of the smoother data makes easier to the autoregressive density to capture the true data distribution. The benefits of this strategy are empirically proved and shown in the experiments."],"string":"[\n \".** Autoregressive models have demonstrate their potential utility for modeling images and other types of complex data with high flexibility (particularly in density estimation). However, its sampling ability is not that good as explained in the paper. Authors show that one of the main weaknesses of autoregressive models comes from the propagation of mistakes due to the mismatch of conditionals. Inspired in the promising results of randomized smoothing in adversarial models (Cohen et al. 2019), authors propose a similar strategy. The addition of Gaussian noise and posterior modeling of the smoother data makes easier to the autoregressive density to capture the true data distribution. The benefits of this strategy are empirically proved and shown in the experiments.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"While autoregressive models excel at image compression, their sample quality is often lacking. Although not realistic, generated images often have high likelihood according to the model, resembling the case of adversarial examples. Inspired by a successful adversarial defense method, we incorporate randomized smoothing into autoregressive generative modeling. We first model a smoothed version of the data distribution, and then reverse the smoothing process to recover the original data distribution. This procedure drastically improves the sample quality of existing autoregressive models on several synthetic and real-world image datasets while obtaining competitive likelihoods on synthetic datasets."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"Chenlin Meng"},{"affiliations":[],"name":"Jiaming Song"},{"affiliations":[],"name":"Yang Song"},{"affiliations":[],"name":"Shengjia Zhao"},{"affiliations":[],"name":"Stefano Ermon"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"Chenlin Meng\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Jiaming Song\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Yang Song\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Shengjia Zhao\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Stefano Ermon\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Guillaume Alain","Yoshua Bengio"],"title":"What regularized auto-encoders learn from the data-generating distribution","venue":"The Journal 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],\n \"title\": \"Large sample analysis of the median heuristic\",\n \"venue\": \"arXiv preprint arXiv:1707.07269,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Mathieu Germain\",\n \"Karol Gregor\",\n \"Iain Murray\",\n \"Hugo Larochelle\"\n ],\n \"title\": \"Made: Masked autoencoder for distribution estimation\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Martin Heusel\",\n \"Hubert Ramsauer\",\n \"Thomas Unterthiner\",\n \"Bernhard Nessler\",\n \"Sepp Hochreiter\"\n ],\n \"title\": \"Gans trained by a two time-scale update rule converge to a local nash equilibrium\",\n \"venue\": \"arXiv preprint arXiv:1706.08500,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Jonathan Ho\",\n \"Xi Chen\",\n \"Aravind Srinivas\",\n \"Yan Duan\",\n \"Pieter Abbeel\"\n ],\n \"title\": \"Flow++: Improving flowbased generative models with variational dequantization and architecture design\",\n \"venue\": null,\n \"year\": 1902\n },\n {\n \"authors\": [\n \"Jonathan Ho\",\n \"Ajay Jain\",\n \"Pieter Abbeel\"\n ],\n \"title\": \"Denoising diffusion probabilistic models\",\n \"venue\": \"arXiv preprint arXiv:2006.11239,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Aapo Hyvärinen\"\n ],\n \"title\": \"Estimation of non-normalized statistical models by score matching\",\n \"venue\": \"Journal of Machine Learning Research,\",\n \"year\": 2005\n },\n {\n \"authors\": [\n \"Aapo Hyvärinen\"\n ],\n \"title\": \"Optimal approximation of signal priors\",\n \"venue\": \"Neural Computation,\",\n \"year\": 2008\n },\n {\n \"authors\": [\n \"Matti Kääriäinen\"\n ],\n \"title\": \"Lower bounds for reductions\",\n \"venue\": \"In Atomic Learning Workshop,\",\n \"year\": 2006\n },\n {\n \"authors\": [\n \"Durk P Kingma\",\n \"Prafulla Dhariwal\"\n ],\n \"title\": \"Glow: Generative flow with invertible 1x1 convolutions\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2018\n },\n 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Szegedy\",\n \"Wojciech Zaremba\",\n \"Ilya Sutskever\",\n \"Joan Bruna\",\n \"Dumitru Erhan\",\n \"Ian Goodfellow\",\n \"Rob Fergus\"\n ],\n \"title\": \"Intriguing properties of neural networks\",\n \"venue\": \"arXiv preprint arXiv:1312.6199,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Joshua B Tenenbaum\",\n \"Vin De Silva\",\n \"John C Langford\"\n ],\n \"title\": \"A global geometric framework for nonlinear dimensionality reduction\",\n \"venue\": null,\n \"year\": 2000\n },\n {\n \"authors\": [\n \"Lucas Theis\",\n \"Aäron van den Oord\",\n \"Matthias Bethge\"\n ],\n \"title\": \"A note on the evaluation of generative models\",\n \"venue\": \"arXiv preprint arXiv:1511.01844,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Benigno Uria\",\n \"Iain Murray\",\n \"Hugo Larochelle\"\n ],\n \"title\": \"Rnade: The real-valued neural autoregressive density-estimator\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2013\n },\n {\n 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INTRODUCTION","text":"Autoregressive models have exhibited promising results in a variety of downstream tasks. For instance, they have shown success in compressing images (Minnen et al., 2018), synthesizing speech (Oord et al., 2016a) and modeling complex decision rules in games (Vinyals et al., 2019). However, the sample quality of autoregressive models on real-world image datasets is still lacking.\nPoor sample quality might be explained by the manifold hypothesis: many real world data distributions (e.g. natural images) lie in the vicinity of a low-dimensional manifold (Belkin & Niyogi, 2003), leading to complicated densities with sharp transitions (i.e. high Lipschitz constants), which are known to be difficult to model for density models such as normalizing flows (Cornish et al., 2019). Since each conditional of an autoregressive model is a 1-dimensional normalizing flow (given a fixed context of previous pixels), a high Lipschitz constant will likely hinder learning of autoregressive models.\nAnother reason for poor sample quality is the “compounding error” issue in autoregressive modeling. To see this, we note that an autoregressive model relies on the previously generated context to make a prediction; once a mistake is made, the model is likely to make another mistake which compounds (Kääriäinen, 2006), eventually resulting in questionable and unrealistic samples. Intuitively, one would expect the model to assign low-likelihoods to such unrealistic images, however, this is not always the case. In fact, the generated samples, although appearing unrealistic, often are assigned high-likelihoods by the autoregressive model, resembling an “adversarial example” (Szegedy et al., 2013; Biggio et al., 2013), an input that causes the model to output an incorrect answer with high confidence.\nInspired by the recent success of randomized smoothing techniques in adversarial defense (Cohen et al., 2019), we propose to apply randomized smoothing to autoregressive generative modeling. More specifically, we propose to address a density estimation problem via a two-stage process. Unlike Cohen et al. (2019) which applies smoothing to the model to make it more robust, we apply smoothing to the data distribution. Specifically, we convolve a symmetric and stationary noise distribution with the data distribution to obtain a new “smoother” distribution. In the first stage, we model the smoothed version of the data distribution using an autoregressive model. In the second stage, we reverse the smoothing process—a procedure which can also be understood as “denoising”—by either applying a gradient-based denoising approach (Alain & Bengio, 2014) or introducing another conditional autoregressive model to recover the original data distribution from the smoothed one. By choosing an appropriate smoothing distribution, we aim to make each step easier than the original learning problem: smoothing facilitates learning in the first stage by making the input distribution\nfully supported without sharp transitions in the density function; generating a sample given a noisy one is easier than generating a sample from scratch.\nWe show with extensive experimental results that our approach is able to drastically improve the sample quality of current autoregressive models on several synthetic datasets and real-world image datasets, while obtaining competitive likelihoods on synthetic datasets. We empirically demonstrate that our method can also be applied to density estimation, image inpainting, and image denoising."},{"heading":"2 BACKGROUND","text":"We consider a density estimation problem. Given D-dimensional i.i.d samples {x1,x2, ...,xN} from a continuous data distribution pdata(x), the goal is to approximate pdata(x) with a model pθ(x) parameterized by θ. A commonly used approach for density estimation is maximum likelihood estimation (MLE), where the objective is to maximize L(θ) , 1N ∑N i=1 log pθ(xi)."},{"heading":"2.1 AUTOREGRESSIVE MODELS","text":"An autoregressive model (Larochelle & Murray, 2011; Salimans et al., 2017) decomposes a joint distribution pθ(x) into the product of univariate conditionals:\npθ(x) = D∏ i=1 pθ(xi|x<i), (1)\nwhere xi stands for the i-th component of x, and x<i refers to the components with indices smaller than i. In general, an autoregressive model parameterizes each conditional pθ(xi|x<i) using a prespecified density function (e.g. mixture of logistics). This bounds the capacity of the model by limiting the number of modes for each conditional.\nAlthough autoregressive models have achieved top likelihoods amongst all types of density based models, their sample quality is still lacking compared to energy-based models (Du & Mordatch, 2019) and score-based models (Song & Ermon, 2019). We believe this can be caused by the following two reasons."},{"heading":"2.2 MANIFOLD HYPOTHESIS","text":"Several existing methods (Roweis & Saul, 2000; Tenenbaum et al., 2000) rely on the manifold hypothesis, i.e. that real-world high-dimensional data tends to lie on a low-dimensional manifold (Narayanan & Mitter, 2010). If the manifold hypothesis is true, then the density of the data distribution is not well defined in the ambient space; if the manifold hypothesis holds only approximately and the data lies in the vicinity of a manifold, then only points that are very close to the manifold would have high density, while all other points would have close to zero density. Thus we may expect the data density around the manifold to have large first-order derivatives, i.e. the density function has a high Lipschitz constant (if not infinity).\nTo see this, let us consider a 2-d example where the data distribution is a thin ring distribution (almost a unit circle) formed by rotating the 1-d Gaussian distribution N (1, 0.012) around the origin. The density function of the ring has a high Lipschitz constant near the “boundary”. Let us focus on a data point travelling along the diagonal as shown in the leftmost panel in figure 2. We plot the first-order\ndirectional derivatives of the density for the point as it approaches the boundary from the inside, then lands on the ring, and finally moves outside the ring (see figure 2). As we can see, when the point is far from the boundary, the derivative has a small magnitude. When the point moves closer to the boundary, the magnitude increases and changes significantly near the boundary even with small displacements in the trajectory. However, once the point has landed on the ring, the magnitude starts to decrease. As it gradually moves off the ring, the magnitude first increases and then decreases just like when the point approached the boundary from the inside. It has been observed that certain likelihood models, such as normalizing flows, exhibit pathological behaviors on data distributions whose densities have high Lipschitz constants (Cornish et al., 2019). Since each conditional of an autoregressive model is a 1-d normalizing flow given a fixed context, a high Lipschitz constant on data density could also hinder learning of autoregressive models."},{"heading":"2.3 COMPOUNDING ERRORS IN AUTOREGRESSIVE MODELING","text":"Autoregressive models can also be susceptible to compounding errors from the conditional distributions (Lamb et al., 2016) during sampling time. We notice that an autoregressive model pθ(x) learns the joint density pdata(x) by matching each of the conditional pθ(xi|x<i) with pdata(xi|x<i). In practice, we typically have access to a limited amount of training data, which makes it hard for an autoregressive model to capture all the conditional distributions correctly due to the curse of dimensionality. During sampling, since a prediction is made based on the previously generated context, once a mistake is made at a previous step, the model is likely to make more mistakes in the later steps, eventually generating a sample x̂ that is far from being an actual image, but is mistakenly assigned a high-likelihood by the model.\nThe generated image x̂, being unrealistic but assigned a high-likelihood, resembles an adversarial example, i.e., an input that causes the model to make mistakes. Recent works (Cohen et al., 2019) in adversarial defense have shown that random noise can be used to improve the model’s robustness to adversarial perturbations — a process during which adversarial examples that are close to actual data are generated to fool the model. We hypothesize that such approach can also be applied to improve an autoregressive modeling process by making the model less vulnerable to compounding errors occurred during density estimation. Inspired by the success of randomized smoothing in adversarial defense (Cohen et al., 2019), we propose to apply smoothing to autoregressive modeling to address the problems mentioned above."},{"heading":"3 GENERATIVE MODELS WITH DISTRIBUTION SMOOTHING","text":"In the following, we propose to decompose a density estimation task into a smoothed data modeling problem followed by an inverse smoothing problem where we recover the true data density from the smoothed one."},{"heading":"3.1 RANDOMIZED SMOOTHING PROCESS","text":"Unlike Cohen et al. (2019) where randomized smoothing is applied to a model, we apply smoothing directly to the data distribution pdata(x). To do this, we introduce a smoothing distribution q(x̃|x) — a distribution that is symmetric and stationary (e.g. a Gaussian or Laplacian kernel) — and convolve it with pdata(x) to obtain a new distribution q(x̃) , ∫ q(x̃|x)pdata(x)dx. When q(x̃|x) is a normal distribution, this convolution process is equivalent to perturbing the data distribution with Gaussian\nnoise, which, intuitively, will make the data distribution smoother. In the following, we formally prove that convolving a 1-d distribution pdata(x) with a suitable noise can indeed “smooth” pdata(x). Theorem 1. Given a continuous and bounded 1-d distribution pdata(x) that is supported on R, for any 1-d distribution q(x̃|x) that is symmetric (i.e. q(x̃|x) = q(x|x̃)), stationary (i.e. translation invariant) and satisfies limx→∞ pdata(x)q(x|x̃) = 0 for any given x̃, we have Lip(q(x̃)) ≤ Lip(pdata(x)), where q(x̃) , ∫ q(x̃|x)pdata(x)dx and Lip(·) denotes the Lipschitz constant of the given 1-d function.\nTheorem 1 shows that convolving a 1-d data distribution pdata(x) with a suitable noise distribution q(x̃|x) (e.g.N (x̃|x, σ2)) can reduce the Lipschitzness (i.e. increase the smoothness) of pdata(x). We provide the proof of Theorem 1 in Appendix A.\nGiven pdata(x) with a high Lipschitz constant, we empirically verify that density estimation becomes an easier task on the smoothed distribution q(x̃) than directly on pdata(x). To see this, we visualize a 1-d example in figure 3a, where we want to model a ten-mode data distribution with a mixture of logistics model. If our model has three logistic components, there is almost no way for the model, which only has three modes, to perfectly fit this data distribution, which has ten separate modes with sharp transitions. The model, after training (see figure 3a), mistakenly assigns a much higher density to the low density regions between nearby modes. If we convolve the data distribution with q(x̃|x) = N (x̃|x, 0.52), the new distribution becomes smoother (see figure 3b) and can be captured reasonably well by the same mixture of logistics model with only three modes (see figure 3b). Comparing the same model’s performance on the two density estimation tasks, we can see that the model is doing a better job at modeling the smoothed version of the data distribution than the original data distribution, which has a high Lipschitz constant.\nThis smoothing process can also be understood as a regularization term for the original maximum likelihood objective (on the un-smoothed data distribution), encouraging the learned model to be smooth, as formalized by the following statement: Proposition 1 (Informal). Assume that the symmetric and stationary smoothing distribution q(x̃|x) has small variance and negligible higher order moments, then\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] ≈ Epdata(x) [ log pθ(x) + η\n2 ∑ i ∂2 log pθ ∂x2i\n] ,\nfor some constant η.\nProposition 1 shows that our smoothing process provides a regularization effect on the original objective Epdata(x)[log pθ(x)] when no noise is added, where the regularization aims to maximize η 2 ∑ i ∂2 log pθ ∂x2i\n. Since samples from pdata should be close to a local maximum of the model, this encourages the second order gradients computed at a data point x to become closer to zero (if it were positive then x will not be a local maximum), creating a smoothing effect. This extra term is also the trace of the score function (up to a multiplicative constant) that can be found in the score matching objective (Hyvärinen, 2005), which is closely related to many denoising methods (Vincent, 2011; Hyvärinen, 2008). This regularization effect can, intuitively, increase the generalization capability of the model. In fact, it has been demonstrated empirically that training with noise can lead to improvements in network generalization (Sietsma & Dow, 1991; Bishop, 1995). Our argument is also\nsimilar to that used in (Bishop, 1995) except that we consider a more general generative modeling case as opposed to supervised learning with squared error. We provide the formal statement and proof of Proposition 1 in Appendix A."},{"heading":"3.2 AUTOREGRESSIVE DISTRIBUTION SMOOTHING MODELS","text":"Motivated by the previous 1-d example, instead of directly modeling pdata(x), which can have a high Lipschitz constant, we propose to first train an autoregressive model on the smoothed version of the data distribution q(x̃). Although the smoothing process makes the distribution easier to learn, it also introduces bias. Thus, we need an extra step to debias the learned distribution by reverting the smoothing process.\nIf our goal is to generate approximate samples for pdata(x), when q(x̃|x) = N (x̃|x, σ2I) and σ is small, we can use the gradient of pθ(x̃) for denoising (Alain & Bengio, 2014). More specifically, given smoothed samples x̃ from pθ(x̃), we can “denoise” samples via: x̄ = x̃ + σ2∇x̃ log pθ(x̃), (2) which only requires the knowledge of pθ(x̃) and the ability to sample from it. However, this approach does not provide a likelihood estimate and Eq. (2) only works when q(x̃|x) is Gaussian (though alternative denoising updates for other smoothing processes could be derived under the Empirical Bayes framework (Raphan & Simoncelli, 2011)). Although Eq. (2) could provide reasonable denoising results when the smoothing distribution has a small variance, x̄ obtained in this way is only a point estimation of x̄ = E[x|x̃] and does not capture the uncertainty of p(x|x̃). To invert more general smoothing distributions (beyond Gaussians) and to obtain likelihood estimations, we introduce a second autoregressive model pθ(x|x̃). The parameterized joint density pθ(x, x̃) can then be computed as pθ(x, x̃) = pθ(x|x̃)pθ(x̃). To obtain our approximation of pdata(x), we need to integrate over x̃ on the joint distribution pθ(x, x̃) to obtain pθ(x) =∫ pθ(x, x̃)dx̃, which is in general intractable. However, we can easily obtain an evidence lower bound (ELBO): log pθ(x) ≥ Eq(x̃|x)[log pθ(x̃)]− Eq(x̃|x)[log q(x̃|x)] + Eq(x̃|x)[log pθ(x|x̃)]. (3)\nNote that when q(x̃|x) is fixed, the entropy term Eq(x̃|x)[log q(x̃|x)] is a constant with respect to the optimization parameters. Maximizing ELBO on pdata(x) is then equivalent to maximizing:\nJ(θ) = Epdata(x) [ Eq(x̃|x)[log pθ(x̃)] + Eq(x̃|x)[log pθ(x|x̃)] ] . (4)\nFrom equation 4, we can see that optimizing the two models pθ(x̃) and pθ(x|x̃) separately via maximum likelihood estimation is equivalent to optimizing J(θ)."},{"heading":"3.3 TRADEOFF IN MODELING","text":"In general, there is a trade-off between the difficulty of modeling pθ(x̃) and pθ(x|x̃). To see this, let us consider two extreme cases for the variance of q(x̃|x) — when q(x̃|x) has a zero variance and an infinite variance. When q(x̃|x) has a zero variance, q(x̃|x) is a distribution with all its probability mass at x, meaning that no noise is added to the data distribution. In this case, modeling the smoothed distribution would be equivalent to modeling pdata(x), which can be hard as discussed above. The reverse smoothing process, however, would be easy since pθ(x|x̃) can simply be an identity map to perfectly invert the smoothing process. In the second case when q(x̃|x) has an infinite variance, modeling p(x̃) would be easy because all the information about the original data is lost, and p(x̃) would be close to the smoothing distribution. Modeling p(x|x̃), on the other hand, is equivalent to directly modeling pdata(x), which can be challenging.\nThus, the key here is to appropriately choose a smoothing level so that both q(x̃) and p(x|x̃) can be approximated relatively well by existing autoregressive models. In general, the optimal variance might be hard to find. Although one can train q(x̃|x) by jointly optimizing ELBO, in practice, we find this approach often assigns a very large variance to q(x̃|x), which can trade-off sample quality for better likelihoods on high dimensional image datasets. We find empirically that a pre-specified q(x̃|x) chosen by heuristics (Saremi & Hyvarinen, 2019; Garreau et al., 2017) is able to generate much better samples than training q(x̃|x) via ELBO. In this paper, we will focus on the sample quality and leave the training of q(x̃|x) for future work."},{"heading":"4 EXPERIMENTS","text":"In this section, we demonstrate empirically that by appropriately choosing the smoothness level of randomized smoothing, our approach is able to drastically improve the sample quality of existing autoregressive models on several synthetic and real-world datasets while retaining competitive likelihoods on synthetic datasets. We also present results on image inpainting in Appendix C.2."},{"heading":"4.1 CHOOSING THE SMOOTHING DISTRIBUTION","text":"To help us build insights into the selection of the smoothing distribution q(x̃|x), we first focus on a 1-d multi-modal distribution (see figure 4 leftmost panel). We use model-based methods to invert the smoothed distribution and provide analysis on “single-step denoising” in Appendix B.1. We start with the exploration of three different types of smoothing distributions – Gaussian distribution, Laplace distribution, and uniform distribution. For each type of distribution, we perform a grid search to find the optimal variance. Since our approach requires the modeling of both pθ(x̃) and pθ(x|x̃), we stack x̃ and x together, and use a MADE model (Germain et al., 2015) with a mixture of two logistic components to parameterize pθ(x̃) and pθ(x|x̃) at the same time. For the baseline model, we train a mixture of logistics model directly on pdata(x). We compare the results in the middle two panels in figure 4.\nWe find that although the baseline with eight logistic components has the capacity to perfectly model the multi-modal data distribution, which has six modes, the baseline model still fails to do so. We believe this can be caused by optimization or initialization issues for modeling a distribution with a high Lipschitz constant. Our method, on the other hand, demonstrates more robustness by successfully modeling the different modes in the data distribution even when using only two mixture components for both pθ(x̃) and pθ(x|x̃). For all the three types of smoothing distributions, we observe a reverse U-shape correlation between the variance of q(x̃|x) and ELBO values — with ELBO first increasing as the variance increases and then decreasing as the variance grows beyond a certain point. The results match our discussion on the trade-off between modeling pθ(x̃) and pθ(x|x̃) in Section 3.3. We notice from the empirical results that Gaussian smoothing is able to obtain better ELBO than the other two distributions. Thus, we will use q(x̃|x) = N (x̃|x, σ2I) for the later experiments."},{"heading":"4.2 2-D SYNTHETIC DATASETS","text":"In this section, we consider two challenging 2-d multi-modal synthetic datasets (see figure 5). We focus on model-based denoising methods and present discussion on “single-step denoising” in Appendix B.2. We use a MADE model with comparable number of total parameters for both the baseline and our approach. For the baseline, we train the MADE model directly on the data. For\nour randomized smoothing model, we choose q(x̃|x) = N (x̃|x, 0.32I) to be the smoothing distribution. We observe that with this randomized smoothing approach, our model is able to generate better samples than the baseline (according to a human observer) even when using less logistic components (see figure 5). We provide more analysis on the model’s performance in Appendix B.2. We also provide the negative log-likelihoods in Tab. 1."},{"heading":"4.3 IMAGE EXPERIMENTS","text":"In this section, we focus on three common image datasets, namely MNIST, CIFAR-10 (Krizhevsky et al., 2009) and CelebA (Liu et al., 2015). We select q(x̃|x) = N (x̃|x, σ2I) to be the smoothing distribution. We use PixelCNN++ (Salimans et al., 2017) as the model architecture for both pθ(x̃) and pθ(x|x̃). We provide more details about settings in Appendix C.\nImage generation. For image datasets, we select the σ of q(x̃|x) = N (x̃|x, σ2I) according to analysis in (Saremi & Hyvarinen, 2019) (see Appendix C for more details). Since q(x̃|x) is a Gaussian distribution, we can apply “single-step denoising” to reverse the smoothing process for samples drawn from pθ(x̃). In this case, the model pθ(x|x̃) is not required for sampling since the gradient of pθ(x̃) can be used to denoise samples (also from pθ(x̃)) (see equation 2). We present smoothed samples from pθ(x̃), reversed smoothing samples processed by “single-step denoising” and processed by pθ(x|x̃) in figure 6. For comparison, we also present samples from a PixelCNN++ with parameters comparable to the sum of total parameters of pθ(x̃) and pθ(x|x̃). We find that by using this randomized smoothing approach, we are able to drastically improve the sample quality of PixelCNN++ (see the rightmost panel in figure 6). We note that with only pθ(x̃), a PixelCNN++ optimized on the smoothed data, we already obtain more realistic samples compared to the original PixelCNN++ method. However, pθ(x|x̃) is needed to compute the likelihood lower bounds. We report the sample quality evaluated by Fenchel Inception Distance (FID (Heusel et al., 2017)), Kernel Inception Distance (KID (Bińkowski et al., 2018)), and Inception scores (Salimans et al., 2016) in Tab. 2. Although our method obtains better samples compared to the original PixelCNN++, our model has worse likelihoods as evaluated in BPDs. We believe this is because likelihood and sample quality are not always directly correlated as discussed in Theis et al. (2015). We also tried training the variance for q(x̃|x) by jointly optimizing ELBO. Although training the variance can produce better likelihoods, it does not generate samples with comparable quality as our method (i.e. choosing variance by heuristics). Thus, it is hard to conclusively determine what is the best way of choosing q(x̃|x). We provide more image samples in Appendix C.4 and nearest neighbors analysis in Appendix C.5."},{"heading":"5 ADDITIONAL EXPERIMENTS ON NORMALIZING FLOWS","text":"In this section, we demonstrate empirically on 2-d synthetic datasets that randomized smoothing techniques can also be applied to improve the sample quality of normalizing flow models (Rezende & Mohamed, 2015). We focus on RealNVP (Dinh et al., 2016). We compare the RealNVP model trained with randomized smoothing, where we use pθ(x|x̃) (also a RealNVP) to revert the smoothing process, with a RealNVP trained with the original method but with comparable number of parameters. We observe that smoothing is able to improve sample quality on the datasets we consider (see figure 7) while also obtaining competitive likelihoods. On the checkerboard dataset, our method has negative log-likelihoods 3.64 while the original RealNVP has 3.72; on the Olympics dataset, our method has negative log-likelihoods 1.32 while the original RealNVP has 1.80. This example demonstrates that randomized smoothing techniques can also be applied to normalizing flow models."},{"heading":"6 RELATED WORK","text":"Our approach shares some similarities with denoising autoencoders (DAE, Vincent et al. (2008)) which recovers a clean observation from a corrupted one. However, unlike DAE which has a train-\nable encoder and a fixed prior distribution, our approach fixes the encoder and models the prior using an autoregressive model. Generative stochastic networks (GSN, Bengio et al. (2014)) use DAEs to train a Markov chain whose equilibrium distribution matches the data distribution. However, GSN needs to start the chain from a sample that is very close to the training distribution. Denoising diffusion model (Sohl-Dickstein et al., 2015; Ho et al., 2020) and NCSN (Song & Ermon (2019; 2020)) address the issue of GSNs by considering a sequence of distributions corresponding to data corrupted with various noise levels. By setting multiple noise levels that are close to each other, the sample from the previous level can serve as a proper initialization for the sampling process at the next level. This way, the model can start from a distribution that is easy to model and gradually move to the desired distribution. However, due to the large number of noise levels, such approaches require many steps for the chain to converge to the right data distribution.\nIn this paper, we instead propose to use only one level of smoothing by modeling each step with a powerful autoregressive model instead of deterministic autoencoders. Motivated by the success of “randomized smoothing” techniques in adversarial defense (Cohen et al., 2019), we perform randomized smoothing directly on the data distribution. Unlike denoising score matching (Vincent, 2011), a technique closely related to denoising diffusion models and NCSN, which requires the perturbed noise to be a Gaussian distribution, we are able to work with different noise distributions.\nOur smoothing method is also relevant to “dequantization” approaches that are common in normalizing flow models, where the discrete data distribution is converted to a continuous one by adding continuous noise (Uria et al., 2013; Ho et al., 2019). However the added noise for “dequantization” in flows is often indistinguishable to human eyes, and the reverse “dequantization” process is often ignored. In contrast, we consider noise scales that are significantly larger and thus a denoising process is required.\nOur method is also related to “quantization” approaches which reduce the number of “significant” bits that are modeled by a generative model (Kingma & Dhariwal, 2018; Menick & Kalchbrenner, 2018). For instance, Glow (Kingma & Dhariwal, 2018) only models the 5 most significant bits of an image, which improves the visual quality of samples but decreases color fidelity. SPN (Menick & Kalchbrenner, 2018) introduces another network to predict the remaining bits conditioned on the 3 most significant bits already modeled. Modeling the most significant bits can be understood as capturing a data distribution perturbed by bit-wise correlated noise, similar to modeling smoothed data in our method. Modeling the remaining bits conditioned on the most significant ones in SPN is then similar to denoising. However, unlike these quantization approaches which process an image at the “significant” bits level, we apply continuous data independent Gaussian noise to the entire image with a different motivation to smooth the data density function."},{"heading":"7 DISCUSSION","text":"In this paper, we propose to incorporate randomized smoothing techniques into autoregressive modeling. By choosing the smoothness level appropriately, this seemingly simple approach is able to drastically improve the sample quality of existing autoregressive models on several synthetic and real-world datasets while retaining reasonable likelihoods. Our work provides insights into how recent adversarial defense techniques can be leveraged to building more robust generative models. Since we apply randomized smoothing technique directly to the target data distribution other than the model, we believe our approach is also applicable to other generative models such as variational autoencoders (VAEs) and generative adversarial networks (GANs)."},{"heading":"ACKNOWLEDGEMENTS","text":"The authors would like to thank Kristy Choi for reviewing the draft of the paper. This research was supported by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1-2145), AFOSR (FA9550-19-1-0024), ARO, and Amazon AWS."},{"heading":"A PROOFS","text":"Theorem 1. Given a continuous and bounded 1-d distribution pdata(x) that is supported on R, for any 1-d distribution q(x̃|x) that is symmetric (i.e. q(x̃|x) = q(x|x̃)), stationary (i.e. translation invariant) and satisfies limx→∞ pdata(x)q(x|x̃) = 0 for any given x̃, we have Lip(q(x̃)) ≤ Lip(pdata(x)), where q(x̃) , ∫ q(x̃|x)pdata(x)dx and Lip(·) denotes the Lipschitz constant of the given 1-d function.\nProof. First, we have that:\n|∇xp(x)| = |p(x)∇x log p(x)| ≤ Lip(p) (5) and if we assume symmetry, i.e. q(x|x̃) = q(x̃|x) then by integration by parts we have:\n∇x̃q(x̃) = Ep(x)[∇x̃q(x̃|x)] = Ep(x)[∇xq(x|x̃)] = −Ep(x)[q(x|x̃)∇x log p(x)] (6) Therefore,\nLip(q) = max x̃ |−Ep(x)[q(x|x̃)∇x log p(x)]| = max x̃ | ∑ x q(x|x̃)p(x)∇x log p(x)| (7)\n≤ max x̃ | ∑ x q(x|x̃)Lip(p)| = Lip(p) max x̃ | ∑ x q(x|x̃)| = Lip(p) (8)\nwhich proves the result.\nProposition 1 (Formal). Given a D-dimensional data distribution pdata(x) and model distribution pθ(x), assume that the smoothing distribution q(x̃|x) satisfies:\n• log pθ is infinitely differentiable on the support of pθ(x)\n• q(x̃|x) is symmetric (i.e. q(x̃|x) = q(x|x̃))\n• q(x̃|x) is stationary (i.e. translation invariant)\n• q(x̃|x) is bounded and fully supported on RD\n• q(x̃|x) is element-wise independent\n• Eq(x̃|x)[(x̃− x)2] is bounded, and Eq(x̃|x)[(x̃i − xi)2] = η at each dimension i.\nDenote = x̃− x, then\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] = Epdata(x) [ log pθ(x) + η\n2 ∑ i ∂2 log pθ ∂x2i\n] + ∫ ∫ o( 2)pdata(x)p( )dxd ,\nwhere o( 2) : RD → R is a function of such that lim →0 o( 2) 2 = 0. Thus when∫ ∫\no( 2)pdata(x)p( )dxd → 0, we have\nEpdata(x)Eq(x̃|x)[log pθ(x̃)]→ Epdata(x) [ log pθ(x) + η\n2 ∑ i ∂2 log pθ ∂x2i\n] . (2)\nProof. To see this, we first note that the new training objective for the smoothed data distribution is Epdata(x)Eq(x̃|x)[log pθ(x̃)]. Let = x̃ − x, because of the assumptions we have, the PDF function q(x̃|x) can be reparameterized as p( ) which satisfies: p is bounded and fully supported on RD; p is element-wise independent and Ep( )[ 2i ] = Ep( i)[ 2i ] = η at each dimension i (i = 1, ..., D). Then we have\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] = ∫ ∫ log pθ(x + )pdata(x)p( )dxd , (9)\nUsing Taylor expansion, we have:\nlog pθ(x + ) = log pθ(x) + ∑ i i ∂ log pθ ∂xi + 1 2 ∑ i,j i j ∂2 log pθ ∂xi∂xj + o( 2).\nSince is independent of x and∫ ∞ −∞ id i = 0, ∫ ∞ −∞ ∫ ∞ −∞ i jd id j = δi,jη,\nwhere δi,j is the Kronecker delta function, the right hand side of Equation 9 becomes Epdata(x)[log pθ(x)] + ∫ ∫ ( 1\n2 ∑ i 2i ∂2 log pθ ∂x2i + o( 2)\n) pdata(x)p( )dxd (10)\n= Epdata(x) [ log pθ(x) + η\n2 ∑ i ∂2 log pθ ∂x2i\n] + ∫ ∫ o( 2)pdata(x)p( )dxd . (11)\nWhen ∫ ∫ o( 2)pdata(x)p( )dxd → 0, (12)\nwe have\nEpdata(x)Eq(x̃|x)[log pθ(x̃)]→ Epdata(x) [ log pθ(x) + η\n2 ∑ i ∂2 log pθ ∂x2i\n] . (13)\nwhere the second term on the right hand side serves as a regularization for the original objective Epdata(x)[log pθ(x)]."},{"heading":"B DENOISING EXPERIMENTS","text":"B.1 ANALYSIS ON 1-D DENOISING\nTo provide more insights into denoising, we first study “single-step denoising” (see equation 2) on a 1-d dataset. We choose the data distribution to be a two mixture of Gaussian distribution 0.5N (−0.3, 0.12) + 0.5N (0.3, 0.12) and the smoothing distribution to be q(x̃|x) = N (x̃|x, 0.32) (see figure 8a). Since the convolution of two Gaussian distributions is also a Gaussian distribution, the smoothed data is a mixture of Gaussian distribution given by 0.5N (−0.3, 0.12 + 0.32) + 0.5N (0.3, 0.12 + 0.32). The ground truth of ∇x̃ log p(x̃) can then be calculated in closed form. Thus, given the smoothed data x̃, we can calculate the ground truth ∇x̃ log p(x̃) in equation 2 and obtain x̄ using “single-step denoising”. We visualize the denoising results in figure 8b. We find that the low density region between the two modes in pdata(x) are not modeled properly in figure 8b. However, this is very expected since “single-step denosing” uses x̄ = E[x|x̃] as the substitute for the denoised result. When the smoothing distribution has a large variance (like in figure 8a where the smoothed data has merged into a one mode distribution), datapoints like x̃0 in the middle low density region of pdata(x) can have high density in the smoothed distribution. Since x̃0, as well as other points in the middle low density region of pdata(x), can come from both modes of pdata(x) with high probability before the smoothing process (see figure 11a), the denoised x̄ = E[x|x̃ = x̃0] can still be located in the middle low density region (see figure 8b). Since a large proportion of the smoothed data is located in the middle low density region of pdata(x), we would expect certain proportion of the density to remain in the low density region after “single-step denoising” just as shown in figure 8b. However, when the smoothing distribution has a smaller variance, “single-step denoising” can achieve much better denoising results (see figure 9, where we use q(x̃|x) = N (x̃|x, 0.12)). Although denoising can be easier when the smoothing distribution has a smaller variance, modeling the smoothed distribution could be harder as we discussed before.\nIn general, the right denoising results should be samples coming from p(x|x̃), which is the reason why samples from pθ(x|x̃) (i.e. introducing the model pθ(x|x̃)) is more ideal than using Eθ[x|x̃] as a denoising substitute (i.e. “single-step denoising”). In general, the capacity of the denoising model pθ(x|x̃) also matters in terms of denoising results. Let us again consider the datapoint x̃0 shown in figure 8a. If the invert smoothing model pθ(x|x̃) is a one mode logistic distribution, due to the mode covering property of maximum likelihood estimation, given the smoothed observation x̃0, the best the model can do is to center its only mode at x̃0 for approximating p(x|x̃ = x̃0) (see figure 10c). Thus, like x̃0, the smoothed datapoints at the low density region between the two modes of pdata(x)\nare still likely to remain between the two modes after denoising (see figure 10b). To solve this issue, we can increase the capacity of pθ(x|x̃) by making it a two mixture of logistics. In this case, the distribution pθ(x|x̃ = x̃0) can be captured in a better way (see figure 11c and figure 11a). After the invert smoothing process, like x̃0, most smoothed datapoints in the low density can be mapped to one of the two high density modes (see figure 11b), resulting in much better denoising effects.\nB.2 ANALYSIS ON 2-D DENOISING\nOn the 2-d Olympics dataset in section 4.2, we find that the intersections between rings can be poorly modeled with the proposed smoothing approach when only two mixture of logistics are used (see figure 12e). We believe this can be caused if the denoising model is not flexible enough to capture the distribution p(x|x̃). More specifically, we note that the ground truth distribution for p(x|x̃) at the intersections of the rings is a highly complicated distribution and can be hard to capture using our model which only has two mixtures of logistics for each dimension. If we increase the flexibility of pθ(x|x̃) by using three or four mixtures of logistics components (note that we still use fewer mixture components than the MADE baseline and we use comparable number of parameters), the intersection of the rings can be modeled in an improved way (see figure 12).\nWe also provide “single-step denoising” results for the experiments in Section 4.2 (see figure 13), where we use the same smoothing distribution, and the MADE model with three mixture components as used in section 4.2. We note that “single-step denoising” results are not very good, which is also expected. As discussed in section B.1, when the smoothing distribution has a relatively large variance, Eθ[x|x̃] is not a good approximation for the denoised result, and we want the denoised sample to come from the distribution pθ(x|x̃), in which case introducing a denoising model pθ(x|x̃) could be a better option. Although we could select q(x̃|x) to have a smaller variance so that “single-step denoing” could work reasonably well, but modeling p(x̃) in this case could be more challenging.\nC IMAGE EXPERIMENTS\nC.1 SETTINGS\nFor the image experiments, we first rescale images to [−1, 1] and then perturb the images with q(x̃|x) = N (x̃|x, σ2I). We use σ = 0.5 for MNIST and σ = 0.3 for both CIFAR-10 and CelebA. The selection of σ is mainly based on analysis in (Saremi & Hyvarinen, 2019). More specifically, given an image, we consider the median value of the Euclidean distance between two data points in a dataset, and then divide it by 2 √ D, where D is the dimension of the data. This provides us with a way of selecting the variance of q(x̃|x), when q(x̃|x) is a Gaussian distribution. We find this selection of variance able to generate reasonably well samples in practice. We train all the models with Adam optimizer with learning rate 0.0002. To model pθ(x|x̃), we stack x̃ and x together at the second dimension to obtain x̂ = [x̃,x], which ensures that x̃ comes before x in the pixel ordering. For instance, this stacking would provide an image x̂ with size 1 × (2 × 28) × 28 on a MNIST image, and an image with size 3 × (2 × 32) × 32 on a CIFAR-10 image. Since PixelCNN++ consists of convolutional layers, we can directly feed x̂ into the default architecture without modifying the model architecture. As the latter pixels of the input only depend on the previous pixels in an autoregressive model and x̃ comes before x, we can parameterize pθ(x|x̃) by computing the likelihoods only on x using the outputs from the autoregressive model.\nC.2 IMAGE INPAINTING\nSince both pθ(x̃) and pθ(x|x̃) are parameterized by an autoregressive model, we can also perform image inpainting using our method. We present the inpainting results on CIFAR-10 in figure 14a and CelebA in figure 14b, where the bottom half of the input image is being inpainted.\nC.3 IMAGE DENOISING\nWe notice that the reverse smoothing process can also be understood as a denoising process. Besides the “single-step denoising” approach shown above, we can also apply pθ(x|x̃) to denoise images. To visualize the denoising performance, we sample xtest from the test set and perturb xtest with q(x̃|x) to obtain a noisy sample x̃test. We feed x̃test into pθ(x|x̃ = x̃test) and draw samples from the model. We visualize the results in figure 15. As we can see, the model exhibits reasonable denoising results, which shows that the autoregressive model is capable of learning the data distribution when conditioned on the smoothed data.\nC.4 MORE SAMPLES\nC.5 NEAREST NEIGHBORS\nC.6 ABLATION STUDIES\nIn this section, we show that gradient-based “single-step denoising” will not improve sample qualities without performing randomized smoothing. To see this, we draw samples from a PixelCNN++ pθ(x) trained directly on pdata(x) (i.e. without smoothing). We perform “single-step denoising” update defined as x = x + σ2∇x log pθ(x). (14) We explore various values for σ, and report the results in figure 26. This shows that “single-step denoising” alone (without randomized smoothing) will not improve sample quality of PixelCNN++."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Autoregressive models have exhibited promising results in a variety of downstream tasks. For instance, they have shown success in compressing images (Minnen et al., 2018), synthesizing speech (Oord et al., 2016a) and modeling complex decision rules in games (Vinyals et al., 2019). However, the sample quality of autoregressive models on real-world image datasets is still lacking.\\nPoor sample quality might be explained by the manifold hypothesis: many real world data distributions (e.g. natural images) lie in the vicinity of a low-dimensional manifold (Belkin & Niyogi, 2003), leading to complicated densities with sharp transitions (i.e. high Lipschitz constants), which are known to be difficult to model for density models such as normalizing flows (Cornish et al., 2019). Since each conditional of an autoregressive model is a 1-dimensional normalizing flow (given a fixed context of previous pixels), a high Lipschitz constant will likely hinder learning of autoregressive models.\\nAnother reason for poor sample quality is the “compounding error” issue in autoregressive modeling. To see this, we note that an autoregressive model relies on the previously generated context to make a prediction; once a mistake is made, the model is likely to make another mistake which compounds (Kääriäinen, 2006), eventually resulting in questionable and unrealistic samples. Intuitively, one would expect the model to assign low-likelihoods to such unrealistic images, however, this is not always the case. In fact, the generated samples, although appearing unrealistic, often are assigned high-likelihoods by the autoregressive model, resembling an “adversarial example” (Szegedy et al., 2013; Biggio et al., 2013), an input that causes the model to output an incorrect answer with high confidence.\\nInspired by the recent success of randomized smoothing techniques in adversarial defense (Cohen et al., 2019), we propose to apply randomized smoothing to autoregressive generative modeling. More specifically, we propose to address a density estimation problem via a two-stage process. Unlike Cohen et al. (2019) which applies smoothing to the model to make it more robust, we apply smoothing to the data distribution. Specifically, we convolve a symmetric and stationary noise distribution with the data distribution to obtain a new “smoother” distribution. In the first stage, we model the smoothed version of the data distribution using an autoregressive model. In the second stage, we reverse the smoothing process—a procedure which can also be understood as “denoising”—by either applying a gradient-based denoising approach (Alain & Bengio, 2014) or introducing another conditional autoregressive model to recover the original data distribution from the smoothed one. By choosing an appropriate smoothing distribution, we aim to make each step easier than the original learning problem: smoothing facilitates learning in the first stage by making the input distribution\\nfully supported without sharp transitions in the density function; generating a sample given a noisy one is easier than generating a sample from scratch.\\nWe show with extensive experimental results that our approach is able to drastically improve the sample quality of current autoregressive models on several synthetic datasets and real-world image datasets, while obtaining competitive likelihoods on synthetic datasets. We empirically demonstrate that our method can also be applied to density estimation, image inpainting, and image denoising.\"\n },\n {\n \"heading\": \"2 BACKGROUND\",\n \"text\": \"We consider a density estimation problem. Given D-dimensional i.i.d samples {x1,x2, ...,xN} from a continuous data distribution pdata(x), the goal is to approximate pdata(x) with a model pθ(x) parameterized by θ. A commonly used approach for density estimation is maximum likelihood estimation (MLE), where the objective is to maximize L(θ) , 1N ∑N i=1 log pθ(xi).\"\n },\n {\n \"heading\": \"2.1 AUTOREGRESSIVE MODELS\",\n \"text\": \"An autoregressive model (Larochelle & Murray, 2011; Salimans et al., 2017) decomposes a joint distribution pθ(x) into the product of univariate conditionals:\\npθ(x) = D∏ i=1 pθ(xi|x<i), (1)\\nwhere xi stands for the i-th component of x, and x<i refers to the components with indices smaller than i. In general, an autoregressive model parameterizes each conditional pθ(xi|x<i) using a prespecified density function (e.g. mixture of logistics). This bounds the capacity of the model by limiting the number of modes for each conditional.\\nAlthough autoregressive models have achieved top likelihoods amongst all types of density based models, their sample quality is still lacking compared to energy-based models (Du & Mordatch, 2019) and score-based models (Song & Ermon, 2019). We believe this can be caused by the following two reasons.\"\n },\n {\n \"heading\": \"2.2 MANIFOLD HYPOTHESIS\",\n \"text\": \"Several existing methods (Roweis & Saul, 2000; Tenenbaum et al., 2000) rely on the manifold hypothesis, i.e. that real-world high-dimensional data tends to lie on a low-dimensional manifold (Narayanan & Mitter, 2010). If the manifold hypothesis is true, then the density of the data distribution is not well defined in the ambient space; if the manifold hypothesis holds only approximately and the data lies in the vicinity of a manifold, then only points that are very close to the manifold would have high density, while all other points would have close to zero density. Thus we may expect the data density around the manifold to have large first-order derivatives, i.e. the density function has a high Lipschitz constant (if not infinity).\\nTo see this, let us consider a 2-d example where the data distribution is a thin ring distribution (almost a unit circle) formed by rotating the 1-d Gaussian distribution N (1, 0.012) around the origin. The density function of the ring has a high Lipschitz constant near the “boundary”. Let us focus on a data point travelling along the diagonal as shown in the leftmost panel in figure 2. We plot the first-order\\ndirectional derivatives of the density for the point as it approaches the boundary from the inside, then lands on the ring, and finally moves outside the ring (see figure 2). As we can see, when the point is far from the boundary, the derivative has a small magnitude. When the point moves closer to the boundary, the magnitude increases and changes significantly near the boundary even with small displacements in the trajectory. However, once the point has landed on the ring, the magnitude starts to decrease. As it gradually moves off the ring, the magnitude first increases and then decreases just like when the point approached the boundary from the inside. It has been observed that certain likelihood models, such as normalizing flows, exhibit pathological behaviors on data distributions whose densities have high Lipschitz constants (Cornish et al., 2019). Since each conditional of an autoregressive model is a 1-d normalizing flow given a fixed context, a high Lipschitz constant on data density could also hinder learning of autoregressive models.\"\n },\n {\n \"heading\": \"2.3 COMPOUNDING ERRORS IN AUTOREGRESSIVE MODELING\",\n \"text\": \"Autoregressive models can also be susceptible to compounding errors from the conditional distributions (Lamb et al., 2016) during sampling time. We notice that an autoregressive model pθ(x) learns the joint density pdata(x) by matching each of the conditional pθ(xi|x<i) with pdata(xi|x<i). In practice, we typically have access to a limited amount of training data, which makes it hard for an autoregressive model to capture all the conditional distributions correctly due to the curse of dimensionality. During sampling, since a prediction is made based on the previously generated context, once a mistake is made at a previous step, the model is likely to make more mistakes in the later steps, eventually generating a sample x̂ that is far from being an actual image, but is mistakenly assigned a high-likelihood by the model.\\nThe generated image x̂, being unrealistic but assigned a high-likelihood, resembles an adversarial example, i.e., an input that causes the model to make mistakes. Recent works (Cohen et al., 2019) in adversarial defense have shown that random noise can be used to improve the model’s robustness to adversarial perturbations — a process during which adversarial examples that are close to actual data are generated to fool the model. We hypothesize that such approach can also be applied to improve an autoregressive modeling process by making the model less vulnerable to compounding errors occurred during density estimation. Inspired by the success of randomized smoothing in adversarial defense (Cohen et al., 2019), we propose to apply smoothing to autoregressive modeling to address the problems mentioned above.\"\n },\n {\n \"heading\": \"3 GENERATIVE MODELS WITH DISTRIBUTION SMOOTHING\",\n \"text\": \"In the following, we propose to decompose a density estimation task into a smoothed data modeling problem followed by an inverse smoothing problem where we recover the true data density from the smoothed one.\"\n },\n {\n \"heading\": \"3.1 RANDOMIZED SMOOTHING PROCESS\",\n \"text\": \"Unlike Cohen et al. (2019) where randomized smoothing is applied to a model, we apply smoothing directly to the data distribution pdata(x). To do this, we introduce a smoothing distribution q(x̃|x) — a distribution that is symmetric and stationary (e.g. a Gaussian or Laplacian kernel) — and convolve it with pdata(x) to obtain a new distribution q(x̃) , ∫ q(x̃|x)pdata(x)dx. When q(x̃|x) is a normal distribution, this convolution process is equivalent to perturbing the data distribution with Gaussian\\nnoise, which, intuitively, will make the data distribution smoother. In the following, we formally prove that convolving a 1-d distribution pdata(x) with a suitable noise can indeed “smooth” pdata(x). Theorem 1. Given a continuous and bounded 1-d distribution pdata(x) that is supported on R, for any 1-d distribution q(x̃|x) that is symmetric (i.e. q(x̃|x) = q(x|x̃)), stationary (i.e. translation invariant) and satisfies limx→∞ pdata(x)q(x|x̃) = 0 for any given x̃, we have Lip(q(x̃)) ≤ Lip(pdata(x)), where q(x̃) , ∫ q(x̃|x)pdata(x)dx and Lip(·) denotes the Lipschitz constant of the given 1-d function.\\nTheorem 1 shows that convolving a 1-d data distribution pdata(x) with a suitable noise distribution q(x̃|x) (e.g.N (x̃|x, σ2)) can reduce the Lipschitzness (i.e. increase the smoothness) of pdata(x). We provide the proof of Theorem 1 in Appendix A.\\nGiven pdata(x) with a high Lipschitz constant, we empirically verify that density estimation becomes an easier task on the smoothed distribution q(x̃) than directly on pdata(x). To see this, we visualize a 1-d example in figure 3a, where we want to model a ten-mode data distribution with a mixture of logistics model. If our model has three logistic components, there is almost no way for the model, which only has three modes, to perfectly fit this data distribution, which has ten separate modes with sharp transitions. The model, after training (see figure 3a), mistakenly assigns a much higher density to the low density regions between nearby modes. If we convolve the data distribution with q(x̃|x) = N (x̃|x, 0.52), the new distribution becomes smoother (see figure 3b) and can be captured reasonably well by the same mixture of logistics model with only three modes (see figure 3b). Comparing the same model’s performance on the two density estimation tasks, we can see that the model is doing a better job at modeling the smoothed version of the data distribution than the original data distribution, which has a high Lipschitz constant.\\nThis smoothing process can also be understood as a regularization term for the original maximum likelihood objective (on the un-smoothed data distribution), encouraging the learned model to be smooth, as formalized by the following statement: Proposition 1 (Informal). Assume that the symmetric and stationary smoothing distribution q(x̃|x) has small variance and negligible higher order moments, then\\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] ≈ Epdata(x) [ log pθ(x) + η\\n2 ∑ i ∂2 log pθ ∂x2i\\n] ,\\nfor some constant η.\\nProposition 1 shows that our smoothing process provides a regularization effect on the original objective Epdata(x)[log pθ(x)] when no noise is added, where the regularization aims to maximize η 2 ∑ i ∂2 log pθ ∂x2i\\n. Since samples from pdata should be close to a local maximum of the model, this encourages the second order gradients computed at a data point x to become closer to zero (if it were positive then x will not be a local maximum), creating a smoothing effect. This extra term is also the trace of the score function (up to a multiplicative constant) that can be found in the score matching objective (Hyvärinen, 2005), which is closely related to many denoising methods (Vincent, 2011; Hyvärinen, 2008). This regularization effect can, intuitively, increase the generalization capability of the model. In fact, it has been demonstrated empirically that training with noise can lead to improvements in network generalization (Sietsma & Dow, 1991; Bishop, 1995). Our argument is also\\nsimilar to that used in (Bishop, 1995) except that we consider a more general generative modeling case as opposed to supervised learning with squared error. We provide the formal statement and proof of Proposition 1 in Appendix A.\"\n },\n {\n \"heading\": \"3.2 AUTOREGRESSIVE DISTRIBUTION SMOOTHING MODELS\",\n \"text\": \"Motivated by the previous 1-d example, instead of directly modeling pdata(x), which can have a high Lipschitz constant, we propose to first train an autoregressive model on the smoothed version of the data distribution q(x̃). Although the smoothing process makes the distribution easier to learn, it also introduces bias. Thus, we need an extra step to debias the learned distribution by reverting the smoothing process.\\nIf our goal is to generate approximate samples for pdata(x), when q(x̃|x) = N (x̃|x, σ2I) and σ is small, we can use the gradient of pθ(x̃) for denoising (Alain & Bengio, 2014). More specifically, given smoothed samples x̃ from pθ(x̃), we can “denoise” samples via: x̄ = x̃ + σ2∇x̃ log pθ(x̃), (2) which only requires the knowledge of pθ(x̃) and the ability to sample from it. However, this approach does not provide a likelihood estimate and Eq. (2) only works when q(x̃|x) is Gaussian (though alternative denoising updates for other smoothing processes could be derived under the Empirical Bayes framework (Raphan & Simoncelli, 2011)). Although Eq. (2) could provide reasonable denoising results when the smoothing distribution has a small variance, x̄ obtained in this way is only a point estimation of x̄ = E[x|x̃] and does not capture the uncertainty of p(x|x̃). To invert more general smoothing distributions (beyond Gaussians) and to obtain likelihood estimations, we introduce a second autoregressive model pθ(x|x̃). The parameterized joint density pθ(x, x̃) can then be computed as pθ(x, x̃) = pθ(x|x̃)pθ(x̃). To obtain our approximation of pdata(x), we need to integrate over x̃ on the joint distribution pθ(x, x̃) to obtain pθ(x) =∫ pθ(x, x̃)dx̃, which is in general intractable. However, we can easily obtain an evidence lower bound (ELBO): log pθ(x) ≥ Eq(x̃|x)[log pθ(x̃)]− Eq(x̃|x)[log q(x̃|x)] + Eq(x̃|x)[log pθ(x|x̃)]. (3)\\nNote that when q(x̃|x) is fixed, the entropy term Eq(x̃|x)[log q(x̃|x)] is a constant with respect to the optimization parameters. Maximizing ELBO on pdata(x) is then equivalent to maximizing:\\nJ(θ) = Epdata(x) [ Eq(x̃|x)[log pθ(x̃)] + Eq(x̃|x)[log pθ(x|x̃)] ] . (4)\\nFrom equation 4, we can see that optimizing the two models pθ(x̃) and pθ(x|x̃) separately via maximum likelihood estimation is equivalent to optimizing J(θ).\"\n },\n {\n \"heading\": \"3.3 TRADEOFF IN MODELING\",\n \"text\": \"In general, there is a trade-off between the difficulty of modeling pθ(x̃) and pθ(x|x̃). To see this, let us consider two extreme cases for the variance of q(x̃|x) — when q(x̃|x) has a zero variance and an infinite variance. When q(x̃|x) has a zero variance, q(x̃|x) is a distribution with all its probability mass at x, meaning that no noise is added to the data distribution. In this case, modeling the smoothed distribution would be equivalent to modeling pdata(x), which can be hard as discussed above. The reverse smoothing process, however, would be easy since pθ(x|x̃) can simply be an identity map to perfectly invert the smoothing process. In the second case when q(x̃|x) has an infinite variance, modeling p(x̃) would be easy because all the information about the original data is lost, and p(x̃) would be close to the smoothing distribution. Modeling p(x|x̃), on the other hand, is equivalent to directly modeling pdata(x), which can be challenging.\\nThus, the key here is to appropriately choose a smoothing level so that both q(x̃) and p(x|x̃) can be approximated relatively well by existing autoregressive models. In general, the optimal variance might be hard to find. Although one can train q(x̃|x) by jointly optimizing ELBO, in practice, we find this approach often assigns a very large variance to q(x̃|x), which can trade-off sample quality for better likelihoods on high dimensional image datasets. We find empirically that a pre-specified q(x̃|x) chosen by heuristics (Saremi & Hyvarinen, 2019; Garreau et al., 2017) is able to generate much better samples than training q(x̃|x) via ELBO. In this paper, we will focus on the sample quality and leave the training of q(x̃|x) for future work.\"\n },\n {\n \"heading\": \"4 EXPERIMENTS\",\n \"text\": \"In this section, we demonstrate empirically that by appropriately choosing the smoothness level of randomized smoothing, our approach is able to drastically improve the sample quality of existing autoregressive models on several synthetic and real-world datasets while retaining competitive likelihoods on synthetic datasets. We also present results on image inpainting in Appendix C.2.\"\n },\n {\n \"heading\": \"4.1 CHOOSING THE SMOOTHING DISTRIBUTION\",\n \"text\": \"To help us build insights into the selection of the smoothing distribution q(x̃|x), we first focus on a 1-d multi-modal distribution (see figure 4 leftmost panel). We use model-based methods to invert the smoothed distribution and provide analysis on “single-step denoising” in Appendix B.1. We start with the exploration of three different types of smoothing distributions – Gaussian distribution, Laplace distribution, and uniform distribution. For each type of distribution, we perform a grid search to find the optimal variance. Since our approach requires the modeling of both pθ(x̃) and pθ(x|x̃), we stack x̃ and x together, and use a MADE model (Germain et al., 2015) with a mixture of two logistic components to parameterize pθ(x̃) and pθ(x|x̃) at the same time. For the baseline model, we train a mixture of logistics model directly on pdata(x). We compare the results in the middle two panels in figure 4.\\nWe find that although the baseline with eight logistic components has the capacity to perfectly model the multi-modal data distribution, which has six modes, the baseline model still fails to do so. We believe this can be caused by optimization or initialization issues for modeling a distribution with a high Lipschitz constant. Our method, on the other hand, demonstrates more robustness by successfully modeling the different modes in the data distribution even when using only two mixture components for both pθ(x̃) and pθ(x|x̃). For all the three types of smoothing distributions, we observe a reverse U-shape correlation between the variance of q(x̃|x) and ELBO values — with ELBO first increasing as the variance increases and then decreasing as the variance grows beyond a certain point. The results match our discussion on the trade-off between modeling pθ(x̃) and pθ(x|x̃) in Section 3.3. We notice from the empirical results that Gaussian smoothing is able to obtain better ELBO than the other two distributions. Thus, we will use q(x̃|x) = N (x̃|x, σ2I) for the later experiments.\"\n },\n {\n \"heading\": \"4.2 2-D SYNTHETIC DATASETS\",\n \"text\": \"In this section, we consider two challenging 2-d multi-modal synthetic datasets (see figure 5). We focus on model-based denoising methods and present discussion on “single-step denoising” in Appendix B.2. We use a MADE model with comparable number of total parameters for both the baseline and our approach. For the baseline, we train the MADE model directly on the data. For\\nour randomized smoothing model, we choose q(x̃|x) = N (x̃|x, 0.32I) to be the smoothing distribution. We observe that with this randomized smoothing approach, our model is able to generate better samples than the baseline (according to a human observer) even when using less logistic components (see figure 5). We provide more analysis on the model’s performance in Appendix B.2. We also provide the negative log-likelihoods in Tab. 1.\"\n },\n {\n \"heading\": \"4.3 IMAGE EXPERIMENTS\",\n \"text\": \"In this section, we focus on three common image datasets, namely MNIST, CIFAR-10 (Krizhevsky et al., 2009) and CelebA (Liu et al., 2015). We select q(x̃|x) = N (x̃|x, σ2I) to be the smoothing distribution. We use PixelCNN++ (Salimans et al., 2017) as the model architecture for both pθ(x̃) and pθ(x|x̃). We provide more details about settings in Appendix C.\\nImage generation. For image datasets, we select the σ of q(x̃|x) = N (x̃|x, σ2I) according to analysis in (Saremi & Hyvarinen, 2019) (see Appendix C for more details). Since q(x̃|x) is a Gaussian distribution, we can apply “single-step denoising” to reverse the smoothing process for samples drawn from pθ(x̃). In this case, the model pθ(x|x̃) is not required for sampling since the gradient of pθ(x̃) can be used to denoise samples (also from pθ(x̃)) (see equation 2). We present smoothed samples from pθ(x̃), reversed smoothing samples processed by “single-step denoising” and processed by pθ(x|x̃) in figure 6. For comparison, we also present samples from a PixelCNN++ with parameters comparable to the sum of total parameters of pθ(x̃) and pθ(x|x̃). We find that by using this randomized smoothing approach, we are able to drastically improve the sample quality of PixelCNN++ (see the rightmost panel in figure 6). We note that with only pθ(x̃), a PixelCNN++ optimized on the smoothed data, we already obtain more realistic samples compared to the original PixelCNN++ method. However, pθ(x|x̃) is needed to compute the likelihood lower bounds. We report the sample quality evaluated by Fenchel Inception Distance (FID (Heusel et al., 2017)), Kernel Inception Distance (KID (Bińkowski et al., 2018)), and Inception scores (Salimans et al., 2016) in Tab. 2. Although our method obtains better samples compared to the original PixelCNN++, our model has worse likelihoods as evaluated in BPDs. We believe this is because likelihood and sample quality are not always directly correlated as discussed in Theis et al. (2015). We also tried training the variance for q(x̃|x) by jointly optimizing ELBO. Although training the variance can produce better likelihoods, it does not generate samples with comparable quality as our method (i.e. choosing variance by heuristics). Thus, it is hard to conclusively determine what is the best way of choosing q(x̃|x). We provide more image samples in Appendix C.4 and nearest neighbors analysis in Appendix C.5.\"\n },\n {\n \"heading\": \"5 ADDITIONAL EXPERIMENTS ON NORMALIZING FLOWS\",\n \"text\": \"In this section, we demonstrate empirically on 2-d synthetic datasets that randomized smoothing techniques can also be applied to improve the sample quality of normalizing flow models (Rezende & Mohamed, 2015). We focus on RealNVP (Dinh et al., 2016). We compare the RealNVP model trained with randomized smoothing, where we use pθ(x|x̃) (also a RealNVP) to revert the smoothing process, with a RealNVP trained with the original method but with comparable number of parameters. We observe that smoothing is able to improve sample quality on the datasets we consider (see figure 7) while also obtaining competitive likelihoods. On the checkerboard dataset, our method has negative log-likelihoods 3.64 while the original RealNVP has 3.72; on the Olympics dataset, our method has negative log-likelihoods 1.32 while the original RealNVP has 1.80. This example demonstrates that randomized smoothing techniques can also be applied to normalizing flow models.\"\n },\n {\n \"heading\": \"6 RELATED WORK\",\n \"text\": \"Our approach shares some similarities with denoising autoencoders (DAE, Vincent et al. (2008)) which recovers a clean observation from a corrupted one. However, unlike DAE which has a train-\\nable encoder and a fixed prior distribution, our approach fixes the encoder and models the prior using an autoregressive model. Generative stochastic networks (GSN, Bengio et al. (2014)) use DAEs to train a Markov chain whose equilibrium distribution matches the data distribution. However, GSN needs to start the chain from a sample that is very close to the training distribution. Denoising diffusion model (Sohl-Dickstein et al., 2015; Ho et al., 2020) and NCSN (Song & Ermon (2019; 2020)) address the issue of GSNs by considering a sequence of distributions corresponding to data corrupted with various noise levels. By setting multiple noise levels that are close to each other, the sample from the previous level can serve as a proper initialization for the sampling process at the next level. This way, the model can start from a distribution that is easy to model and gradually move to the desired distribution. However, due to the large number of noise levels, such approaches require many steps for the chain to converge to the right data distribution.\\nIn this paper, we instead propose to use only one level of smoothing by modeling each step with a powerful autoregressive model instead of deterministic autoencoders. Motivated by the success of “randomized smoothing” techniques in adversarial defense (Cohen et al., 2019), we perform randomized smoothing directly on the data distribution. Unlike denoising score matching (Vincent, 2011), a technique closely related to denoising diffusion models and NCSN, which requires the perturbed noise to be a Gaussian distribution, we are able to work with different noise distributions.\\nOur smoothing method is also relevant to “dequantization” approaches that are common in normalizing flow models, where the discrete data distribution is converted to a continuous one by adding continuous noise (Uria et al., 2013; Ho et al., 2019). However the added noise for “dequantization” in flows is often indistinguishable to human eyes, and the reverse “dequantization” process is often ignored. In contrast, we consider noise scales that are significantly larger and thus a denoising process is required.\\nOur method is also related to “quantization” approaches which reduce the number of “significant” bits that are modeled by a generative model (Kingma & Dhariwal, 2018; Menick & Kalchbrenner, 2018). For instance, Glow (Kingma & Dhariwal, 2018) only models the 5 most significant bits of an image, which improves the visual quality of samples but decreases color fidelity. SPN (Menick & Kalchbrenner, 2018) introduces another network to predict the remaining bits conditioned on the 3 most significant bits already modeled. Modeling the most significant bits can be understood as capturing a data distribution perturbed by bit-wise correlated noise, similar to modeling smoothed data in our method. Modeling the remaining bits conditioned on the most significant ones in SPN is then similar to denoising. However, unlike these quantization approaches which process an image at the “significant” bits level, we apply continuous data independent Gaussian noise to the entire image with a different motivation to smooth the data density function.\"\n },\n {\n \"heading\": \"7 DISCUSSION\",\n \"text\": \"In this paper, we propose to incorporate randomized smoothing techniques into autoregressive modeling. By choosing the smoothness level appropriately, this seemingly simple approach is able to drastically improve the sample quality of existing autoregressive models on several synthetic and real-world datasets while retaining reasonable likelihoods. Our work provides insights into how recent adversarial defense techniques can be leveraged to building more robust generative models. Since we apply randomized smoothing technique directly to the target data distribution other than the model, we believe our approach is also applicable to other generative models such as variational autoencoders (VAEs) and generative adversarial networks (GANs).\"\n },\n {\n \"heading\": \"ACKNOWLEDGEMENTS\",\n \"text\": \"The authors would like to thank Kristy Choi for reviewing the draft of the paper. This research was supported by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1-2145), AFOSR (FA9550-19-1-0024), ARO, and Amazon AWS.\"\n },\n {\n \"heading\": \"A PROOFS\",\n \"text\": \"Theorem 1. Given a continuous and bounded 1-d distribution pdata(x) that is supported on R, for any 1-d distribution q(x̃|x) that is symmetric (i.e. q(x̃|x) = q(x|x̃)), stationary (i.e. translation invariant) and satisfies limx→∞ pdata(x)q(x|x̃) = 0 for any given x̃, we have Lip(q(x̃)) ≤ Lip(pdata(x)), where q(x̃) , ∫ q(x̃|x)pdata(x)dx and Lip(·) denotes the Lipschitz constant of the given 1-d function.\\nProof. First, we have that:\\n|∇xp(x)| = |p(x)∇x log p(x)| ≤ Lip(p) (5) and if we assume symmetry, i.e. q(x|x̃) = q(x̃|x) then by integration by parts we have:\\n∇x̃q(x̃) = Ep(x)[∇x̃q(x̃|x)] = Ep(x)[∇xq(x|x̃)] = −Ep(x)[q(x|x̃)∇x log p(x)] (6) Therefore,\\nLip(q) = max x̃ |−Ep(x)[q(x|x̃)∇x log p(x)]| = max x̃ | ∑ x q(x|x̃)p(x)∇x log p(x)| (7)\\n≤ max x̃ | ∑ x q(x|x̃)Lip(p)| = Lip(p) max x̃ | ∑ x q(x|x̃)| = Lip(p) (8)\\nwhich proves the result.\\nProposition 1 (Formal). Given a D-dimensional data distribution pdata(x) and model distribution pθ(x), assume that the smoothing distribution q(x̃|x) satisfies:\\n• log pθ is infinitely differentiable on the support of pθ(x)\\n• q(x̃|x) is symmetric (i.e. q(x̃|x) = q(x|x̃))\\n• q(x̃|x) is stationary (i.e. translation invariant)\\n• q(x̃|x) is bounded and fully supported on RD\\n• q(x̃|x) is element-wise independent\\n• Eq(x̃|x)[(x̃− x)2] is bounded, and Eq(x̃|x)[(x̃i − xi)2] = η at each dimension i.\\nDenote = x̃− x, then\\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] = Epdata(x) [ log pθ(x) + η\\n2 ∑ i ∂2 log pθ ∂x2i\\n] + ∫ ∫ o( 2)pdata(x)p( )dxd ,\\nwhere o( 2) : RD → R is a function of such that lim →0 o( 2) 2 = 0. Thus when∫ ∫\\no( 2)pdata(x)p( )dxd → 0, we have\\nEpdata(x)Eq(x̃|x)[log pθ(x̃)]→ Epdata(x) [ log pθ(x) + η\\n2 ∑ i ∂2 log pθ ∂x2i\\n] . (2)\\nProof. To see this, we first note that the new training objective for the smoothed data distribution is Epdata(x)Eq(x̃|x)[log pθ(x̃)]. Let = x̃ − x, because of the assumptions we have, the PDF function q(x̃|x) can be reparameterized as p( ) which satisfies: p is bounded and fully supported on RD; p is element-wise independent and Ep( )[ 2i ] = Ep( i)[ 2i ] = η at each dimension i (i = 1, ..., D). Then we have\\nEpdata(x)Eq(x̃|x)[log pθ(x̃)] = ∫ ∫ log pθ(x + )pdata(x)p( )dxd , (9)\\nUsing Taylor expansion, we have:\\nlog pθ(x + ) = log pθ(x) + ∑ i i ∂ log pθ ∂xi + 1 2 ∑ i,j i j ∂2 log pθ ∂xi∂xj + o( 2).\\nSince is independent of x and∫ ∞ −∞ id i = 0, ∫ ∞ −∞ ∫ ∞ −∞ i jd id j = δi,jη,\\nwhere δi,j is the Kronecker delta function, the right hand side of Equation 9 becomes Epdata(x)[log pθ(x)] + ∫ ∫ ( 1\\n2 ∑ i 2i ∂2 log pθ ∂x2i + o( 2)\\n) pdata(x)p( )dxd (10)\\n= Epdata(x) [ log pθ(x) + η\\n2 ∑ i ∂2 log pθ ∂x2i\\n] + ∫ ∫ o( 2)pdata(x)p( )dxd . (11)\\nWhen ∫ ∫ o( 2)pdata(x)p( )dxd → 0, (12)\\nwe have\\nEpdata(x)Eq(x̃|x)[log pθ(x̃)]→ Epdata(x) [ log pθ(x) + η\\n2 ∑ i ∂2 log pθ ∂x2i\\n] . (13)\\nwhere the second term on the right hand side serves as a regularization for the original objective Epdata(x)[log pθ(x)].\"\n },\n {\n \"heading\": \"B DENOISING EXPERIMENTS\",\n \"text\": \"B.1 ANALYSIS ON 1-D DENOISING\\nTo provide more insights into denoising, we first study “single-step denoising” (see equation 2) on a 1-d dataset. We choose the data distribution to be a two mixture of Gaussian distribution 0.5N (−0.3, 0.12) + 0.5N (0.3, 0.12) and the smoothing distribution to be q(x̃|x) = N (x̃|x, 0.32) (see figure 8a). Since the convolution of two Gaussian distributions is also a Gaussian distribution, the smoothed data is a mixture of Gaussian distribution given by 0.5N (−0.3, 0.12 + 0.32) + 0.5N (0.3, 0.12 + 0.32). The ground truth of ∇x̃ log p(x̃) can then be calculated in closed form. Thus, given the smoothed data x̃, we can calculate the ground truth ∇x̃ log p(x̃) in equation 2 and obtain x̄ using “single-step denoising”. We visualize the denoising results in figure 8b. We find that the low density region between the two modes in pdata(x) are not modeled properly in figure 8b. However, this is very expected since “single-step denosing” uses x̄ = E[x|x̃] as the substitute for the denoised result. When the smoothing distribution has a large variance (like in figure 8a where the smoothed data has merged into a one mode distribution), datapoints like x̃0 in the middle low density region of pdata(x) can have high density in the smoothed distribution. Since x̃0, as well as other points in the middle low density region of pdata(x), can come from both modes of pdata(x) with high probability before the smoothing process (see figure 11a), the denoised x̄ = E[x|x̃ = x̃0] can still be located in the middle low density region (see figure 8b). Since a large proportion of the smoothed data is located in the middle low density region of pdata(x), we would expect certain proportion of the density to remain in the low density region after “single-step denoising” just as shown in figure 8b. However, when the smoothing distribution has a smaller variance, “single-step denoising” can achieve much better denoising results (see figure 9, where we use q(x̃|x) = N (x̃|x, 0.12)). Although denoising can be easier when the smoothing distribution has a smaller variance, modeling the smoothed distribution could be harder as we discussed before.\\nIn general, the right denoising results should be samples coming from p(x|x̃), which is the reason why samples from pθ(x|x̃) (i.e. introducing the model pθ(x|x̃)) is more ideal than using Eθ[x|x̃] as a denoising substitute (i.e. “single-step denoising”). In general, the capacity of the denoising model pθ(x|x̃) also matters in terms of denoising results. Let us again consider the datapoint x̃0 shown in figure 8a. If the invert smoothing model pθ(x|x̃) is a one mode logistic distribution, due to the mode covering property of maximum likelihood estimation, given the smoothed observation x̃0, the best the model can do is to center its only mode at x̃0 for approximating p(x|x̃ = x̃0) (see figure 10c). Thus, like x̃0, the smoothed datapoints at the low density region between the two modes of pdata(x)\\nare still likely to remain between the two modes after denoising (see figure 10b). To solve this issue, we can increase the capacity of pθ(x|x̃) by making it a two mixture of logistics. In this case, the distribution pθ(x|x̃ = x̃0) can be captured in a better way (see figure 11c and figure 11a). After the invert smoothing process, like x̃0, most smoothed datapoints in the low density can be mapped to one of the two high density modes (see figure 11b), resulting in much better denoising effects.\\nB.2 ANALYSIS ON 2-D DENOISING\\nOn the 2-d Olympics dataset in section 4.2, we find that the intersections between rings can be poorly modeled with the proposed smoothing approach when only two mixture of logistics are used (see figure 12e). We believe this can be caused if the denoising model is not flexible enough to capture the distribution p(x|x̃). More specifically, we note that the ground truth distribution for p(x|x̃) at the intersections of the rings is a highly complicated distribution and can be hard to capture using our model which only has two mixtures of logistics for each dimension. If we increase the flexibility of pθ(x|x̃) by using three or four mixtures of logistics components (note that we still use fewer mixture components than the MADE baseline and we use comparable number of parameters), the intersection of the rings can be modeled in an improved way (see figure 12).\\nWe also provide “single-step denoising” results for the experiments in Section 4.2 (see figure 13), where we use the same smoothing distribution, and the MADE model with three mixture components as used in section 4.2. We note that “single-step denoising” results are not very good, which is also expected. As discussed in section B.1, when the smoothing distribution has a relatively large variance, Eθ[x|x̃] is not a good approximation for the denoised result, and we want the denoised sample to come from the distribution pθ(x|x̃), in which case introducing a denoising model pθ(x|x̃) could be a better option. Although we could select q(x̃|x) to have a smaller variance so that “single-step denoing” could work reasonably well, but modeling p(x̃) in this case could be more challenging.\\nC IMAGE EXPERIMENTS\\nC.1 SETTINGS\\nFor the image experiments, we first rescale images to [−1, 1] and then perturb the images with q(x̃|x) = N (x̃|x, σ2I). We use σ = 0.5 for MNIST and σ = 0.3 for both CIFAR-10 and CelebA. The selection of σ is mainly based on analysis in (Saremi & Hyvarinen, 2019). More specifically, given an image, we consider the median value of the Euclidean distance between two data points in a dataset, and then divide it by 2 √ D, where D is the dimension of the data. This provides us with a way of selecting the variance of q(x̃|x), when q(x̃|x) is a Gaussian distribution. We find this selection of variance able to generate reasonably well samples in practice. We train all the models with Adam optimizer with learning rate 0.0002. To model pθ(x|x̃), we stack x̃ and x together at the second dimension to obtain x̂ = [x̃,x], which ensures that x̃ comes before x in the pixel ordering. For instance, this stacking would provide an image x̂ with size 1 × (2 × 28) × 28 on a MNIST image, and an image with size 3 × (2 × 32) × 32 on a CIFAR-10 image. Since PixelCNN++ consists of convolutional layers, we can directly feed x̂ into the default architecture without modifying the model architecture. As the latter pixels of the input only depend on the previous pixels in an autoregressive model and x̃ comes before x, we can parameterize pθ(x|x̃) by computing the likelihoods only on x using the outputs from the autoregressive model.\\nC.2 IMAGE INPAINTING\\nSince both pθ(x̃) and pθ(x|x̃) are parameterized by an autoregressive model, we can also perform image inpainting using our method. We present the inpainting results on CIFAR-10 in figure 14a and CelebA in figure 14b, where the bottom half of the input image is being inpainted.\\nC.3 IMAGE DENOISING\\nWe notice that the reverse smoothing process can also be understood as a denoising process. Besides the “single-step denoising” approach shown above, we can also apply pθ(x|x̃) to denoise images. To visualize the denoising performance, we sample xtest from the test set and perturb xtest with q(x̃|x) to obtain a noisy sample x̃test. We feed x̃test into pθ(x|x̃ = x̃test) and draw samples from the model. We visualize the results in figure 15. As we can see, the model exhibits reasonable denoising results, which shows that the autoregressive model is capable of learning the data distribution when conditioned on the smoothed data.\\nC.4 MORE SAMPLES\\nC.5 NEAREST NEIGHBORS\\nC.6 ABLATION STUDIES\\nIn this section, we show that gradient-based “single-step denoising” will not improve sample qualities without performing randomized smoothing. To see this, we draw samples from a PixelCNN++ pθ(x) trained directly on pdata(x) (i.e. without smoothing). We perform “single-step denoising” update defined as x = x + σ2∇x log pθ(x). (14) We explore various values for σ, and report the results in figure 26. This shows that “single-step denoising” alone (without randomized smoothing) will not improve sample quality of PixelCNN++.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2021,"string":"2,021"},"title":{"kind":"string","value":"IMPROVED AUTOREGRESSIVE MODELING WITH DISTRIBUTION SMOOTHING"}}},{"rowIdx":92,"cells":{"paper_id":{"kind":"string","value":"SP:5918a2c105a901f8de4bba248dc283a476d9beac"},"summaries":{"kind":"list like","value":["This work considers an important problem of generating adversarial examples to attack a black-box model. The paper proposes a new approach to consider an adversarial example as a result of a sequence of pixel changes from a benign instance. Therefore, the adversarial generation problem can be considered as a bandit problem, and thus we can leverage Bayesian optimization to search for an instance that maximize the changes on the loss function through a sequence of pixel changes. The evaluation is comprehensive, and demonstrates that fewer number of black-box queries are needed to achieve a higher attack success rate."],"string":"[\n \"This work considers an important problem of generating adversarial examples to attack a black-box model. The paper proposes a new approach to consider an adversarial example as a result of a sequence of pixel changes from a benign instance. Therefore, the adversarial generation problem can be considered as a bandit problem, and thus we can leverage Bayesian optimization to search for an instance that maximize the changes on the loss function through a sequence of pixel changes. The evaluation is comprehensive, and demonstrates that fewer number of black-box queries are needed to achieve a higher attack success rate.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching over the structured space can be approximated by a time-varying contextual bandits problem, where the attacker takes feature of the associated arm to make modifications of the input, and receives an immediate reward as the reduction of the loss function. The time-varying contextual bandits problem can then be solved by a Bayesian optimization procedure, which can take advantage of the features of the structured action space. The experiments on ImageNet and the Google Cloud Vision API demonstrate that the proposed method achieves the state of the art success rates and query efficiencies for both undefended and defended models."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Abdullah Al-Dujaili","Una-May O’Reilly"],"title":"Sign bits are all you need for black-box attacks","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Abdullah Al-Dujaili","Una-May O’Reilly"],"title":"Sign bits are all you need for black-box attacks","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Maksym Andriushchenko","Francesco Croce","Nicolas Flammarion","Matthias Hein"],"title":"Square attack: a query-efficient black-box adversarial attack via random search","venue":null,"year":2020},{"authors":["Wieland Brendel","Jonas Rauber","Matthias Bethge"],"title":"Decision-based adversarial attacks: Reliable attacks against black-box machine learning models","venue":"arXiv preprint arXiv:1712.04248,","year":2017},{"authors":["Nicholas Carlini","David Wagner"],"title":"Towards evaluating the robustness of neural networks","venue":"IEEE Symposium on Security and Privacy (SP),","year":2017},{"authors":["Jianbo Chen","Michael I Jordan","Martin J Wainwright"],"title":"Hopskipjumpattack: A query-efficient decision-based attack","venue":"IEEE Symposium on Security and Privacy (SP),","year":2020},{"authors":["Pin-Yu Chen","Huan Zhang","Yash Sharma","Jinfeng Yi","Cho-Jui Hsieh"],"title":"Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models","venue":"In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security,","year":2017},{"authors":["Kun Dong","David Eriksson","Hannes Nickisch","David Bindel","Andrew G Wilson"],"title":"Scalable log determinants for gaussian process kernel learning","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["Jiawei Du","Hu Zhang","Joey Tianyi Zhou","Yi Yang","Jiashi Feng"],"title":"Query-efficient meta attack to deep neural networks","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Jacob Gardner","Geoff Pleiss","Kilian Q Weinberger","David Bindel","Andrew G Wilson"],"title":"Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration","venue":"In Advances in Neural Information Processing Systems,","year":2018},{"authors":["Chuan Guo","Mayank Rana","Moustapha Cisse","Laurens van der Maaten"],"title":"Countering adversarial images using input transformations","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["Nikolaus Hansen"],"title":"The cma evolution strategy: A tutorial","venue":"arXiv preprint arXiv:1604.00772,","year":2016},{"authors":["Kaiming He","Xiangyu Zhang","Shaoqing Ren","Jian Sun"],"title":"Deep residual learning for image recognition","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2016},{"authors":["Gao Huang","Zhuang Liu","Laurens Van Der Maaten","Kilian Q Weinberger"],"title":"Densely connected convolutional networks","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2017},{"authors":["Zhichao Huang","Tong Zhang"],"title":"Black-box adversarial attack with transferable model-based embedding","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Andrew Ilyas","Logan Engstrom","Anish Athalye","Jessy Lin"],"title":"Black-box adversarial attacks with limited queries and information","venue":"Proceedings of the 35th International Conference on Machine Learning,","year":2018},{"authors":["Andrew Ilyas","Logan Engstrom","Aleksander Madry"],"title":"Prior convictions: Black-box adversarial attacks with bandits and priors","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Alexey Kurakin","Ian Goodfellow","Samy Bengio"],"title":"Adversarial machine learning at scale","venue":"arXiv preprint arXiv:1611.01236,","year":2016},{"authors":["Huichen Li","Xiaojun Xu","Xiaolu Zhang","Shuang Yang","Bo Li"],"title":"Qeba: Query-efficient boundarybased blackbox attack","venue":"In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition,","year":2020},{"authors":["Yandong Li","Lijun Li","Liqiang Wang","Tong Zhang","Boqing Gong"],"title":"Nattack: Learning the distributions of adversarial examples for an improved black-box attack on deep neural networks","venue":null,"year":1905},{"authors":["Yanpei Liu","Xinyun Chen","Chang Liu","Dawn Song"],"title":"Delving into transferable adversarial examples and black-box attacks","venue":"arXiv preprint arXiv:1611.02770,","year":2016},{"authors":["Laurent Meunier","Jamal Atif","Olivier Teytaud"],"title":"Yet another but more efficient black-box adversarial attack: tiling and evolution strategies","venue":null,"year":1910},{"authors":["Jonas Mockus","Vytautas Tiesis","Antanas Zilinskas"],"title":"The application of bayesian methods for seeking the extremum","venue":"Towards global optimization,","year":1978},{"authors":["Seungyong Moon","Gaon An","Hyun Oh Song"],"title":"Parsimonious black-box adversarial attacks via efficient combinatorial optimization","venue":"arXiv preprint arXiv:1905.06635,","year":2019},{"authors":["Nicolas Papernot","Patrick McDaniel","Ian Goodfellow"],"title":"Transferability in machine learning: from phenomena to black-box attacks using adversarial samples","venue":"arXiv preprint arXiv:1605.07277,","year":2016},{"authors":["Carl Edward Rasmussen"],"title":"Gaussian processes in machine learning","venue":"In Summer School on Machine Learning,","year":2003},{"authors":["Binxin Ru","Adam Cobb","Arno Blaas","Yarin Gal"],"title":"Bayesopt adversarial attack","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Olga Russakovsky","Jia Deng","Hao Su","Jonathan Krause","Sanjeev Satheesh","Sean Ma","Zhiheng Huang","Andrej Karpathy","Aditya Khosla","Michael Bernstein"],"title":"Imagenet large scale visual recognition challenge","venue":"International journal of computer vision,","year":2015},{"authors":["Satya Narayan Shukla","Anit Kumar Sahu","Devin Willmott","J Zico Kolter"],"title":"Black-box adversarial attacks with bayesian optimization","venue":null,"year":1909},{"authors":["Karen Simonyan","Andrew Zisserman"],"title":"Very deep convolutional networks for large-scale image recognition","venue":"arXiv preprint arXiv:1409.1556,","year":2014},{"authors":["Christian Szegedy","Wojciech Zaremba","Ilya Sutskever","Joan Bruna","Dumitru Erhan","Ian Goodfellow","Rob Fergus"],"title":"Intriguing properties of neural networks","venue":"In International Conference on Learning Representations,","year":2014},{"authors":["Chun-Chen Tu","Paishun Ting","Pin-Yu Chen","Sijia Liu","Huan Zhang","Jinfeng Yi","Cho-Jui Hsieh","Shin-Ming Cheng"],"title":"Autozoom: Autoencoder-based zeroth order optimization method for attacking black-box neural networks","venue":"arXiv preprint arXiv:1805.11770,","year":2018},{"authors":["Daan Wierstra","Tom Schaul","Jan Peters","Jürgen Schmidhuber"],"title":"Natural evolution strategies","venue":"IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence),","year":2008},{"authors":["Cihang Xie","Yuxin Wu","Laurens van der Maaten","Alan Yuille","Kaiming He"],"title":"Feature denoising for improving adversarial robustness","venue":"arXiv preprint arXiv:1812.03411,","year":2018}],"string":"[\n {\n \"authors\": [\n \"Abdullah Al-Dujaili\",\n \"Una-May O’Reilly\"\n ],\n \"title\": \"Sign bits are all you need for black-box attacks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Abdullah Al-Dujaili\",\n \"Una-May O’Reilly\"\n ],\n \"title\": \"Sign bits are all you need for black-box attacks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Maksym Andriushchenko\",\n \"Francesco Croce\",\n \"Nicolas Flammarion\",\n \"Matthias Hein\"\n ],\n \"title\": \"Square attack: a query-efficient black-box adversarial attack via random search\",\n \"venue\": null,\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Wieland Brendel\",\n \"Jonas Rauber\",\n \"Matthias Bethge\"\n ],\n \"title\": \"Decision-based adversarial attacks: Reliable attacks against black-box machine learning models\",\n \"venue\": \"arXiv preprint arXiv:1712.04248,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Nicholas Carlini\",\n \"David Wagner\"\n ],\n \"title\": \"Towards evaluating the robustness of neural networks\",\n \"venue\": \"IEEE Symposium on Security and Privacy (SP),\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Jianbo Chen\",\n \"Michael I Jordan\",\n \"Martin J Wainwright\"\n ],\n \"title\": \"Hopskipjumpattack: A query-efficient decision-based attack\",\n \"venue\": \"IEEE Symposium on Security and Privacy (SP),\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Pin-Yu Chen\",\n \"Huan Zhang\",\n \"Yash Sharma\",\n \"Jinfeng Yi\",\n \"Cho-Jui Hsieh\"\n ],\n \"title\": \"Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models\",\n \"venue\": \"In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Kun Dong\",\n \"David Eriksson\",\n \"Hannes Nickisch\",\n \"David Bindel\",\n \"Andrew G Wilson\"\n ],\n \"title\": \"Scalable log determinants for gaussian process kernel learning\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Jiawei Du\",\n \"Hu Zhang\",\n \"Joey Tianyi Zhou\",\n \"Yi Yang\",\n \"Jiashi Feng\"\n ],\n \"title\": \"Query-efficient meta attack to deep neural networks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Jacob Gardner\",\n \"Geoff Pleiss\",\n \"Kilian Q Weinberger\",\n \"David Bindel\",\n \"Andrew G Wilson\"\n ],\n \"title\": \"Gpytorch: Blackbox matrix-matrix gaussian process inference with gpu acceleration\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Chuan Guo\",\n \"Mayank Rana\",\n \"Moustapha Cisse\",\n \"Laurens van der Maaten\"\n ],\n \"title\": \"Countering adversarial images using input transformations\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Nikolaus Hansen\"\n ],\n \"title\": \"The cma evolution strategy: A tutorial\",\n \"venue\": \"arXiv preprint arXiv:1604.00772,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Kaiming He\",\n \"Xiangyu Zhang\",\n \"Shaoqing Ren\",\n \"Jian Sun\"\n ],\n \"title\": \"Deep residual learning for image recognition\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Gao Huang\",\n \"Zhuang Liu\",\n \"Laurens Van Der Maaten\",\n \"Kilian Q Weinberger\"\n ],\n \"title\": \"Densely connected convolutional networks\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Zhichao Huang\",\n \"Tong Zhang\"\n ],\n \"title\": \"Black-box adversarial attack with transferable model-based embedding\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Andrew Ilyas\",\n \"Logan Engstrom\",\n \"Anish Athalye\",\n \"Jessy Lin\"\n ],\n \"title\": \"Black-box adversarial attacks with limited queries and information\",\n \"venue\": \"Proceedings of the 35th International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Andrew Ilyas\",\n \"Logan Engstrom\",\n \"Aleksander Madry\"\n ],\n \"title\": \"Prior convictions: Black-box adversarial attacks with bandits and priors\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Alexey Kurakin\",\n \"Ian Goodfellow\",\n \"Samy Bengio\"\n ],\n \"title\": \"Adversarial machine learning at scale\",\n \"venue\": \"arXiv preprint arXiv:1611.01236,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Huichen Li\",\n \"Xiaojun Xu\",\n \"Xiaolu Zhang\",\n \"Shuang Yang\",\n \"Bo Li\"\n ],\n \"title\": \"Qeba: Query-efficient boundarybased blackbox attack\",\n \"venue\": \"In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Yandong Li\",\n \"Lijun Li\",\n \"Liqiang Wang\",\n \"Tong Zhang\",\n \"Boqing Gong\"\n ],\n \"title\": \"Nattack: Learning the distributions of adversarial examples for an improved black-box attack on deep neural networks\",\n \"venue\": null,\n \"year\": 1905\n },\n {\n \"authors\": [\n \"Yanpei Liu\",\n \"Xinyun Chen\",\n \"Chang Liu\",\n \"Dawn Song\"\n ],\n \"title\": \"Delving into transferable adversarial examples and black-box attacks\",\n \"venue\": \"arXiv preprint arXiv:1611.02770,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Laurent Meunier\",\n \"Jamal Atif\",\n \"Olivier Teytaud\"\n ],\n \"title\": \"Yet another but more efficient black-box adversarial attack: tiling and evolution strategies\",\n \"venue\": null,\n \"year\": 1910\n },\n {\n \"authors\": [\n \"Jonas Mockus\",\n \"Vytautas Tiesis\",\n \"Antanas Zilinskas\"\n ],\n \"title\": \"The application of bayesian methods for seeking the extremum\",\n \"venue\": \"Towards global optimization,\",\n \"year\": 1978\n },\n {\n \"authors\": [\n \"Seungyong Moon\",\n \"Gaon An\",\n \"Hyun Oh Song\"\n ],\n \"title\": \"Parsimonious black-box adversarial attacks via efficient combinatorial optimization\",\n \"venue\": \"arXiv preprint arXiv:1905.06635,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Nicolas Papernot\",\n \"Patrick McDaniel\",\n \"Ian Goodfellow\"\n ],\n \"title\": \"Transferability in machine learning: from phenomena to black-box attacks using adversarial samples\",\n \"venue\": \"arXiv preprint arXiv:1605.07277,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Carl Edward Rasmussen\"\n ],\n \"title\": \"Gaussian processes in machine learning\",\n \"venue\": \"In Summer School on Machine Learning,\",\n \"year\": 2003\n },\n {\n \"authors\": [\n \"Binxin Ru\",\n \"Adam Cobb\",\n \"Arno Blaas\",\n \"Yarin Gal\"\n ],\n \"title\": \"Bayesopt adversarial attack\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Olga Russakovsky\",\n \"Jia Deng\",\n \"Hao Su\",\n \"Jonathan Krause\",\n \"Sanjeev Satheesh\",\n \"Sean Ma\",\n \"Zhiheng Huang\",\n \"Andrej Karpathy\",\n \"Aditya Khosla\",\n \"Michael Bernstein\"\n ],\n \"title\": \"Imagenet large scale visual recognition challenge\",\n \"venue\": \"International journal of computer vision,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Satya Narayan Shukla\",\n \"Anit Kumar Sahu\",\n \"Devin Willmott\",\n \"J Zico Kolter\"\n ],\n \"title\": \"Black-box adversarial attacks with bayesian optimization\",\n \"venue\": null,\n \"year\": 1909\n },\n {\n \"authors\": [\n \"Karen Simonyan\",\n \"Andrew Zisserman\"\n ],\n \"title\": \"Very deep convolutional networks for large-scale image recognition\",\n \"venue\": \"arXiv preprint arXiv:1409.1556,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Christian Szegedy\",\n \"Wojciech Zaremba\",\n \"Ilya Sutskever\",\n \"Joan Bruna\",\n \"Dumitru Erhan\",\n \"Ian Goodfellow\",\n \"Rob Fergus\"\n ],\n \"title\": \"Intriguing properties of neural networks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Chun-Chen Tu\",\n \"Paishun Ting\",\n \"Pin-Yu Chen\",\n \"Sijia Liu\",\n \"Huan Zhang\",\n \"Jinfeng Yi\",\n \"Cho-Jui Hsieh\",\n \"Shin-Ming Cheng\"\n ],\n \"title\": \"Autozoom: Autoencoder-based zeroth order optimization method for attacking black-box neural networks\",\n \"venue\": \"arXiv preprint arXiv:1805.11770,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Daan Wierstra\",\n \"Tom Schaul\",\n \"Jan Peters\",\n \"Jürgen Schmidhuber\"\n ],\n \"title\": \"Natural evolution strategies\",\n \"venue\": \"IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence),\",\n \"year\": 2008\n },\n {\n \"authors\": [\n \"Cihang Xie\",\n \"Yuxin Wu\",\n \"Laurens van der Maaten\",\n \"Alan Yuille\",\n \"Kaiming He\"\n ],\n \"title\": \"Feature denoising for improving adversarial robustness\",\n \"venue\": \"arXiv preprint arXiv:1812.03411,\",\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":null,"text":"We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching over the structured space can be approximated by a time-varying contextual bandits problem, where the attacker takes feature of the associated arm to make modifications of the input, and receives an immediate reward as the reduction of the loss function. The time-varying contextual bandits problem can then be solved by a Bayesian optimization procedure, which can take advantage of the features of the structured action space. The experiments on ImageNet and the Google Cloud Vision API demonstrate that the proposed method achieves the state of the art success rates and query efficiencies for both undefended and defended models."},{"heading":"1 INTRODUCTION","text":"Although deep learning has many applications, it is known that neural networks are vulnerable to adversarial examples, which are small perturbations of inputs that can fool neural networks into making wrong predictions (Szegedy et al., 2014). While adversarial noise can easily be found when the neural models are known (referred to as white-box attack) (Kurakin et al., 2016). However, in real world scenarios models are often unknown, this situation is referred to as black-box attack.\nSome methods (Liu et al., 2016; Papernot et al., 2016) use the transfer-based attack, which generates adversarial examples on a substitute model and transfer the adversarial noise to the target model. However, the transferability is limited and its effectiveness relies highly on the similarity between the networks (Huang & Zhang, 2020). If two networks are very different, transfer-based methods will have low success rates.\nIn practice, most computer vision API such as the Google Cloud Vision API allow users to access the scores or probabilities of the classification results. Therefore, the attacker may query the black-box model and perform zeroth order optimization to find an adversarial example without the knowledge of the target model. Due to the availability of scores, this scenario is called score-based attack.\nThere have been a line of studies on black-box attack which directly estimate the gradient direction of the underlying model, and apply (stochastic) gradient descent to the input image (Ilyas et al., 2018; 2019; Chen et al., 2017; Huang & Zhang, 2020; Tu et al., 2018; Li et al., 2019). In this paper, we take another approach and formulate score-based attack as a time-varying contextual bandits problem. At each state, the attacker may change the adversarial perturbation and get the reward as the reduction of the loss. And the attacker would receive some features about the arms before making the decision. By limiting the action space to image blocks, the associated bandits problem exhibits local correlation structures and the slow varying property suitable for learning. Therefore, we may use the location and other features of the blocks to estimate the reward for the future selection of the actions.\nUsing the above insights, we propose a new method called CorrAttack, which utilizes the local correlation structure and the slow varying property of the underlying bandits problem. CorrAttack uses Bayesian optimization with Gaussian process regression (Rasmussen, 2003) to model the correlation and select optimal actions. A forgetting strategy is added to the algorithm so that the Gaussian process regression can handle the time-varying changes. CorrAttack can effectively find\nblocks with the largest rewards. The resulting method achieves much lower numbers of average queries and higher success rates than prior methods with a similar action space (Moon et al., 2019).\nIt is worth noting that BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019) also employ Bayesian optimization for score-based attack. However, their Gaussian process regression directly models the loss as a function of the image, whose dimension can be more than one thousand. Therefore, their speed is slow especially for BayesOpt, which uses slow additive kernel. CorrAttack, on the other hand, searches over a much limited action space and models the reward as a function of the low dimensional feature. Therefore, the optimization of CorrAttack is more efficient, and the method is significantly faster than BayesOpt.\nWe summarize the contributions of this work as follows:\n1. We formulate the score-based adversarial attack as a time-varying contextual bandits, and show that the reward function has slow varying properties. In our new formulation, the attacker could take advantage of the features to model the reward of the arms with learning techniques. Compared to the traditional approach, the use of learning in the proposed framework greatly improves the efficiency of searching over optimal actions.\n2. We propose a new method, CorrAttack, which uses Bayesian optimization with Gaussian process regression to learn the reward of each action, by using the feature of the arms.\n3. The experiments show that CorrAttack achieves the state of the art performance on ImageNet and Google Cloud Vision API for both defended and undefended models."},{"heading":"2 RELATED WORK","text":"There have been a line of works focusing on black-box adversarial attack. Here, we give a brief review of various existing methods.\nTransfer-Based Attack Transfer-based attack assumes the transferability of adversarial examples across different neural networks. It starts with a substitute model that is in the same domain as the target model. The adversaries can be easily generated on the white-box substitute model, and be transferred to attack the target model (Papernot et al., 2016). The approach, however, depends highly on the similarity of the networks. If two networks are distinct, the success rate of transferred attack would rapidly decrease (Huang & Zhang, 2020). Besides, we may not access the data for training the substitute model in practice.\nScore-based Attack Many approaches estimate the gradient with the output scores of the target network. However, the high dimensionality of input images makes naive coordinate-wise search impossible as it requires millions of queries. ZOO (Chen et al., 2017) is an early work of gradient estimation, which estimates the gradient of an image block and perform block-wise gradient descent. NES (Wierstra et al., 2008) and CMA-ES (Hansen, 2016) are two evolution strategies that can perform query efficient score-based attack Ilyas et al. (2018); Meunier et al. (2019). Instead of the gradient itself, SignHunter (Al-Dujaili & O’Reilly, 2020a) just estimates the sign of gradient to reduce the complexity. AutoZOOM (Tu et al., 2018) uses bilinear transformation or autoencoder to reduce the sampling space and accelerate the optimization process. In the same spirit, data prior can be used to improve query efficiency (Ilyas et al., 2019). Besides, MetaAttack (Du et al., 2020) takes a meta learning approach to learn gradient patterns from prior information, which reduces queries for attacking targeted model.\nMany zeroth order optimization methods for black-box attacks rely on gradient estimation. However, there are some research works using gradient free methods to perform black-box attack. BayesOpt and Bayes-Attack (Ru et al., 2020; Shukla et al., 2019) employ Bayesian optimization to find the adversarial examples. They use Gaussian process regression on the embedding and apply bilinear transformation to resize the embedding to the size of image. Although the bilinear transformation could alleviate the high dimensionality of images, the dimension of their embeddings are still in the thousands, which makes Bayesian optimization very ineffective and computationally expensive. A different method, PARSI, poses the attack on `∞ norm as a discrete optimization problem over {−ε, ε}d (Moon et al., 2019). It uses a Lazy-Greedy algorithm to search over the space {−ε, ε}d to find an adversarial example. SimBA (Guo et al., 2018) also employs a discrete search space targeted at `2 norm.\nDecision-based Attack Decision-based attack assumes the attacker could only get the output label of the model. Boundary Attack and its variants (Brendel et al., 2017; Chen et al., 2020; Li et al., 2020) are designed for the setting. However, the information received by the attacker is much smaller than score-based attack, and it would take many more queries than score-based attack to successfully attack an image."},{"heading":"3 PRELIMINARIES","text":"A Gaussian process (Rasmussen, 2003) is a prior distribution defined on some bounded set Z , and is determined by a mean function µ : Z → R and a covariance kernel κ : Z × Z → R. Given n observations Dn = {(zi, f(zi))}ni=1, the prior distribution on f(z1:n) is\nf(z1:n) ∼ Normal(µ0(z1:n), κ0(z1:n, z1:n)), (1) where we use compact notation for functions applied to collections of input points: z1:n indicates the sequence z1, · · · , zn, f(z1:n) = [f(z1), · · · , f(zn)], µ0(z1:n) = [µ0(z1), · · · , µ0(zn)], κ0(z1:n, z1:n) = [κ0(z1, z1), · · · , κ0(z1, zn); · · · ;κ0(zn, z1), · · · , κ0(zn, zn); ]. Now we wish to infer the value of f(z) at some new point z, the posterior process f(z)|Dn is also a Gaussian process (GP) with mean µn and covariance σ2n:\nf(z)|Dn ∼ Normal(µn(z), σ2n(z)), (2) µn(z) = κ0(z, z1:n)κ0(z1:n, z1:n)\n−1(f(z1:n)− µ0(z1:n)) + µ0(z), σ2n(z) = κ0(z, z)− κ0(z, z1:n)κ0(z1:n, z1:n)−1κ0(z1:n, z).\nAs a optimization method to maximize a function f , Bayesian optimization models the function to make decisions about where to evaluate the next point z. Assuming we already obtained observations Dt−1 = {(zi, f(zi))}t−1i=1 , to determine the next point zt for evaluation, we first use the posterior GP to define an acquisition function ϕt : Z → R, which models the utility of evaluating f(z) for any z ∈ Z . We then evaluate f(zt) with\nzt = arg max Z ϕt(z). (3)\nIn this work, we use the expected improvement (EI) acquisition function (Mockus et al., 1978) ϕt(z) = √ σ2n(z)(γ(z)Φ(γ(z)) + φ(γ(z))) with γ(z) =\nµn(z)− f(zbest)√ σ2n(z) , (4)\nwhich measures the expected improvement over the current best value zbest = arg maxzi f(zi) according to the posterior GP. Here Φ(·) and φ(·) are the cdf and pdf of N (0, I) respectively."},{"heading":"4 SCORE-BASED BLACK-BOX ATTACK","text":"Suppose a classifier F (x) has input x and label y. An un-targeted adversarial example xadv satisfies:\narg max j∈{1,···C}\nF (xadv)j 6= y and ‖xadv − x‖p ≤ ε, (5)\nwhere C is the number of classes. While an adversarial example for targeted attack means the maximum position of F (x) should be the targeted class q: arg maxj∈{1,···C} F (xadv)j = q. In order to find xadv , we may optimize a surrogate loss function `(x, y) (e.g hinge loss).\nIn this work, we consider adversarial attack as a time-varying contextual bandits problem. At each time t, we observe a state xt which is a modification of the original input x0. Before taking arm at ∈ A ⊂ Rd, we could observe the feature z of arms. And at would modify state xt to xt+1 according to\nxt+1 = arg min s∈{xt+at,xt} `(ΠBp(x,ε) (s) , y) (6)\nwith reward function r(xt, at) = `(xt+1, y)− `(xt, y) and the checking step tries to remove negative reward. In this frame, we would like to estimate the reward r(xt, at) with feature zt using learning, and then pick at to maximize the reward. Observe that\nr(xt, at) ≈ ∇x`(xt, y)>(xt+1 − xt), (7)\nwhere the gradient ∇xt`(xt, y) is unknown. It follows from the formulation that we may rewrite r(xt, at) as a function r(xt, at) ≈ f(xt, xt+1 − xt). Since in general, we make small steps from one iteration to the next iteration, δt(at) = xt+1 − xt is small. We may approximate the reward with fixed gradient locally with\nf(xt, δt) = f̃t(at),\nWe may consider the learning of reward as a time-varying contextual bandits problem with reward function f̃t(at) for arm at at time t. Since xt+1 − xt is small, this time-varying bandits has slowvarying property: the function f̃t changes slowly from time t to time t+ 1.\nIn the proposed framework, our goal is to learn the time-varying bandits reward f̃t(at) with feature zt. We use Gaussian process regression to model the reward function using recent historic data since the reward function is slow-varying, and describe the details in the subsequent sections.\nWe note that the most general action space contains all at ∈ Rd, where d is the number of image pixels. However, it is impossible to explore the arms in such a large space. In this work, we choose a specific class of actions A = {ai}ni=1, n is the image blocks of different sizes. It covers the space of the adversarial perturbations while maintaining good complexity. We also find the location and the PCA of the blocks a good component of the feature z associated with the arm. Besides, modifying a block only affects the state locally. Therefore the reward function remains similar after state changes."},{"heading":"4.1 STRUCTURED SEARCH WITH GAUSSIAN PROCESS REGRESSION AND BAYESIAN OPTIMIZATION","text":"Define the block size as b, we divide the image into several blocks E = {e000, e001, · · · , ehwc}, where the block is b× b square of pixels and (h,w, c) = (height/b,width/b, channel). Each block eijk is associated with the feature zeijk such as the location of the block.\nSuppose we have time-varying bandits with state xt and unknown reward function f̃t at time t. By taking the action aeijk , we change the individual block eijk of xt and get xt+1 with reward f̃t(aeijk). We consider two ways of taking action aeijk on block eijk : CorrAttackDiff and CorrAttackFlip.\nFinite Difference CorrAttackDiff: For action aeijk , the attacker will query `(xt + ηeijk, y) and `(xt − ηeijk, y), and choose\naeijk = arg min s∈{ηeijk,−ηeijk} `(xt + s, y). (8)\nThe action space A = {aeijk |eijk ∈ E}. In our framework, the bandits problem can also be regarded as learning the conditional gradient over actions. That is, when η is small, we try to choose action at with\nat = arg min eijk∈E\ne>ijk∇xt`(xt, y) (9)\nwhich is the conditional gradient over the set of blocks.\nDiscrete Approximation CorrAttackFlip: In general, adversarial attack with `∞ budget can be formulated as constrained optimization with ‖xadv − x‖∞ ≤ . However, PARSI (Moon et al., 2019) limits the space to {−ε,+ε}d, which leads to better performance for black-box attack (Moon et al., 2019). The continuous optimization problem becomes a discrete optimization problems as follows:\nmaximize `(xadv, y) =⇒ maximize `(xadv, y) (10)\nsubject to ‖xadv − x‖∞ ≤ subject to xadv − x ∈ { ,− }d.\nFollowing PARSI, we consider two stages to perform structured search. When flipping ε to −ε, aeijk changes the block to −ε and A = {−2εeijk|eijk ∈ E}. When changing −ε to ε, A = {2εeijk|eijk ∈ E} instead. Gaussian Process (GP) Regression: We model the difference function\ngt(at) = `(ΠBp(x,ε) (xt + at) , y)− `(xt, y) (11)\ninstead of the reward function f̃t(at) ≥ 0, as the difference function could be negative, providing more information about the negative arms in A. We would collect historic actions with feature and difference {zk, gk(ak))}tk=1 and learn the difference to make choices at a later stage. At each time t, we use the Gaussian process regression to model the correlation between the features zeijk and use Bayesian optimization to select the next action. More specifically, the same as eq. (2), we let\ngt(aeijk)|Dt ∼ Normal(µt(zeijk), σ2t (zeijk)), (12) where Dt = {zk, gk(ak))}tk=t−τ is the difference of evaluated blocks et−τ :t with feature zet−τ:t and τ is a parameter to forget old samples. Then we use EI acquisition function to pick up the next action at+1 in A. More specifically, the same as eq. (4), we let\nat+1 = arg max A\n( √ σ2t (zeijk)(γ(zeijk)Φ(γ(zeijk)) + φ(γ(zeijk)))) (13)\nAs the difference function gt is varying, we take two strategies in Algorithm 2 to update the previous samples to make sure GP regression learns the current difference function well. The first strategy is to remove old samples in Dt. Even if the bandits are slowly varying, the difference function will change significantly after a significant number of rounds. Therefore, we need to forget samples before t− τ . The second strategy is to remove samples near the last block eitjtkt in Dt. As we discuss later, the difference function may change significantly in a local region near the last selected block. Therefore previous samples in this local region will be inaccurate. The resulting algorithm for CorrAttack is shown in Algorithm 1, which mainly follows standard procedure of Bayesian optimization.\nAlgorithm 1 CorrAttack Require: Loss function `(·, ·), Input x0 and its label y, Action space A = {aeijk |eijk ∈ E}, Parameter c, τ ,\nα 1: Build set D0 = {(zeipjpkp , g0(aeipjpkp ))} m p=1 using latin hypercube sampling from A 2: repeat 3: Fit the parameter of Normal(µt(zeijk ), σ 2 t (zeijk )) on Dt according to Equation (12) 4: Calculate acquisition function ϕt(zeijk ) and according to Equation (13) 5: Select aeitjtkt = arg maxA ϕt(zeijk ) according to Equation (13) 6: xt+1 = arg mins∈{xt+aeitjtkt ,xt} `(ΠBp(x,ε) (s) , y) 7: Update sample set Dt with Algorithm 2 Dt+1 = UPDATESAMPLES(Dt, xt, xt+1, eitjtkt , gt+1(aeitjtkt ), τ, α) 8: until maxA ϕt(zeijk ) < c 9: return xT ;\nAlgorithm 2 Update Samples Require: Sample set Dt, State xt, xt+1, Block eitjtkt , Difference gt+1(aeitjtkt ), Paramter τ , α 1: if xt+1 6= xt then 2: Dt+1 = Dt \\ {(zeijk , g) ∈ Dt||i− it|+ |j − jt| ≤ α} 3: else 4: Dt+1 = Dt ∪ {(zeitjtkt , gt+1(aeitjtkt ))} 5: end if 6: Remove the earliest sample from D if the cardinality |D| > τ 7: return Dt+1"},{"heading":"4.2 FEATURES AND SLOW VARYING PROPERTY","text":"Features of Contextual Bandits: We use a four dimensional vector as the feature zeijk : zeijk = (i, j, k, pca) (14)\nwhere i, j, k is the location of the block. And pca is the first component of PCA decomposition of [x0(e000), x0(e001), · · ·x0(ehwc)]. x0(eijk) means the block of natural image at the given position. The reward function depends on the gradient in Equation (7). It has been shown that the gradient ∇x`(x, y) has local dependencies (Ilyas et al., 2019). Suppose two coordinates eijk and elpq are close, then∇x`(x, y)ijk ≈ ∇x`(x, y)lpq. We consider the finite difference of the block eijk\n∆t(eijk) = ` (xt + ηeijk, y)− ` (xt − ηeijk, y) ≈ 2ηe>ijk∇xt` (xt, y) (15)\nwhere η is a small step size. When η is small, the reward can be approximated by the average of the gradients around a small region, which also has local dependencies. In fact, the local structure of the reward will also be persevered when the block size and η is large. Figure 1 shows one example of the finite difference ∆t(eijk) obtained on ImageNet dataset with ResNet50. This shows blocks with closer locations are more likely to have similar reward. Therefore, we add the location of the block as the feature so that it uses historic data to find the arm with the largest reward.\nIn addition to the location of the difference, we may add other features. The block of the image itself forms a strong feature for the regression, but the dimension of the block is too high for GP regression. Therefore, we use PCA to lower the dimension and add the first component into the feature vector.\nSlow Varying Property In addition to the local dependencies of finite difference, the difference would also be slow varying if we just change a small region of xt. Let xt+1 = xt− ηeitjtkt , Figure 2 shows the difference of ∆t(eijk) and ∆t+1(eijk), which is centralized in a small region near eitjtkt . Reward function is based on the finite difference, which also has the slow varying property. It could be explained by the local property of convolution. When η is small, the finite difference can be approximated with gradient and the local Hessian:\n∆t+1(eijk)−∆t(eijk) ≈ η2e>ijk∇2xt`(xt, y)eitjtkt (16) The difference is much smaller than ∆t(eijk). Today’s neural networks are built with stacks of convolutions and non-linear operations. Since these operations are localized in a small region, the Hessian of a neural network is also localized and the reward function only changes near eitjtkt ."},{"heading":"4.3 HIERARCHICAL BAYESIAN OPTIMIZATION SEARCH","text":"Recent black-box approaches (Chen et al., 2017; Moon et al., 2019) exploit the hierarchical image structure for query efficiency. Following these approaches, we take a hierarchical approach and perform the accelerated local search in Algorithm 1 from a coarse grid (large blocks) to a fine grid (smaller blocks). The algorithm for hierarchical attack iteratively performs Algorithm 1 at one block size, and then divides the blocks into smaller sizes. At each block size, we build a Gaussian process to model the difference function, and perform structured search with the blocks until maxA ϕt(zeijk) < c. When dividing the blocks into smaller sizes, we will build a new block set E with action aeijk and new feature zeijk , but keep the xt in last block size as x0 in new block size. Define the stage as S = {0, 1, · · · , s} and initial block size as b. The block at stage s is b2s × b 2s square of pixels and (h,w, c) = (2s ∗ height/b, 2s ∗ width/b, channel). The overall hierarchical accelerated local search algorithm is shown in Appendix A. It is important to note that most of the attacks terminate in the early stages and rarely need to run on fine scales."},{"heading":"5 EXPERIMENTS","text":"We evaluated the number of queries versus the success rates of CorrAttack on both undefended and defended network on ImageNet (Russakovsky et al., 2015). Moreover, we attacked Google Cloud Vision API to show that CorrAttack can generalize to a true black-box model.\nWe used the common hinge loss proposed in the CW attack (Carlini & Wagner, 2017). We compared two versions of CorrAttack : CorrAttackDiff and CorrAttackFlip, to ZOO (Chen et al., 2017), NES (Ilyas et al., 2018), NAttack (Li et al., 2019), Bandits (Ilyas et al., 2019), PARSI (Moon et al., 2019), Square Attack (Andriushchenko et al., 2020), SignHunter(Al-Dujaili & O’Reilly, 2020b), BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019). We only test adversarial attack on `∞ norm. The details of the Gaussian processes regression and the hyperparameters of CorrAttack are given in the Appendix B. We shall mention that CorrAttack is not sensitive to the hyperparameters. The hyperparameters of other methods follow those suggested by the original papers."},{"heading":"5.1 UNDEFENDED NETWORK","text":"We randomly select 1000 images from the validation set of ImageNet and only attack correctly classified images. The query efficiency of CorrAttack is tested on VGG16 (Simonyan & Zisserman, 2014), Resnet50 (He et al., 2016) and Densenet121 (Huang et al., 2017), which are the most commonly used network structures. We set ε = 0.05 and the query limit to be 10000 except for BayesOpt and Bayes-Attack. For targeted attacks, we randomly choose the target class for each image and the target classes are maintained the same for the evaluation of different algorithms. The results are shown in Table 1 and 2. CorrAttackFlip outperforms other methods by a large margin.\nAs BayesOpt and Bayes-Attack takes tens of thousands of hours to attack 1000 images, we compare them with CorrAttackFlip only on 20 images and un-targeted attack. The query limit is also reduced to 1000 as the time for BayesOpt and Bayes-Attack quickly increases as more samples add into the Gaussian distribution. The time comparison between three models is shown in Appendix C.6.\nOptimality of Bayesian optimization Appendix C.1 shows the rank the actions found by CorrAttack. The attacker could find the action with large reward quickly.\nVarying ε We also test the algorithms at different budget of adversarial perturbations at ε = 0.04 and ε = 0.06 on Resnet50. As it is shown in Appendix C.2, CorrAttack shows a consistently better performance at different ε.\nAblation study on random choices Appendix C.3 shows the ablation study of random version of CorrAttackDiff and CorrAttackFlip. In both cases, Bayesian optimization helps to gain better query efficiency.\nAblation study on hierarchical attack We perform ablation study on the hierarchical attack and the result is shown in Appendix C.4. Hierarchical structure accelerates the CorrAttack and eliminates the sensitivity of choosing initial block size.\nAblation study on features Appendix C.5 demonstrates how the feature of the contextual bandits affects the performance of attack. PCA would help to improve the efficiency of attack."},{"heading":"5.2 DEFENDED NETWORK","text":"To evaluate the effectiveness of CorrAttack on adversarially defended networks, we tested our method on one of the SOTA robust model (Xie et al., 2018) on ImageNet. The weight is downloaded from Github1. \"ResneXt DenoiseAll\" is chosen as the target model as it achieves the best performance. We set ε = 0.05 and the maximum number of queries is 10000. As BayesOpt runs very slowly, the attack is also performed on 10 images and the query limit is 1000. The result is shown in Table 3. CorrAttackFlip still outperforms other methods."},{"heading":"5.3 GOOGLE CLOUD VISION API","text":"We also attacked Google Cloud Vision API, a real world black-box model for classification. The target is to remove the top-1 label out of the classification output. We choose 10 images for the ImageNet dataset and set the query limit to be 500 due to high cost to use the API. We compare CorrAttackFlip with NAttack, BayesOpt and PARSI. The result is shown in Table 4. We also show one example of the classification output in Appendix C.9\n1https://github.com/facebookresearch/ImageNet-Adversarial-Training"},{"heading":"6 CONCLUSION AND FUTURE WORK","text":"We formulate the score-based adversarial attack as a time-varying contextual bandits and propose a new method CorrAttack. By performing structured search on the blocks of the image, the bandits has the slow varying property. CorrAttack takes advantage of the the features of the arm, and uses Bayesian optimization with Gaussian process regression to learn the reward function. The experiment shows that CorrAttack can quickly find the action with large reward and CorrAttack achieves superior query efficiency and success rate on ImageNet and Google Cloud Vision API.\nWe only include basic features for learning the bandits. Other features like embedding from the transfer-based attack Huang & Zhang (2020) may be taken into account in the future work. While our work only focuses on adversarial attack on `∞ norm, the same contextual bandits formulation could be generalized to other `p norm to improve query efficiency. Besides, defense against CorrAttack may be achieved with adversarial training on CorrAttack , but it may not be able to defend other attacks in the meantime."},{"heading":"A ALGORITHM","text":"Algorithm 3 Split Block Require: Set of blocks E, Block size b, E′ = ∅ 1: for each block e ∈ E do 2: Split the block e into 4 blocks {e1, e2, e3, e4} with size b/2 3: E′ ← E′ ∪ {e1, e2, e3, e4} 4: end for 5: return E′;\nAlgorithm 4 Hierarchical CorrAttackDiff Require: Loss function `(·, ·), Input image x and its label y, Initial Block size b, Set of blocks E containing all\nblocks of the image, Threshold c, τ , α, Step size η, Adversarial budget ε 1: x0 = x 2: repeat 3: Choose A = {aeijk |eijk ∈ E} with Equation (8) 4: Run CorrAttack on current block size x = CORRATTACK (`(·, ·), x, y, A, c, τ, α) 5: if b > 1 then 6: Split the blocks into finer blocks using Algorithm 3 E = SPLITBLOCK(E, b) 7: b← b/2 8: end if 9: until ` converges\n10: return xK ;\nAlgorithm 5 Hierarchical CorrAttackFlip Require: Loss function `(·, ·), Input image x and its label y, Block size b, Set of blocks E containing all blocks\nof the image, Threshold c, τ , α, Adversarial budget ε 1: x0 = x 2: for eijk ∈ E do 3: Randomly draw v from {−ε, ε} 4: x0[eijk] = v + x0[eijk] 5: end for 6: repeat 7: An = {2εeijk ∈ E|e>ijk(xk − x) < 0} 8: Run CorrAttack flipping −ε to ε x̃k = CORRATTACK (`(·, ·), xk, y, An, c, τ, α) 9: Ap = {−2εeijk ∈ E|e>ijk(x̃k − x) > 0} 10: Run CorrAttack flipping ε to −ε xk+1 = CORRATTACK (`(·, ·), x̃k, y, Ap, c, τ, α) 11: if b > 1 then 12: Split the blocks into finer blocks using Algorithm 3 E = SPLITBLOCK(E, b) 13: b← b/2 14: end if 15: until ` converges 16: return xK ;"},{"heading":"B DETAILS OF EXPERIMENT SETTING","text":"We use the hinge loss for all the experiments. For un-targeted attacks,\n`untarget(x, y) = max { F (x)y −max\nj 6=y F (x)j ,−ω\n} (17)\nand for targeted attacks,\n`target(x, y) = max { max j F (x)j − F (x)t,−ω } . (18)\nHere F represents the logits of the network outputs, t is the target class, and ω denotes the margin. The image will be projected into the ε-ball. Besides, the value of the image will be clipped to range [0, 1]."},{"heading":"B.1 GAUSSIAN PROCESS REGRESSION AND BYAESIAN OPTIMIZATION","text":"We further provide details on both the computational scaling and modeling setup for the GP regression.\nTo address computational issues, we use GPyTorch (Gardner et al., 2018) for scalable GP regression. GPyTorch follows (Dong et al., 2017) to solve linear systems using the conjugate gradient (CG) method and approximates the log-determinant via the Lanczos process. Without GPyTorch, running BO with a GP regression for more than a few thousand evaluations would be infeasible as classical approaches to GP regression scale cubically in the number of data points.\nOn the modeling side, the GP is parameterized using a Matérn-5/2 kernel with ARD and a constant mean function for all experiments. The GP hyperparameters are fitted before proposing a new batch by optimizing the log-marginal likelihood. The domain is rescaled to [0, 1]d and the function values are standardized before fitting the GP regression. We use a Matérn-5/2 kernel with ARD for CorrAttack and use the following bounds for the hyperparameters: (length scale) λi ∈ [0.005, 2.0 ], (output scale) λ′i ∈ [0.05, 20.0], (noise variance) σ2 ∈ [0.0005, 0.1]."},{"heading":"B.2 HYPERPARAMETERS","text":"For CorrAttack in Algorithm 4 and Algorithm 5, we set the initial block size b to be 32 and the step size η for CorrAttackDiff is 0.03. In Algorithm 1, we use the initial sampling ratio m = 0.03n at the start point for Gaussian process regression, the threshold c = 10−4 to decide when to stop the search of current block size. In Algorithm 2, the threshold is different for different block size. For CorrAttackFlip, α = 1, 1, 2, 2, 3 for block size 32, 16, 8, 4, 2 and for CorrAttackDiff, α = 0, 0, 1, 1, 2 for block size 32, 16, 8, 4, 2. We set τ = 3m = 0.09n to remove the earliest samples from D once |D| > α. The Adam optimizer is used to optimize the mean µ and covariance κ of Gaussian process, where the iteration is 1 and the learning rate is 0.1.\nFor PARSI, the block size is set to 32 as CorrAttack , other hyperparameters are the same as the original paper.\nFor Bandits, Bayes-Attack and BayesOpt, the hyperparameters are the same as the original paper.\nWe optimize the hyperparameters for ZOO, NES. For un-targeted attack on NES, we set the sample size to be 50, learning rate to be 0.1. For targeted attack on NES, the sample size is also 50 and the learning rate is 0.05. The learning is decay by 50% if the loss doesn’t decrease for 20 iterations.\nFor NAttack, we set the hyperparameters the same as NES and add momentum and learning rate decay, which are not mentioned in the original paper.\nFor ZOO, we set the learning rate to 1.0 and sample size to be 50. Other setting follows the original paper."},{"heading":"C ADDITIONAL EXPERIMENTS","text":""},{"heading":"C.1 OPTIMALITY OF BAYESIAN OPTIMIZATION","text":"Figure 3 shows the reward function that the Bayesian optimization could find in the action set. CorrAttack could find the action with high reward within just a few queries. It shows that the Gaussian process regression could model the correlation of the reward function and the Bayesian optimization could use it to optimize the time-varying contextual bandits.\nC.2 VARYING THE ADVERSARIAL BUDGET\nWe test CorrAttack on different adversarial budget on ImageNet for both un-targeted attack and targeted attack. Table 5 and Table 6 show the success rate and average queries for ε = 0.04, 0.05, 0.06. CorrAttackFlip achieves the best performance among all methods."},{"heading":"C.3 ABLATION STUDY ON RANDOM CHOICES","text":"Table 7 and Table 8 show the ablation study on the strategy to choose action xt+1 in the line 6 of Algorithm 1. The process of Bayesian optimization helps to accelerate the optimization. As targeted attack is more complicated and requires larger number of queries, CorrAttack has more advantage in this scenario."},{"heading":"C.4 ABLATION STUDY ON HIERARCHICAL ATTACK","text":"We perform un-targeted attack on Resnet50 as shown in Table 10. Hierarchical attack lowers the average queries and improves the query efficiency. Besides, hierarchical attack avoids the problem of choosing block size. As shown in Table 10, block size for non-hierarchical is essential for the performance."},{"heading":"C.5 ABLATION STUDY ON FEATURES","text":"Table 11 shows the success rate and average queries for CorrAttackwith different features. We perform ablation study on the features of the contextual bandits. One contains just the location of the\nblock and the other contains both the location and the PCA feature. PCA helps the learning process of the reward and achieve higher success rate and lower number of queries. PCA feature achieves significant improvement on CorrAttackFlip. We may find more useful features in the future."},{"heading":"C.6 COMPARISON BETWEEN CORRATTACKFLIP , BAYESOPT AND BAYES-ATTACK","text":"The main difference between BayesOpt and Bayes-Attack is using different types of GP regression (Standard GP for Bayes-Attack and Additive GP for BayesOpt), so we will consider these two models as a group when comparing with our model CorrAttack.\nDifference between CorrAttack, BayesOpt and Bayes-Attack: For l∞ attacks, assume there are no hierarchical structure, we have blocksE = {e000, e001, · · · , ehwc}, where the block is b×b square of pixels and (h,w, c) = (height/b,width/b, channel). CorrAttack, BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019) all try to search the adversarial noise on E with perturbation δ ∈ [− , ]d where d = h× w × c, the perturbation of block eijk at time t is δteijk .\nBayesOpt and Bayes-Attack use a GP regression directly on δ ∈ [− , ]d (all blocks),\nf(δ)|Dn ∼ Normal(µn(δ), σ2n(δ)). (19)\nCorrAttack define an action spaceA and use a standard GP regression on features zeijk = (i, j, k, pca) (single block),\ngt(aeijk)|Dt ∼ Normal(µt(zeijk), σ2t (zeijk)). (20)\nAt each iteration, in BayesOpt and Bayes-Attack, the changes of overall perturbation is\nδt − δt−1 = {δte000 ∪ δ t e001 ∪ · · · ∪ δ t ehwc } − {δt−1e000 ∪ δ t−1 e001 ∪ · · · ∪ δ t−1 ehwc }. (21)\nHowever, in CorrAttack,\nδt − δt−1 = δteijk − δ t−1 eijk . (22)\nIn conclusion, BayesOpt and Bayes-Attack view each block as a dimension, try to search the overall perturbation directly. CorrAttack defines a low dimension feature space, keep an overall perturbation and try to search an action on single block.\nTime complexity and running time: The time complexity of fitting GP regression is O(dn2) where d is the dimension of input and n is the number of samples. And the dimension for CorrAttack (d = 4 for zeijk = (i, j, k, pca)) is much smaller than BayesOpt and Bayes-Attack (d = 6912 if h = w = 48, c = 3). Moreover, we can convert the continuous search space of BayesOpt and Bayes-Attack from [− , ]6912 to discrete search space E = {e000, e001, · · · , ehwc}, whose number is only 6912, smaller search space could save the computation time of acquisition function.\nWe compare the running time for CorrAttackFlip with BayesOpt and Bayes-Attack on 20 images from ImageNet. Table 12 shows the running time for the un-targeted attack. We use PyTorch2 to develop these two models. All experiments were conducted on a personal workstation with 28 Intel(R) Xeon(R) Gold 5120 2.20GHz CPUs, an NVIDIA GeForce RTX2080Ti 11GB GPU and 252G memory.\nBayesOpt models the loss function with a very high dimensional Gaussian process. The decomposition of additive kernel also needs to be restarted several times. Even though we try to optimize the speed of BayesOpt with GPU acceleration, it is still very slow and takes hundreds of times more computational resources than CorrAttack .\nBayes-Attack could be regarded as a simpler version of BayesOpt, which does not add additive kernel. We do not evaluate it on targeted task (when query>1000) since GP inference time grows fast as evaluated query increases, e.g. For Bayes-Attack, when 150 <query< 200, Time=1.6s/query; 800 <query< 1000, Time = 10.5s/query. CorrAttack solves this problem with Time=0.1s/query even when query reaches 10000. Since we forget the previous samples before t − τ , our input sample n will be smaller than τ . The forgetting technique can not be applied into the Bayes-Attack and BayesOpt since they are searching the perturbation of all blocks so each sample needs to be remembered."},{"heading":"C.7 GROWING CURVE OF SUCCESS RATE","text":"The number of average queries is sometimes misleading due to the the heavy tail distribution of queries. Therefore in Figure 4, we plot the success rates at different query levels to show the detailed behaviors of different attacks. It shows that CorrAttack is much more efficient than other methods at all query levels.\n2https://pytorch.org/\nC.8 VISUALIZATION OF LOCAL PROPERTY AND SLOW VARYING PROPERTY\nFigure 5 shows more examples of finite difference for different network architectures and different dataset. They all have local correlation structure as shown in Figure 1. And Figure 6 shows more examples like Figure 2, the slow varying properties exist for different architectures and different datasets."},{"heading":"C.9 GOOGLE CLOUD VISION API","text":"Figure 7 shows the example of attacking the Google Cloud Vision API. CorrAttackFlip and PARSI successfully change the classification result. BeyesOpt, however, can not remove the top 1 classification result out of the output.\nC.10 VISUALIZATION OF ADVERSARIAL EXAMPLES"}],"string":"[\n {\n \"heading\": null,\n \"text\": \"We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching over the structured space can be approximated by a time-varying contextual bandits problem, where the attacker takes feature of the associated arm to make modifications of the input, and receives an immediate reward as the reduction of the loss function. The time-varying contextual bandits problem can then be solved by a Bayesian optimization procedure, which can take advantage of the features of the structured action space. The experiments on ImageNet and the Google Cloud Vision API demonstrate that the proposed method achieves the state of the art success rates and query efficiencies for both undefended and defended models.\"\n },\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Although deep learning has many applications, it is known that neural networks are vulnerable to adversarial examples, which are small perturbations of inputs that can fool neural networks into making wrong predictions (Szegedy et al., 2014). While adversarial noise can easily be found when the neural models are known (referred to as white-box attack) (Kurakin et al., 2016). However, in real world scenarios models are often unknown, this situation is referred to as black-box attack.\\nSome methods (Liu et al., 2016; Papernot et al., 2016) use the transfer-based attack, which generates adversarial examples on a substitute model and transfer the adversarial noise to the target model. However, the transferability is limited and its effectiveness relies highly on the similarity between the networks (Huang & Zhang, 2020). If two networks are very different, transfer-based methods will have low success rates.\\nIn practice, most computer vision API such as the Google Cloud Vision API allow users to access the scores or probabilities of the classification results. Therefore, the attacker may query the black-box model and perform zeroth order optimization to find an adversarial example without the knowledge of the target model. Due to the availability of scores, this scenario is called score-based attack.\\nThere have been a line of studies on black-box attack which directly estimate the gradient direction of the underlying model, and apply (stochastic) gradient descent to the input image (Ilyas et al., 2018; 2019; Chen et al., 2017; Huang & Zhang, 2020; Tu et al., 2018; Li et al., 2019). In this paper, we take another approach and formulate score-based attack as a time-varying contextual bandits problem. At each state, the attacker may change the adversarial perturbation and get the reward as the reduction of the loss. And the attacker would receive some features about the arms before making the decision. By limiting the action space to image blocks, the associated bandits problem exhibits local correlation structures and the slow varying property suitable for learning. Therefore, we may use the location and other features of the blocks to estimate the reward for the future selection of the actions.\\nUsing the above insights, we propose a new method called CorrAttack, which utilizes the local correlation structure and the slow varying property of the underlying bandits problem. CorrAttack uses Bayesian optimization with Gaussian process regression (Rasmussen, 2003) to model the correlation and select optimal actions. A forgetting strategy is added to the algorithm so that the Gaussian process regression can handle the time-varying changes. CorrAttack can effectively find\\nblocks with the largest rewards. The resulting method achieves much lower numbers of average queries and higher success rates than prior methods with a similar action space (Moon et al., 2019).\\nIt is worth noting that BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019) also employ Bayesian optimization for score-based attack. However, their Gaussian process regression directly models the loss as a function of the image, whose dimension can be more than one thousand. Therefore, their speed is slow especially for BayesOpt, which uses slow additive kernel. CorrAttack, on the other hand, searches over a much limited action space and models the reward as a function of the low dimensional feature. Therefore, the optimization of CorrAttack is more efficient, and the method is significantly faster than BayesOpt.\\nWe summarize the contributions of this work as follows:\\n1. We formulate the score-based adversarial attack as a time-varying contextual bandits, and show that the reward function has slow varying properties. In our new formulation, the attacker could take advantage of the features to model the reward of the arms with learning techniques. Compared to the traditional approach, the use of learning in the proposed framework greatly improves the efficiency of searching over optimal actions.\\n2. We propose a new method, CorrAttack, which uses Bayesian optimization with Gaussian process regression to learn the reward of each action, by using the feature of the arms.\\n3. The experiments show that CorrAttack achieves the state of the art performance on ImageNet and Google Cloud Vision API for both defended and undefended models.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"There have been a line of works focusing on black-box adversarial attack. Here, we give a brief review of various existing methods.\\nTransfer-Based Attack Transfer-based attack assumes the transferability of adversarial examples across different neural networks. It starts with a substitute model that is in the same domain as the target model. The adversaries can be easily generated on the white-box substitute model, and be transferred to attack the target model (Papernot et al., 2016). The approach, however, depends highly on the similarity of the networks. If two networks are distinct, the success rate of transferred attack would rapidly decrease (Huang & Zhang, 2020). Besides, we may not access the data for training the substitute model in practice.\\nScore-based Attack Many approaches estimate the gradient with the output scores of the target network. However, the high dimensionality of input images makes naive coordinate-wise search impossible as it requires millions of queries. ZOO (Chen et al., 2017) is an early work of gradient estimation, which estimates the gradient of an image block and perform block-wise gradient descent. NES (Wierstra et al., 2008) and CMA-ES (Hansen, 2016) are two evolution strategies that can perform query efficient score-based attack Ilyas et al. (2018); Meunier et al. (2019). Instead of the gradient itself, SignHunter (Al-Dujaili & O’Reilly, 2020a) just estimates the sign of gradient to reduce the complexity. AutoZOOM (Tu et al., 2018) uses bilinear transformation or autoencoder to reduce the sampling space and accelerate the optimization process. In the same spirit, data prior can be used to improve query efficiency (Ilyas et al., 2019). Besides, MetaAttack (Du et al., 2020) takes a meta learning approach to learn gradient patterns from prior information, which reduces queries for attacking targeted model.\\nMany zeroth order optimization methods for black-box attacks rely on gradient estimation. However, there are some research works using gradient free methods to perform black-box attack. BayesOpt and Bayes-Attack (Ru et al., 2020; Shukla et al., 2019) employ Bayesian optimization to find the adversarial examples. They use Gaussian process regression on the embedding and apply bilinear transformation to resize the embedding to the size of image. Although the bilinear transformation could alleviate the high dimensionality of images, the dimension of their embeddings are still in the thousands, which makes Bayesian optimization very ineffective and computationally expensive. A different method, PARSI, poses the attack on `∞ norm as a discrete optimization problem over {−ε, ε}d (Moon et al., 2019). It uses a Lazy-Greedy algorithm to search over the space {−ε, ε}d to find an adversarial example. SimBA (Guo et al., 2018) also employs a discrete search space targeted at `2 norm.\\nDecision-based Attack Decision-based attack assumes the attacker could only get the output label of the model. Boundary Attack and its variants (Brendel et al., 2017; Chen et al., 2020; Li et al., 2020) are designed for the setting. However, the information received by the attacker is much smaller than score-based attack, and it would take many more queries than score-based attack to successfully attack an image.\"\n },\n {\n \"heading\": \"3 PRELIMINARIES\",\n \"text\": \"A Gaussian process (Rasmussen, 2003) is a prior distribution defined on some bounded set Z , and is determined by a mean function µ : Z → R and a covariance kernel κ : Z × Z → R. Given n observations Dn = {(zi, f(zi))}ni=1, the prior distribution on f(z1:n) is\\nf(z1:n) ∼ Normal(µ0(z1:n), κ0(z1:n, z1:n)), (1) where we use compact notation for functions applied to collections of input points: z1:n indicates the sequence z1, · · · , zn, f(z1:n) = [f(z1), · · · , f(zn)], µ0(z1:n) = [µ0(z1), · · · , µ0(zn)], κ0(z1:n, z1:n) = [κ0(z1, z1), · · · , κ0(z1, zn); · · · ;κ0(zn, z1), · · · , κ0(zn, zn); ]. Now we wish to infer the value of f(z) at some new point z, the posterior process f(z)|Dn is also a Gaussian process (GP) with mean µn and covariance σ2n:\\nf(z)|Dn ∼ Normal(µn(z), σ2n(z)), (2) µn(z) = κ0(z, z1:n)κ0(z1:n, z1:n)\\n−1(f(z1:n)− µ0(z1:n)) + µ0(z), σ2n(z) = κ0(z, z)− κ0(z, z1:n)κ0(z1:n, z1:n)−1κ0(z1:n, z).\\nAs a optimization method to maximize a function f , Bayesian optimization models the function to make decisions about where to evaluate the next point z. Assuming we already obtained observations Dt−1 = {(zi, f(zi))}t−1i=1 , to determine the next point zt for evaluation, we first use the posterior GP to define an acquisition function ϕt : Z → R, which models the utility of evaluating f(z) for any z ∈ Z . We then evaluate f(zt) with\\nzt = arg max Z ϕt(z). (3)\\nIn this work, we use the expected improvement (EI) acquisition function (Mockus et al., 1978) ϕt(z) = √ σ2n(z)(γ(z)Φ(γ(z)) + φ(γ(z))) with γ(z) =\\nµn(z)− f(zbest)√ σ2n(z) , (4)\\nwhich measures the expected improvement over the current best value zbest = arg maxzi f(zi) according to the posterior GP. Here Φ(·) and φ(·) are the cdf and pdf of N (0, I) respectively.\"\n },\n {\n \"heading\": \"4 SCORE-BASED BLACK-BOX ATTACK\",\n \"text\": \"Suppose a classifier F (x) has input x and label y. An un-targeted adversarial example xadv satisfies:\\narg max j∈{1,···C}\\nF (xadv)j 6= y and ‖xadv − x‖p ≤ ε, (5)\\nwhere C is the number of classes. While an adversarial example for targeted attack means the maximum position of F (x) should be the targeted class q: arg maxj∈{1,···C} F (xadv)j = q. In order to find xadv , we may optimize a surrogate loss function `(x, y) (e.g hinge loss).\\nIn this work, we consider adversarial attack as a time-varying contextual bandits problem. At each time t, we observe a state xt which is a modification of the original input x0. Before taking arm at ∈ A ⊂ Rd, we could observe the feature z of arms. And at would modify state xt to xt+1 according to\\nxt+1 = arg min s∈{xt+at,xt} `(ΠBp(x,ε) (s) , y) (6)\\nwith reward function r(xt, at) = `(xt+1, y)− `(xt, y) and the checking step tries to remove negative reward. In this frame, we would like to estimate the reward r(xt, at) with feature zt using learning, and then pick at to maximize the reward. Observe that\\nr(xt, at) ≈ ∇x`(xt, y)>(xt+1 − xt), (7)\\nwhere the gradient ∇xt`(xt, y) is unknown. It follows from the formulation that we may rewrite r(xt, at) as a function r(xt, at) ≈ f(xt, xt+1 − xt). Since in general, we make small steps from one iteration to the next iteration, δt(at) = xt+1 − xt is small. We may approximate the reward with fixed gradient locally with\\nf(xt, δt) = f̃t(at),\\nWe may consider the learning of reward as a time-varying contextual bandits problem with reward function f̃t(at) for arm at at time t. Since xt+1 − xt is small, this time-varying bandits has slowvarying property: the function f̃t changes slowly from time t to time t+ 1.\\nIn the proposed framework, our goal is to learn the time-varying bandits reward f̃t(at) with feature zt. We use Gaussian process regression to model the reward function using recent historic data since the reward function is slow-varying, and describe the details in the subsequent sections.\\nWe note that the most general action space contains all at ∈ Rd, where d is the number of image pixels. However, it is impossible to explore the arms in such a large space. In this work, we choose a specific class of actions A = {ai}ni=1, n is the image blocks of different sizes. It covers the space of the adversarial perturbations while maintaining good complexity. We also find the location and the PCA of the blocks a good component of the feature z associated with the arm. Besides, modifying a block only affects the state locally. Therefore the reward function remains similar after state changes.\"\n },\n {\n \"heading\": \"4.1 STRUCTURED SEARCH WITH GAUSSIAN PROCESS REGRESSION AND BAYESIAN OPTIMIZATION\",\n \"text\": \"Define the block size as b, we divide the image into several blocks E = {e000, e001, · · · , ehwc}, where the block is b× b square of pixels and (h,w, c) = (height/b,width/b, channel). Each block eijk is associated with the feature zeijk such as the location of the block.\\nSuppose we have time-varying bandits with state xt and unknown reward function f̃t at time t. By taking the action aeijk , we change the individual block eijk of xt and get xt+1 with reward f̃t(aeijk). We consider two ways of taking action aeijk on block eijk : CorrAttackDiff and CorrAttackFlip.\\nFinite Difference CorrAttackDiff: For action aeijk , the attacker will query `(xt + ηeijk, y) and `(xt − ηeijk, y), and choose\\naeijk = arg min s∈{ηeijk,−ηeijk} `(xt + s, y). (8)\\nThe action space A = {aeijk |eijk ∈ E}. In our framework, the bandits problem can also be regarded as learning the conditional gradient over actions. That is, when η is small, we try to choose action at with\\nat = arg min eijk∈E\\ne>ijk∇xt`(xt, y) (9)\\nwhich is the conditional gradient over the set of blocks.\\nDiscrete Approximation CorrAttackFlip: In general, adversarial attack with `∞ budget can be formulated as constrained optimization with ‖xadv − x‖∞ ≤ . However, PARSI (Moon et al., 2019) limits the space to {−ε,+ε}d, which leads to better performance for black-box attack (Moon et al., 2019). The continuous optimization problem becomes a discrete optimization problems as follows:\\nmaximize `(xadv, y) =⇒ maximize `(xadv, y) (10)\\nsubject to ‖xadv − x‖∞ ≤ subject to xadv − x ∈ { ,− }d.\\nFollowing PARSI, we consider two stages to perform structured search. When flipping ε to −ε, aeijk changes the block to −ε and A = {−2εeijk|eijk ∈ E}. When changing −ε to ε, A = {2εeijk|eijk ∈ E} instead. Gaussian Process (GP) Regression: We model the difference function\\ngt(at) = `(ΠBp(x,ε) (xt + at) , y)− `(xt, y) (11)\\ninstead of the reward function f̃t(at) ≥ 0, as the difference function could be negative, providing more information about the negative arms in A. We would collect historic actions with feature and difference {zk, gk(ak))}tk=1 and learn the difference to make choices at a later stage. At each time t, we use the Gaussian process regression to model the correlation between the features zeijk and use Bayesian optimization to select the next action. More specifically, the same as eq. (2), we let\\ngt(aeijk)|Dt ∼ Normal(µt(zeijk), σ2t (zeijk)), (12) where Dt = {zk, gk(ak))}tk=t−τ is the difference of evaluated blocks et−τ :t with feature zet−τ:t and τ is a parameter to forget old samples. Then we use EI acquisition function to pick up the next action at+1 in A. More specifically, the same as eq. (4), we let\\nat+1 = arg max A\\n( √ σ2t (zeijk)(γ(zeijk)Φ(γ(zeijk)) + φ(γ(zeijk)))) (13)\\nAs the difference function gt is varying, we take two strategies in Algorithm 2 to update the previous samples to make sure GP regression learns the current difference function well. The first strategy is to remove old samples in Dt. Even if the bandits are slowly varying, the difference function will change significantly after a significant number of rounds. Therefore, we need to forget samples before t− τ . The second strategy is to remove samples near the last block eitjtkt in Dt. As we discuss later, the difference function may change significantly in a local region near the last selected block. Therefore previous samples in this local region will be inaccurate. The resulting algorithm for CorrAttack is shown in Algorithm 1, which mainly follows standard procedure of Bayesian optimization.\\nAlgorithm 1 CorrAttack Require: Loss function `(·, ·), Input x0 and its label y, Action space A = {aeijk |eijk ∈ E}, Parameter c, τ ,\\nα 1: Build set D0 = {(zeipjpkp , g0(aeipjpkp ))} m p=1 using latin hypercube sampling from A 2: repeat 3: Fit the parameter of Normal(µt(zeijk ), σ 2 t (zeijk )) on Dt according to Equation (12) 4: Calculate acquisition function ϕt(zeijk ) and according to Equation (13) 5: Select aeitjtkt = arg maxA ϕt(zeijk ) according to Equation (13) 6: xt+1 = arg mins∈{xt+aeitjtkt ,xt} `(ΠBp(x,ε) (s) , y) 7: Update sample set Dt with Algorithm 2 Dt+1 = UPDATESAMPLES(Dt, xt, xt+1, eitjtkt , gt+1(aeitjtkt ), τ, α) 8: until maxA ϕt(zeijk ) < c 9: return xT ;\\nAlgorithm 2 Update Samples Require: Sample set Dt, State xt, xt+1, Block eitjtkt , Difference gt+1(aeitjtkt ), Paramter τ , α 1: if xt+1 6= xt then 2: Dt+1 = Dt \\\\ {(zeijk , g) ∈ Dt||i− it|+ |j − jt| ≤ α} 3: else 4: Dt+1 = Dt ∪ {(zeitjtkt , gt+1(aeitjtkt ))} 5: end if 6: Remove the earliest sample from D if the cardinality |D| > τ 7: return Dt+1\"\n },\n {\n \"heading\": \"4.2 FEATURES AND SLOW VARYING PROPERTY\",\n \"text\": \"Features of Contextual Bandits: We use a four dimensional vector as the feature zeijk : zeijk = (i, j, k, pca) (14)\\nwhere i, j, k is the location of the block. And pca is the first component of PCA decomposition of [x0(e000), x0(e001), · · ·x0(ehwc)]. x0(eijk) means the block of natural image at the given position. The reward function depends on the gradient in Equation (7). It has been shown that the gradient ∇x`(x, y) has local dependencies (Ilyas et al., 2019). Suppose two coordinates eijk and elpq are close, then∇x`(x, y)ijk ≈ ∇x`(x, y)lpq. We consider the finite difference of the block eijk\\n∆t(eijk) = ` (xt + ηeijk, y)− ` (xt − ηeijk, y) ≈ 2ηe>ijk∇xt` (xt, y) (15)\\nwhere η is a small step size. When η is small, the reward can be approximated by the average of the gradients around a small region, which also has local dependencies. In fact, the local structure of the reward will also be persevered when the block size and η is large. Figure 1 shows one example of the finite difference ∆t(eijk) obtained on ImageNet dataset with ResNet50. This shows blocks with closer locations are more likely to have similar reward. Therefore, we add the location of the block as the feature so that it uses historic data to find the arm with the largest reward.\\nIn addition to the location of the difference, we may add other features. The block of the image itself forms a strong feature for the regression, but the dimension of the block is too high for GP regression. Therefore, we use PCA to lower the dimension and add the first component into the feature vector.\\nSlow Varying Property In addition to the local dependencies of finite difference, the difference would also be slow varying if we just change a small region of xt. Let xt+1 = xt− ηeitjtkt , Figure 2 shows the difference of ∆t(eijk) and ∆t+1(eijk), which is centralized in a small region near eitjtkt . Reward function is based on the finite difference, which also has the slow varying property. It could be explained by the local property of convolution. When η is small, the finite difference can be approximated with gradient and the local Hessian:\\n∆t+1(eijk)−∆t(eijk) ≈ η2e>ijk∇2xt`(xt, y)eitjtkt (16) The difference is much smaller than ∆t(eijk). Today’s neural networks are built with stacks of convolutions and non-linear operations. Since these operations are localized in a small region, the Hessian of a neural network is also localized and the reward function only changes near eitjtkt .\"\n },\n {\n \"heading\": \"4.3 HIERARCHICAL BAYESIAN OPTIMIZATION SEARCH\",\n \"text\": \"Recent black-box approaches (Chen et al., 2017; Moon et al., 2019) exploit the hierarchical image structure for query efficiency. Following these approaches, we take a hierarchical approach and perform the accelerated local search in Algorithm 1 from a coarse grid (large blocks) to a fine grid (smaller blocks). The algorithm for hierarchical attack iteratively performs Algorithm 1 at one block size, and then divides the blocks into smaller sizes. At each block size, we build a Gaussian process to model the difference function, and perform structured search with the blocks until maxA ϕt(zeijk) < c. When dividing the blocks into smaller sizes, we will build a new block set E with action aeijk and new feature zeijk , but keep the xt in last block size as x0 in new block size. Define the stage as S = {0, 1, · · · , s} and initial block size as b. The block at stage s is b2s × b 2s square of pixels and (h,w, c) = (2s ∗ height/b, 2s ∗ width/b, channel). The overall hierarchical accelerated local search algorithm is shown in Appendix A. It is important to note that most of the attacks terminate in the early stages and rarely need to run on fine scales.\"\n },\n {\n \"heading\": \"5 EXPERIMENTS\",\n \"text\": \"We evaluated the number of queries versus the success rates of CorrAttack on both undefended and defended network on ImageNet (Russakovsky et al., 2015). Moreover, we attacked Google Cloud Vision API to show that CorrAttack can generalize to a true black-box model.\\nWe used the common hinge loss proposed in the CW attack (Carlini & Wagner, 2017). We compared two versions of CorrAttack : CorrAttackDiff and CorrAttackFlip, to ZOO (Chen et al., 2017), NES (Ilyas et al., 2018), NAttack (Li et al., 2019), Bandits (Ilyas et al., 2019), PARSI (Moon et al., 2019), Square Attack (Andriushchenko et al., 2020), SignHunter(Al-Dujaili & O’Reilly, 2020b), BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019). We only test adversarial attack on `∞ norm. The details of the Gaussian processes regression and the hyperparameters of CorrAttack are given in the Appendix B. We shall mention that CorrAttack is not sensitive to the hyperparameters. The hyperparameters of other methods follow those suggested by the original papers.\"\n },\n {\n \"heading\": \"5.1 UNDEFENDED NETWORK\",\n \"text\": \"We randomly select 1000 images from the validation set of ImageNet and only attack correctly classified images. The query efficiency of CorrAttack is tested on VGG16 (Simonyan & Zisserman, 2014), Resnet50 (He et al., 2016) and Densenet121 (Huang et al., 2017), which are the most commonly used network structures. We set ε = 0.05 and the query limit to be 10000 except for BayesOpt and Bayes-Attack. For targeted attacks, we randomly choose the target class for each image and the target classes are maintained the same for the evaluation of different algorithms. The results are shown in Table 1 and 2. CorrAttackFlip outperforms other methods by a large margin.\\nAs BayesOpt and Bayes-Attack takes tens of thousands of hours to attack 1000 images, we compare them with CorrAttackFlip only on 20 images and un-targeted attack. The query limit is also reduced to 1000 as the time for BayesOpt and Bayes-Attack quickly increases as more samples add into the Gaussian distribution. The time comparison between three models is shown in Appendix C.6.\\nOptimality of Bayesian optimization Appendix C.1 shows the rank the actions found by CorrAttack. The attacker could find the action with large reward quickly.\\nVarying ε We also test the algorithms at different budget of adversarial perturbations at ε = 0.04 and ε = 0.06 on Resnet50. As it is shown in Appendix C.2, CorrAttack shows a consistently better performance at different ε.\\nAblation study on random choices Appendix C.3 shows the ablation study of random version of CorrAttackDiff and CorrAttackFlip. In both cases, Bayesian optimization helps to gain better query efficiency.\\nAblation study on hierarchical attack We perform ablation study on the hierarchical attack and the result is shown in Appendix C.4. Hierarchical structure accelerates the CorrAttack and eliminates the sensitivity of choosing initial block size.\\nAblation study on features Appendix C.5 demonstrates how the feature of the contextual bandits affects the performance of attack. PCA would help to improve the efficiency of attack.\"\n },\n {\n \"heading\": \"5.2 DEFENDED NETWORK\",\n \"text\": \"To evaluate the effectiveness of CorrAttack on adversarially defended networks, we tested our method on one of the SOTA robust model (Xie et al., 2018) on ImageNet. The weight is downloaded from Github1. \\\"ResneXt DenoiseAll\\\" is chosen as the target model as it achieves the best performance. We set ε = 0.05 and the maximum number of queries is 10000. As BayesOpt runs very slowly, the attack is also performed on 10 images and the query limit is 1000. The result is shown in Table 3. CorrAttackFlip still outperforms other methods.\"\n },\n {\n \"heading\": \"5.3 GOOGLE CLOUD VISION API\",\n \"text\": \"We also attacked Google Cloud Vision API, a real world black-box model for classification. The target is to remove the top-1 label out of the classification output. We choose 10 images for the ImageNet dataset and set the query limit to be 500 due to high cost to use the API. We compare CorrAttackFlip with NAttack, BayesOpt and PARSI. The result is shown in Table 4. We also show one example of the classification output in Appendix C.9\\n1https://github.com/facebookresearch/ImageNet-Adversarial-Training\"\n },\n {\n \"heading\": \"6 CONCLUSION AND FUTURE WORK\",\n \"text\": \"We formulate the score-based adversarial attack as a time-varying contextual bandits and propose a new method CorrAttack. By performing structured search on the blocks of the image, the bandits has the slow varying property. CorrAttack takes advantage of the the features of the arm, and uses Bayesian optimization with Gaussian process regression to learn the reward function. The experiment shows that CorrAttack can quickly find the action with large reward and CorrAttack achieves superior query efficiency and success rate on ImageNet and Google Cloud Vision API.\\nWe only include basic features for learning the bandits. Other features like embedding from the transfer-based attack Huang & Zhang (2020) may be taken into account in the future work. While our work only focuses on adversarial attack on `∞ norm, the same contextual bandits formulation could be generalized to other `p norm to improve query efficiency. Besides, defense against CorrAttack may be achieved with adversarial training on CorrAttack , but it may not be able to defend other attacks in the meantime.\"\n },\n {\n \"heading\": \"A ALGORITHM\",\n \"text\": \"Algorithm 3 Split Block Require: Set of blocks E, Block size b, E′ = ∅ 1: for each block e ∈ E do 2: Split the block e into 4 blocks {e1, e2, e3, e4} with size b/2 3: E′ ← E′ ∪ {e1, e2, e3, e4} 4: end for 5: return E′;\\nAlgorithm 4 Hierarchical CorrAttackDiff Require: Loss function `(·, ·), Input image x and its label y, Initial Block size b, Set of blocks E containing all\\nblocks of the image, Threshold c, τ , α, Step size η, Adversarial budget ε 1: x0 = x 2: repeat 3: Choose A = {aeijk |eijk ∈ E} with Equation (8) 4: Run CorrAttack on current block size x = CORRATTACK (`(·, ·), x, y, A, c, τ, α) 5: if b > 1 then 6: Split the blocks into finer blocks using Algorithm 3 E = SPLITBLOCK(E, b) 7: b← b/2 8: end if 9: until ` converges\\n10: return xK ;\\nAlgorithm 5 Hierarchical CorrAttackFlip Require: Loss function `(·, ·), Input image x and its label y, Block size b, Set of blocks E containing all blocks\\nof the image, Threshold c, τ , α, Adversarial budget ε 1: x0 = x 2: for eijk ∈ E do 3: Randomly draw v from {−ε, ε} 4: x0[eijk] = v + x0[eijk] 5: end for 6: repeat 7: An = {2εeijk ∈ E|e>ijk(xk − x) < 0} 8: Run CorrAttack flipping −ε to ε x̃k = CORRATTACK (`(·, ·), xk, y, An, c, τ, α) 9: Ap = {−2εeijk ∈ E|e>ijk(x̃k − x) > 0} 10: Run CorrAttack flipping ε to −ε xk+1 = CORRATTACK (`(·, ·), x̃k, y, Ap, c, τ, α) 11: if b > 1 then 12: Split the blocks into finer blocks using Algorithm 3 E = SPLITBLOCK(E, b) 13: b← b/2 14: end if 15: until ` converges 16: return xK ;\"\n },\n {\n \"heading\": \"B DETAILS OF EXPERIMENT SETTING\",\n \"text\": \"We use the hinge loss for all the experiments. For un-targeted attacks,\\n`untarget(x, y) = max { F (x)y −max\\nj 6=y F (x)j ,−ω\\n} (17)\\nand for targeted attacks,\\n`target(x, y) = max { max j F (x)j − F (x)t,−ω } . (18)\\nHere F represents the logits of the network outputs, t is the target class, and ω denotes the margin. The image will be projected into the ε-ball. Besides, the value of the image will be clipped to range [0, 1].\"\n },\n {\n \"heading\": \"B.1 GAUSSIAN PROCESS REGRESSION AND BYAESIAN OPTIMIZATION\",\n \"text\": \"We further provide details on both the computational scaling and modeling setup for the GP regression.\\nTo address computational issues, we use GPyTorch (Gardner et al., 2018) for scalable GP regression. GPyTorch follows (Dong et al., 2017) to solve linear systems using the conjugate gradient (CG) method and approximates the log-determinant via the Lanczos process. Without GPyTorch, running BO with a GP regression for more than a few thousand evaluations would be infeasible as classical approaches to GP regression scale cubically in the number of data points.\\nOn the modeling side, the GP is parameterized using a Matérn-5/2 kernel with ARD and a constant mean function for all experiments. The GP hyperparameters are fitted before proposing a new batch by optimizing the log-marginal likelihood. The domain is rescaled to [0, 1]d and the function values are standardized before fitting the GP regression. We use a Matérn-5/2 kernel with ARD for CorrAttack and use the following bounds for the hyperparameters: (length scale) λi ∈ [0.005, 2.0 ], (output scale) λ′i ∈ [0.05, 20.0], (noise variance) σ2 ∈ [0.0005, 0.1].\"\n },\n {\n \"heading\": \"B.2 HYPERPARAMETERS\",\n \"text\": \"For CorrAttack in Algorithm 4 and Algorithm 5, we set the initial block size b to be 32 and the step size η for CorrAttackDiff is 0.03. In Algorithm 1, we use the initial sampling ratio m = 0.03n at the start point for Gaussian process regression, the threshold c = 10−4 to decide when to stop the search of current block size. In Algorithm 2, the threshold is different for different block size. For CorrAttackFlip, α = 1, 1, 2, 2, 3 for block size 32, 16, 8, 4, 2 and for CorrAttackDiff, α = 0, 0, 1, 1, 2 for block size 32, 16, 8, 4, 2. We set τ = 3m = 0.09n to remove the earliest samples from D once |D| > α. The Adam optimizer is used to optimize the mean µ and covariance κ of Gaussian process, where the iteration is 1 and the learning rate is 0.1.\\nFor PARSI, the block size is set to 32 as CorrAttack , other hyperparameters are the same as the original paper.\\nFor Bandits, Bayes-Attack and BayesOpt, the hyperparameters are the same as the original paper.\\nWe optimize the hyperparameters for ZOO, NES. For un-targeted attack on NES, we set the sample size to be 50, learning rate to be 0.1. For targeted attack on NES, the sample size is also 50 and the learning rate is 0.05. The learning is decay by 50% if the loss doesn’t decrease for 20 iterations.\\nFor NAttack, we set the hyperparameters the same as NES and add momentum and learning rate decay, which are not mentioned in the original paper.\\nFor ZOO, we set the learning rate to 1.0 and sample size to be 50. Other setting follows the original paper.\"\n },\n {\n \"heading\": \"C ADDITIONAL EXPERIMENTS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"C.1 OPTIMALITY OF BAYESIAN OPTIMIZATION\",\n \"text\": \"Figure 3 shows the reward function that the Bayesian optimization could find in the action set. CorrAttack could find the action with high reward within just a few queries. It shows that the Gaussian process regression could model the correlation of the reward function and the Bayesian optimization could use it to optimize the time-varying contextual bandits.\\nC.2 VARYING THE ADVERSARIAL BUDGET\\nWe test CorrAttack on different adversarial budget on ImageNet for both un-targeted attack and targeted attack. Table 5 and Table 6 show the success rate and average queries for ε = 0.04, 0.05, 0.06. CorrAttackFlip achieves the best performance among all methods.\"\n },\n {\n \"heading\": \"C.3 ABLATION STUDY ON RANDOM CHOICES\",\n \"text\": \"Table 7 and Table 8 show the ablation study on the strategy to choose action xt+1 in the line 6 of Algorithm 1. The process of Bayesian optimization helps to accelerate the optimization. As targeted attack is more complicated and requires larger number of queries, CorrAttack has more advantage in this scenario.\"\n },\n {\n \"heading\": \"C.4 ABLATION STUDY ON HIERARCHICAL ATTACK\",\n \"text\": \"We perform un-targeted attack on Resnet50 as shown in Table 10. Hierarchical attack lowers the average queries and improves the query efficiency. Besides, hierarchical attack avoids the problem of choosing block size. As shown in Table 10, block size for non-hierarchical is essential for the performance.\"\n },\n {\n \"heading\": \"C.5 ABLATION STUDY ON FEATURES\",\n \"text\": \"Table 11 shows the success rate and average queries for CorrAttackwith different features. We perform ablation study on the features of the contextual bandits. One contains just the location of the\\nblock and the other contains both the location and the PCA feature. PCA helps the learning process of the reward and achieve higher success rate and lower number of queries. PCA feature achieves significant improvement on CorrAttackFlip. We may find more useful features in the future.\"\n },\n {\n \"heading\": \"C.6 COMPARISON BETWEEN CORRATTACKFLIP , BAYESOPT AND BAYES-ATTACK\",\n \"text\": \"The main difference between BayesOpt and Bayes-Attack is using different types of GP regression (Standard GP for Bayes-Attack and Additive GP for BayesOpt), so we will consider these two models as a group when comparing with our model CorrAttack.\\nDifference between CorrAttack, BayesOpt and Bayes-Attack: For l∞ attacks, assume there are no hierarchical structure, we have blocksE = {e000, e001, · · · , ehwc}, where the block is b×b square of pixels and (h,w, c) = (height/b,width/b, channel). CorrAttack, BayesOpt (Ru et al., 2020) and Bayes-Attack (Shukla et al., 2019) all try to search the adversarial noise on E with perturbation δ ∈ [− , ]d where d = h× w × c, the perturbation of block eijk at time t is δteijk .\\nBayesOpt and Bayes-Attack use a GP regression directly on δ ∈ [− , ]d (all blocks),\\nf(δ)|Dn ∼ Normal(µn(δ), σ2n(δ)). (19)\\nCorrAttack define an action spaceA and use a standard GP regression on features zeijk = (i, j, k, pca) (single block),\\ngt(aeijk)|Dt ∼ Normal(µt(zeijk), σ2t (zeijk)). (20)\\nAt each iteration, in BayesOpt and Bayes-Attack, the changes of overall perturbation is\\nδt − δt−1 = {δte000 ∪ δ t e001 ∪ · · · ∪ δ t ehwc } − {δt−1e000 ∪ δ t−1 e001 ∪ · · · ∪ δ t−1 ehwc }. (21)\\nHowever, in CorrAttack,\\nδt − δt−1 = δteijk − δ t−1 eijk . (22)\\nIn conclusion, BayesOpt and Bayes-Attack view each block as a dimension, try to search the overall perturbation directly. CorrAttack defines a low dimension feature space, keep an overall perturbation and try to search an action on single block.\\nTime complexity and running time: The time complexity of fitting GP regression is O(dn2) where d is the dimension of input and n is the number of samples. And the dimension for CorrAttack (d = 4 for zeijk = (i, j, k, pca)) is much smaller than BayesOpt and Bayes-Attack (d = 6912 if h = w = 48, c = 3). Moreover, we can convert the continuous search space of BayesOpt and Bayes-Attack from [− , ]6912 to discrete search space E = {e000, e001, · · · , ehwc}, whose number is only 6912, smaller search space could save the computation time of acquisition function.\\nWe compare the running time for CorrAttackFlip with BayesOpt and Bayes-Attack on 20 images from ImageNet. Table 12 shows the running time for the un-targeted attack. We use PyTorch2 to develop these two models. All experiments were conducted on a personal workstation with 28 Intel(R) Xeon(R) Gold 5120 2.20GHz CPUs, an NVIDIA GeForce RTX2080Ti 11GB GPU and 252G memory.\\nBayesOpt models the loss function with a very high dimensional Gaussian process. The decomposition of additive kernel also needs to be restarted several times. Even though we try to optimize the speed of BayesOpt with GPU acceleration, it is still very slow and takes hundreds of times more computational resources than CorrAttack .\\nBayes-Attack could be regarded as a simpler version of BayesOpt, which does not add additive kernel. We do not evaluate it on targeted task (when query>1000) since GP inference time grows fast as evaluated query increases, e.g. For Bayes-Attack, when 150 <query< 200, Time=1.6s/query; 800 <query< 1000, Time = 10.5s/query. CorrAttack solves this problem with Time=0.1s/query even when query reaches 10000. Since we forget the previous samples before t − τ , our input sample n will be smaller than τ . The forgetting technique can not be applied into the Bayes-Attack and BayesOpt since they are searching the perturbation of all blocks so each sample needs to be remembered.\"\n },\n {\n \"heading\": \"C.7 GROWING CURVE OF SUCCESS RATE\",\n \"text\": \"The number of average queries is sometimes misleading due to the the heavy tail distribution of queries. Therefore in Figure 4, we plot the success rates at different query levels to show the detailed behaviors of different attacks. It shows that CorrAttack is much more efficient than other methods at all query levels.\\n2https://pytorch.org/\\nC.8 VISUALIZATION OF LOCAL PROPERTY AND SLOW VARYING PROPERTY\\nFigure 5 shows more examples of finite difference for different network architectures and different dataset. They all have local correlation structure as shown in Figure 1. And Figure 6 shows more examples like Figure 2, the slow varying properties exist for different architectures and different datasets.\"\n },\n {\n \"heading\": \"C.9 GOOGLE CLOUD VISION API\",\n \"text\": \"Figure 7 shows the example of attacking the Google Cloud Vision API. CorrAttackFlip and PARSI successfully change the classification result. BeyesOpt, however, can not remove the top 1 classification result out of the output.\\nC.10 VISUALIZATION OF ADVERSARIAL EXAMPLES\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"string","value":"CORRATTACK: BLACK-BOX ADVERSARIAL ATTACK"}}},{"rowIdx":93,"cells":{"paper_id":{"kind":"string","value":"SP:9403fa2679f18af78aed2e81b75eb39abeb722eb"},"summaries":{"kind":"list like","value":["The paper develops a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. It shows theoretically and empirically that SGD favors flat minima exponentially more than sharp minima. In particular, the paper analyzed the dependence of mean escape time from the valley with the Hessians on local minima and saddle points for both SGD and SGLD, and revealed the exponential dependence of the mean escape time with the sharpness. Experiments on real-world data have verified the theoretical results on the mean escape time. "],"string":"[\n \"The paper develops a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. It shows theoretically and empirically that SGD favors flat minima exponentially more than sharp minima. In particular, the paper analyzed the dependence of mean escape time from the valley with the Hessians on local minima and saddle points for both SGD and SGLD, and revealed the exponential dependence of the mean escape time with the sharpness. Experiments on real-world data have verified the theoretical results on the mean escape time. \"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks in practice. SGD is known to find a flat minimum that often generalizes well. However, it is mathematically unclear how deep learning can select a flat minimum among so many minima. To answer the question quantitatively, we develop a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. To the best of our knowledge, we are the first to theoretically and empirically prove that, benefited from the Hessian-dependent covariance of stochastic gradient noise, SGD favors flat minima exponentially more than sharp minima, while Gradient Descent (GD) with injected white noise favors flat minima only polynomially more than sharp minima. We also reveal that either a small learning rate or large-batch training requires exponentially many iterations to escape from minima in terms of the ratio of the batch size and learning rate. Thus, large-batch training cannot search flat minima efficiently in a realistic computational time."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"DESCENT EXPONEN"},{"affiliations":[],"name":"TIALLY FAVORS"},{"affiliations":[],"name":"FLAT MINIMA"},{"affiliations":[],"name":"Zeke Xie"},{"affiliations":[],"name":"Issei Sato"},{"affiliations":[],"name":"Masashi Sugiyama"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"DESCENT EXPONEN\"\n },\n {\n \"affiliations\": [],\n \"name\": \"TIALLY FAVORS\"\n },\n {\n \"affiliations\": [],\n \"name\": \"FLAT MINIMA\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Zeke Xie\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Issei Sato\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Masashi Sugiyama\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Alessandro Achille","Stefano Soatto"],"title":"Where is the information in a deep neural network","venue":"arXiv preprint arXiv:1905.12213,","year":2019},{"authors":["George B Arfken","Hans J Weber"],"title":"Mathematical methods for physicists","venue":null,"year":1999},{"authors":["Devansh Arpit","Stanisław Jastrzębski","Nicolas Ballas","David Krueger","Emmanuel Bengio","Maxinder S Kanwal","Tegan Maharaj","Asja Fischer","Aaron Courville","Yoshua Bengio"],"title":"A closer look at memorization in deep networks","venue":"In International Conference on Machine Learning,","year":2017},{"authors":["Nils Berglund"],"title":"Kramers’ law: Validity, derivations and generalisations","venue":"Markov Processes and Related Fields,","year":2013},{"authors":["Pratik Chaudhari","Stefano Soatto"],"title":"Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks","venue":"Information Theory and Applications Workshop (ITA),","year":2018},{"authors":["Pratik Chaudhari","Anna Choromanska","Stefano Soatto","Yann LeCun","Carlo Baldassi","Christian Borgs","Jennifer Chayes","Levent Sagun","Riccardo Zecchina"],"title":"Entropy-sgd: Biasing gradient descent into wide valleys","venue":"In International Conference on Learning Representations,","year":2017},{"authors":["William Coffey","Yu P Kalmykov"],"title":"The Langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering, volume 27","venue":"World Scientific,","year":2012},{"authors":["Claudio De Stefano","Marilena Maniaci","Francesco Fontanella","A Scotto di Freca"],"title":"Reliable writer identification in medieval manuscripts through page layout features: The “avila","venue":"bible case. 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Dahl\",\n \"Chris Shallue\",\n \"Roger B Grosse\"\n ],\n \"title\": \"Which algorithmic choices matter at which batch sizes? insights from a noisy quadratic model\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yuchen Zhang\",\n \"Percy Liang\",\n \"Moses Charikar\"\n ],\n \"title\": \"A hitting time analysis of stochastic gradient langevin dynamics\",\n \"venue\": \"In Conference on Learning Theory, pp. 1980–2022,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Huan-Xiang Zhou\"\n ],\n \"title\": \"Rate theories for biologists\",\n \"venue\": \"Quarterly reviews of biophysics,\",\n \"year\": 2010\n },\n {\n \"authors\": [\n \"Zhanxing Zhu\",\n \"Jingfeng Wu\",\n \"Bing Yu\",\n \"Lei Wu\",\n \"Jinwen Ma\"\n ],\n \"title\": \"The anisotropic noise in stochastic gradient descent: Its behavior of escaping from sharp minima and regularization effects\",\n \"venue\": \"In ICML,\",\n \"year\": 2019\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various application areas. Due to the over-parametrization and the highly complex loss landscape of deep networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995; 1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al., 2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al., 2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al., 2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b), and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a). However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat minimum.\nThe diffusion theory is an important theoretical tool to understand how deep learning dynamics works. It helps us model the diffusion process of probability densities of parameters instead of model parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics (SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrzębski et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented a Beyesian perspective on generalization of SGD. Wu et al. (2018) studied the escape problems of\nSGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained a related proposition that describes the mean escape time in terms of a free energy that depends on the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise.\nHowever, the related papers mainly analyzed the diffusion process under parameter-independent and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four contributions:\n• The proposed theory formulates the fundamental roles of gradient noise, batch size, the learning rate, and the Hessian in minima selection. • The SGN covariance is approximately proportional to the Hessian and inverse to batch size. • Either a small learning rate or large-batch training requires exponentially many iterations to\nescape minima in terms of ratio of batch size and learning rate. • To the best of our knowledge, we are the first to theoretically and empirically reveal that\nSGD favors flat minima exponentially more than sharp minima."},{"heading":"2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS","text":"We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We denote the data samples as {xj}mj=1, the model parameters as θ and the loss function over data samples x as L(θ, x). For simplicity, we denote the training loss as L(θ). Following Mandt et al. (2017), we may write SGD dynamics as\nθt+1 = θt − η ∂L̂(θt)\n∂θt = θt − η\n∂L(θt)\n∂θt + ηC(θt)\n1 2 ζt, (1)\nwhere L̂(θ) is the loss of one minibatch, ζt ∼ N (0, I), and C(θ) represents the gradient noise covariance matrix. The classic approach is to model SGN by Gaussian noise, N (0, C(θ)) (Mandt et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018).\nStochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch training, C(θt) 1 2 ζt =\n∂L(θt) ∂θt − ∂L̂(θt)∂θt , which is the difference between gradient descent and stochastic\ngradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the mean of many infinite-variance random variables converges to a stable distribution, while the mean of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in practice, we believe the Gaussian approximation of SGN is reasonable.\nSimsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise. They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed distribution looks closer to a stable distribution than a Gaussian distribution. However, this research line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed “SGN” across n model parameters and regarded “SGN\" as n samples drawn from a single-variant distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic Gaussian noise. In our paper, SGN computed over different minibatches obeys a n-variant Gaussian distribution, which can be parameter-dependent and anisotropic.\nIn Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that\nthe contradicted observations are from the different formulations of gradient noise. Simsekli et al. (2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a n-variant distribution. Our empirical analysis in Figure 1 holds well at least when the batch size B is larger than 16, which is common in practice. Similar empirical evidence can be observed for training ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C.\nPanigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN is best described as Gaussian at-least in the early phases of training. Comparing our results with Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors. First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network matters. Because SGN on some layers/networks is more Gaussian than other layers/networks.\nThe isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameterdependent and anisotropic SGN brings a large improvement over existing parameter-independent and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under parameter-independent anisotropic heavy-tailed noise, when the batch size is too small (B ∼ 1) to apply Central Limit Theorem. We will leave it as future work.\nSGD Dynamics. Let us replace η by dt as unit time. Then the continuous-time dynamics of SGD (Coffey & Kalmykov, 2012) is written as\ndθ = −∂L(θ) ∂θ dt+ [2D(θ)] 1 2 dWt, (2)\nwhere dWt ∼ N (0, Idt) and D(θ) = η2C(θ). We note that the dynamical time t in the continuoustime dynamics is equal to the product of the number of iterations T and the learning rate η: t = ηT . The associated Fokker-Planck Equation is written as\n∂P (θ, t)\n∂t =∇ · [P (θ, t)∇L(θ)] +∇ · ∇D(θ)P (θ, t) (3)\n= ∑ i ∂ ∂θi [ P (θ, t) ∂L(θ) ∂θi ] + ∑ i ∑ j ∂2 ∂θi∂θj Dij(θ)P (θ, t), (4)\nwhere∇ is a nabla operator, and Dij is the element in the ith row and jth column of D. In standard SGLD, the injected gradient noise is fixed and isotropic Gaussian, D = I .\nThe next question is how to formulate the SGN covariance C(θ) for SGD? Based on Smith & Le (2018), we can express the SGN covariance as\nC(θ) = 1\nB 1 m m∑ j=1 ∇L(θ, xj)∇L(θ, xj)> −∇L(θ)∇L(θ)> ≈ 1 Bm m∑ j=1 ∇L(θ, xj)∇L(θ, xj)>.\n(5)\nThe approximation is true near critical points, due to the fact that the gradient noise variance dominates the gradient mean near critical points. We know the observed fisher information matrix satisfies FIM(θ) ≈ H(θ) near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrzębski et al. (2017); Zhu et al. (2019), we obtain\nC(θ) ≈ 1 Bm m∑ j=1 ∇L(θ, xj)∇L(θ, xj)> = 1 B FIM(θ) ≈ 1 B H(θ), (6)\nwhich approximately gives\nD(θ) = η\n2 C(θ) =\nη\n2B H(θ) (7)\nnear minima. It indicates that the SGN covariance C(θ) is approximately proportional to the Hessian H(θ) and inverse to the batch size B. Obviously, we can generalize Equation 7 by D(θ) = ηC(θ)2 = η 2B [H(θ)]\n+ near critical points, when there exist negative eigenvalues in H along some directions. We use [·]+ to denote the positive semidefinite transformation of a symmetric matrix: if we have the eigendecomposation H = U diag(H1, · · · , Hn−1, Hn)U>, then [H]+ = U diag(|H1|, · · · , |Hn−1|, |Hn|)U>. We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a followup paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that, the relation still approximately holds for even the randomly network, which is far from critical points. The correlation is especially high along the flat directions with small-magnitude eigenvalues of the Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian noise approximation all failed to capture this core relation in deep learning dynamics."},{"heading":"3 SGD DIFFUSION THEORY","text":"We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We assume there are two valleys, Sharp Valley a1 and Flat Valley a2, seen in Figure 3. Also Col b is the\nboundary between two valleys. What is the mean escape time for a particle governed by Equation 2 to escape from Sharp Valley a1 to Flat Valley a2? The mean escape time is widely used in related statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019).\nGauss’s Divergence Theorem (Arfken & Weber, 1999; Lipschutz et al., 2009) states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. We respectively denote the mean escape time as τ , the escape rate as γ, and the probability current as J . We apply Gauss’s Divergence Theorem to the Fokker-Planck Equation resulting in\n∇ · [P (θ, t)∇L(θ)] +∇ · ∇D(θ)P (θ, t) = ∂P (θ, t) ∂t = −∇ · J(θ, t). (8)\nThe mean escape time is expressed (Van Kampen, 1992) as\nτ = 1 γ = P (θ ∈ Va)∫ Sa J · dS , (9)\nwhere P (θ ∈ Va) = ∫ Va P (θ)dV is the current probability inside Valley a, J is the probability\ncurrent produced by the probability source P (θ ∈ Va), j = ∫ Sa J · dS is the probability flux (surface integrals of probability current), Sa is the surface (boundary) surrounding Valley a, and Va is the volume surrounded by Sa. We have j = J in the case of one-dimensional escape.\nClassical Assumptions. We state three classical assumptions first for the density diffusion theory. Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring, 1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012), and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers (Jastrzębski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape Problems.\nAssumption 1 (The Second Order Taylor Approximation). The loss function around critical points θ? can be approximately written as\nL(θ) = L(θ?) + g(θ?)(θ − θ?) + 1 2 (θ − θ?)>H(θ?)(θ − θ?).\nAssumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima. Assumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature).\nWe will dive into these two assumptions deeper than previous papers for SGD dynamics. Assumptions 2 and 3 both mean that our diffusion theory can better describe the escape processes that cost more iterations. As this class of “slow” escape processes takes main computational time compared with “fast” escape processes, this class of “slow” escape process is more interesting for training of deep neural networks. Our empirical analysis in Section 4 supports that the escape processes in the\nwide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus, Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B.\nEscape paths. We generalize the concept of critical points into critical paths as the path where 1) the gradient perpendicular to the path direction must be zero, and 2) the second order directional derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs) for escaping must be critical paths. The most possible escape direction at one point must be the direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs, which are strictly flat, is ignorable according to our following analysis.\nThe boundary between Sharp Valley a1 and Flat Valley a2 is the saddle point b. The Hessian at b, Hb, must have only one negative eigenvalue and the corresponding eigenvector is the escape direction. Without losing generality, we first assume that there is only one most possible path through Col b existing between Sharp Valley a1 and Flat Valley a2.\nSGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers are interested in SGLD, when the injected noise dominates SGN as η → 0 in final epochs. Because SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is usually simplified as Gradient Descent with injected white noise, whose behavior is identical to Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a mixture of injected white noise and SGN. Theorem 3.1 (SGLD Escapes Minima). The loss function L(θ) is of class C2 and n-dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the outside of Valley a is\nτ = 1\nγ = 2π √ −det(Hb) det(Ha) 1 |Hbe| exp ( ∆L D ) .\nWe denote that Ha and Hb are the Hessians of the loss function at the minimum a and the saddle point b, ∆L = L(b)− L(a) is the loss barrier height, e indicates the escape direction, and Hbe is the eigenvalue of the Hessian Hb corresponding to the escape direction. The diffusion coefficient D is usually set to 1 in SGLD.\nSGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD is far from the Gibs-Boltzmann distribution, P (θ) = 1Z exp ( −L(θ)D ) . These different characteristics make SGD diffusion behave differently from known physical dynamical systems and much less studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN and injected white noise, as long as the eigenvectors of D(θ) are closely aligned with the eigenvectors of H(θ). Theorem 3.2 (SGD Escapes Minima). The loss function L(θ) is of class C2 and n-dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGD, then the mean escape time from Valley a to the outside of Valley a is\nτ = 2π 1\n|Hbe| exp\n[ 2B∆L\nη\n( s\nHae + (1− s) |Hbe|\n)] ,\nwhere s ∈ (0, 1) is a path-dependent parameter, and Hae and Hbe are, respectively, the eigenvalues of the Hessians at the minimum a and the saddle point b corresponding to the escape direction e.\nMultiple-path escape. Each escape path contributes to the total escape rate. Multiple paths combined together have a total escape rate. If there are multiple parallel from the start valley to the end valley, we can compute the total escape rate easily based on the following computation rule. The computation\nrule is based on the fact that the probability flux integrals are additive. We can easily generalize the mean escape time analysis into the cases that there are multiple parallel escape paths indexed by p. As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the dimensionality directly, higher dimensionality may increase the number of escape paths and loss valleys, and change the spectrum of the Hessians.\nRule 1. If there are multiple MPPs between the start valley and the end valley, then γtotal = ∑ p γp.\nThus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and analyze the escape rates.\nMinima selection. Now, we may formulate the probability of minima selection as Proposition 1. We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and the landscape contain many good modes and bad modes. SGD transits from one mode to another mode during training. The mean escape time of one mode corresponds to the number of iterations which SGD spends on this mode during training, which is naturally proportional to the probability of selecting this mode after training.\nProposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by\nP (θ ∈ Va) = τa∑ v τv ,\nwhere v is the index of valleys, and τv is the mean escape time from Valley v to the outside of Valley v."},{"heading":"4 EMPIRICAL ANALYSIS","text":"In this section, we try to directly validate the escape formulas on real-world datasets. Each escape process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100 times under various gradient noise scales, batch sizes, learning rates, and sharpness.\nHow to compare the escape rates under the same settings with various minima sharpness? Our method is to multiply a rescaling factor √ k to each parameter, and the Hessian will be proportionally rescaled by a factor k. If we let L(θ) = f(θ)→ L(θ) = f( √ kθ), then H(θ) = ∇2f(θ)→ H(θ) = k∇2f(θ). Thus, we can use k to indicate the minima sharpness. The theoretical relations of SGD we try to validate can be formulated as: (1) − log(γ) = O( 1k ), (2) − log(γ) = O(B), and (3) − log(γ) = O( 1η ).\nThe mean escape time analysis of SGD. Styblinski-Tang Function, which has multiple minima and saddle points, is a common test function for nonconvex optimization. We conduct an intuitional 10-dimensional experiment, where the simulations start from a given minimum and terminate when reaching the boundary of the loss valley. The number of iterations is recorded for calculating\nthe escape rate. We also train fully connected networks on four real-world datasets, including a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis (De Stefano et al., 2018; Dua & Graff, 2017). Figure 4 and Figure 5 clearly verifies that the escape rate exponentially depends on the minima sharpness (reflected by k), the batch size, and the learning rate on both test functions and real-world training, which fully supports our theoretical results.\nModel architecture and details: We used fully-connected networks with the depth 2 and the width 10 in Figure 5. The experiments using Logistic Regression and Fully-connected networks with the depth 3 are presented in Appendix E. We leave more experimental details and results in Appendix D.1 and Appendix E.\nThe mean escape time analysis of SGLD. We try to validate γ = O(k) and − log(γ) = O( 1D ) for SGLD (dominated by injected Gaussian noise). Figure 6 shows that SGLD only favors flat minima polynomially more than sharp minima as Theorem 3.1 indicates. Figure 6 also verifies that the injected gradient noise scale exponentially affects flat minima selection."},{"heading":"5 DISCUSSION","text":"SGD favors flat minima exponentially more than sharp minima. We can discover a few interesting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially depends on the eigenvalue of the Hessian at minima along the escape direction, Hae. Thus, SGD favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD comes from the exponential relation of the mean escape time and the minima sharpness. The measure of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness” is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly more than Adam.\nThe ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains why large-batch training can easily get trapped near sharp minima, and increasing the learning rate proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause is large-batch training expects exponentially longer time to escape minima. Note that, as the mean escape time in the theorems is equivalent to the product of the learning rate and the number of iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the batch size and the learning rate. The practical computational time in large-batch training is usually too short to search many enough flat minima. We conjecture that exponentially increasing training iterations may be helpful for large batch training, while this is often too expensive in practice.\nLow dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of overparametrized deep networks are close to zero, while only a small number of eigenvalues are large (Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018).\nAs the escape rate exponentially depends the corresponding eigenvalues, a small number of large eigenvalues means that the process of minima selection mainly happens in the relatively low dimensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics hardly depends on those “meaningless” dimensions with small second order directional derivatives. This novel characteristic of SGD significantly reduces the explorable parameter space around one minimum into a much lower dimensional space.\nHigh-order effects. As we have applied the second-order Taylor approximation near critical points, our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this paper. However, we also argue that the third-order effect is much smaller than the second-order effect under the low temperature assumption in Kramers Escape Problems. We will leave the more refined high-order theory as future work."},{"heading":"6 CONCLUSION","text":"In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to formulate the exponential relation of minima selection to the minima sharpness, the batch size, and the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory not only helps us understand how SGD selects flat minima, but also will provide researchers a powerful theoretical tool to analyze more learning behaviors and design better optimizers in future."},{"heading":"ACKNOWLEDGEMENT","text":"We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study."},{"heading":"A PROOFS","text":""},{"heading":"A.1 PROOF OF THEOREM 3.1","text":"Proof. This proposition is a well known conclusion in statistical physics under Assumption 1, 2 and 3. We still provide an intuitional proof here, and the following proof of SGD Diffusion will closely relate to this proof. We decompose the proof into two steps: 1) compute the probability of locating in valley a, P (θ ∈ Va), and 2) compute the probability flux j = ∫ Sa J · dS.\nWithout losing generality, we first prove the one-dimensional case.\nStep 1: Under Assumption 1, the stationary distribution around minimum a is P (θ) = P (a) exp[−L(θ)−L(a)T ], where T = D. Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. We use the T notation as the temperature parameter in the stationary distribution, and use the D notation as the diffusion coefficient in the dynamics, for their different roles.\nP (θ ∈ Va) (10)\n= ∫ θ∈Va P (θ)dV (11)\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) T ] dθ (12)\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) T ] dθ (13)\n=P (a) (2πT )\n1 2\nH 1 2 a\n. (14)\nStep 2:\nJ =P (θ)∇L(θ) + P (θ)∇D +D∇P (θ) (15)\nJ =P (θ) ( ∇L(θ) +∇D − D\nT ∇L(θ)\n) (16)\n∇D = ( D T − 1 ) ∇L (17)\nApply this result to the Fokker-Planck Equation 4, we have\n∇ · ∇[D(θ)P (θ, t)] (18) =∇ ·D∇P (θ, t) +∇ · [( D T − 1 ) ∇L(θ) ] P (θ, t) (19)\nAnd thus we obtain the Smoluchowski equation and a new form of J\n∂P (θ, t)\n∂t = ∇ ·\n[ D ( 1\nT ∇L(θ) +∇\n) P (θ, t) ] = −∇ · J(θ, t), (20)\nJ(θ) = D exp ( −L(θ) T ) ∇ [ exp ( L(θ) T ) P (θ) ] . (21)\nWe note that the probability density outside Valley a must be zero, P (c) = 0. As we want to compute the probability flux escaping from Valley a in the proof, the probability flux escaping from other valleys into Valley a should be ignored. Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path∫ c\na\n∂\n∂θ\n[ exp ( L(θ)\nT\n) P (θ) ] dθ = ∫ c a − J D exp ( L(θ) T ) dθ (22)\nexp\n( L(θ)\nT\n) P (θ)|ca =− J\nD ∫ c a exp ( L(θ) T ) dθ (23)\n0− exp ( L(a)\nT\n) P (a) =− J\nD ∫ c a exp ( L(θ) T ) dθ (24)\nJ = D exp\n( L(a) T ) P (a)∫ c\na exp ( L(θ) T ) dθ . (25)\nWe move J to the outside of integral based on Gauss’s Divergence Theorem, because J is fixed on the escape path from one minimum to another. As there is no field source on the escape path,∫ V ∇ · J(θ)dV = 0. Then ∇J(θ) = 0. Obviously, only minima are probability sources in deep learning. Under Assumption 3 and the second-order Taylor approximation, we have∫ c a exp ( L(θ) T ) dθ (26)\n= ∫ c a exp [ L(b) + 12 (θ − b) >Hb(θ − b) +O(∆θ3) T ] dθ (27)\n≈ exp ( L(b)\nT )∫ +∞ −∞ exp [ 1 2 (θ − b) >Hb(θ − b) T ] dθ (28)\n= exp\n( L(b)\nT\n)√ 2πT\n|Hb| . (29)\nBased on the results of Step 1 and Step 2, we obtain\nγ =\n∫ Sa J · dS P (θ ∈ Va) =\nJ\nP (θ ∈ Va) (30)\n= DP (a) exp\n( L(a) T ) exp ( L(b) T )√ 2πT |Hb|\n1 P (a) √\n2πT Ha\n(31)\n= D √ Ha|Hb|\n2πT exp\n( −∆Lab\nT\n) (32)\n= √ Ha|Hb| 2π exp ( −∆Lab D ) (33)\nWe generalize the proof of one-dimensional diffusion to high-dimensional diffusion\nStep 1: P (θ ∈ Va) (34)\n= ∫ θ∈Va P (θ)dV (35)\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) T ] dV (36)\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) T ] dV (37)\n=P (a) (2πT )\nn 2\ndet(Ha) 1 2\n(38)\nStep 2: Based on the formula of the one-dimensional probability current and flux, we obtain∫ Sb J · dS (39)\n=Jb ∫ Sb exp [ − 1 2 (θ − b) >H+b (θ − b) T ] dS (40)\n=Jb (2πT )\nn−1 2 ( ∏n−1 i=1 Hbi) 1 2\n(41)\nSo we have\nτ =2π\n√ ∏n−1 i=1 Hbi\ndet(Ha)|Hbe| exp\n( ∆L\nT\n) (42)\n=2π √ −det(Hb) det(Ha) 1 |Hbe| exp ( ∆L D ) . (43)"},{"heading":"A.2 PROOF OF THEOREM 3.2","text":"Proof. We decompose the proof into two steps and analyze the one-dimensional case like before. The following proof is similar to the proof of SGLD except that we make Ta the temperature near the minimum a and Tb the temperature near the saddle point b.\nOne-dimensional SGD Diffusion:\nStep 1: Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points.\nP (θ ∈ Va) (44)\n= ∫ θ∈Va P (θ)dV (45)\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) Ta ] dV (46)\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) Ta ] dθ (47)\n=P (a) (2πTa)\n1 2\nH 1 2 a\n(48)\nStep 2: J =P (θ)∇L(θ) + P (θ)∇D +D∇P (θ) (49)\nJ =P (θ) [ ∇L(θ) +∇D − D\nT ∇L(θ)−DL(θ)∇\n( 1\nT\n)] (50)\nAccording to Equation 7, ∇ (\n1 T\n) is ignorable near the minimum a and the col b, thus\n∇D = ( D T − 1 ) ∇L. (51)\nApply this result to the Fokker-Planck Equation 4, we have\n∇ · ∇[D(θ)P (θ, t)] (52) =∇ ·D∇P (θ, t) +∇ · [( D T − 1 ) ∇L(θ) ] P (θ, t) (53)\nAnd thus we obtain the Smoluchowski equation and a new form of J\n∂P (θ, t)\n∂t = ∇ ·\n[ D ( 1\nT ∇L(θ) +∇\n) P (θ, t) ] = −∇ · J, (54)\nJ = D exp ( −L(θ) T ) ∇ [ exp ( L(θ) T ) P (θ) ] . (55)\nWe note that the Smoluchowski equation is true only near critical points. We assume the point s is the midpoint on the most possible path between a and b, where L(s) = (1− s)L(a) + sL(b). The temperature Ta dominates the path a→ s, while temperature Tb dominates the path s→ b. So we have\n∇ [ exp ( L(θ)− L(s)\nT\n) P (θ) ] = JD−1 exp ( L(θ)− L(s)\nT\n) . (56)\nUnder Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path\nLeft = ∫ c a ∂ ∂θ [exp ( L(θ)− L(s) T ) P (θ)]dθ (57)\n= ∫ s a ∂ ∂θ [ exp ( L(θ)− L(s) Ta ) P (θ) ] dθ (58)\n+ ∫ c s ∂ ∂θ [ exp ( L(θ)− L(s) Tb ) P (θ) ] dθ (59)\n=[P (s)− exp ( L(a)− L(s)\nTa\n) P (a)] + [0− P (s)] (60)\n=− exp ( L(a)− L(s)\nTa\n) P (a) (61)\nRight =− J ∫ c a D−1 exp ( L(θ)− L(s) T ) dθ (62)\nWe move J to the outside of integral based on Gauss’s Divergence Theorem, because J is fixed on the escape path from one minimum to another. As there is no field source on the escape path,∫ V ∇·J(θ)dV = 0 and∇J(θ) = 0. Obviously, only minima are probability sources in deep learning. So we obtain\nJ = exp\n( L(a)−L(s)\nTa ) P (a)∫ c\na D−1 exp\n( L(θ)−L(s)\nT\n) dθ . (63)\nUnder Assumption 3, we have∫ c a D−1 exp ( L(θ)− L(s) T ) dθ (64)\n≈ ∫ c a D−1 exp [ L(b)− L(s) + 12 (θ − b) >Hb(θ − b) Tb ] dθ (65)\n≈D−1b ∫ +∞ −∞ exp [ L(b)− L(s) + 12 (θ − b) >Hb(θ − b) Tb ] dθ (66)\n=D−1b exp\n( L(b)− L(s)\nTb\n)√ 2πTb |Hb| . (67)\nBased on the results of Step 1 and Step 2, we have\nγ =\n∫ Sa J · dS P (θ ∈ Va) =\nJ\nP (θ ∈ Va) (68)\n= P (a) exp\n( L(a)−L(s)\nTa ) D−1b exp ( L(b)−L(s)\nTb\n)√ 2πTb |Hb|\n1 P (a) √\n2πTa Ha\n(69)\n= √ TbHa|Hb| 2π √ Ta exp ( −L(s)− L(a) Ta − L(b)− L(s) Tb ) (70)\n= √ TbHa|Hb| 2π √ Ta exp ( −s∆L Ta − (1− s)∆L Tb ) (71)\nSo we have\nτ = 1\nγ = 2π\n√ Ta\nTbHa|Hb| exp\n( s∆L\nTa + (1− s)∆L Tb\n) . (72)\nIn the case of pure SGN, Ta = η2BHa and Tb = − η 2BHb gives\nτ = 1\nγ = 2π\n1\n|Hb| exp\n[ 2B∆L\nη ( s Ha + (1− s) |Hb| )\n] . (73)\nWe generalize the proof above into the high-dimensional SGD diffusion.\nStep 1: P (θ ∈ Va) (74)\n= ∫ θ∈Va P (θ)dV (75)\n=P (a) ∫ θ∈Va exp [ −1 2 (θ − a)>(D− 1 2 a HaD − 12 a )(θ − a) ] dV (76)\n=P (a) (2π)\nn 2\ndet(D−1a Ha) 1 2\n(77)\nStep 2: Based on the formula of the one-dimensional probability current and flux, we obtain the high-dimensional flux escaping through Col b:∫\nSb\nJ · dS (78)\n=J1d ∫ Sb exp [ −1 2 (θ − b)>[D− 1 2 b HbD − 12 b ] ⊥e(θ − b) ] dS (79)\n=J1d (2π)\nn−1 2 ( ∏ i6=e(D −1 bi Hbi)) 1 2 , (80)\nwhere [·]⊥e indicates the directions perpendicular to the escape direction e. So we have\nγ = 1\n2π\n√ det(HaD −1 a )\n−det(HbD−1b ) |Hbe| exp\n( −s∆L\nTa − (1− s)∆L Tb\n) (81)\nTa and Tb are the eigenvalues of H−1a Da and H −1 b Db corresponding to the escape direction. We know Da = η2BHa and Db = η 2B [Hb] +. As D must be positive semidefinite, we replace Hb = U>b diag(Hb1, · · · , Hb(n−1), Hbe)Ub by its positive semidefinite analog [Hb]+ = U>b diag(Hb1, · · · , Hb(n−1), |Hbe|)Ub. Thus, we have\nτ = 1\nγ = 2π\n1\n|Hbe| exp\n[ 2B∆L\nη\n( s\nHae + (1− s) |Hbe|\n)] . (82)"},{"heading":"A.3 PROOF OF PROPOSITION 1","text":"Proof. A stationary distribution must have a balanced probability flux between valleys. So the probability flux of each valley must be equivalent,\nP (θ ∈ V1)γ12 = P (θ ∈ V2)γ21 (83)\nAs τ = γ−1, it leads to P (θ ∈ Vv) ∝ τv . We normalize the total probability to 1, then we obtain the result."},{"heading":"B ASSUMPTIONS","text":"Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily near saddle points. It means that ∂P (θ,t)∂t = −∇ · J(θ, t) ≈ 0 holds near minima a1 and a2, but not necessarily holds near saddle point b. Quasi-Equilibrium Assumption is actually weaker but more useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt et al., 2017). Under Assumption 2, the probability density P can behave like a stationary distribution only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about density motion can be obtained under the stationary assumption.\nLow Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jastrzębski et al., 2017), and is always justified when ηB is small. Under Assumption 3, the probability densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor approximation, Assumption 1, even more reasonable in SGD diffusion.\nHere, we try to provide a more intuitive explanation about Low Temperature Assumption in the domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The temperature can be interpreted as a real number D. In SGD, we have the temperature as D = η2BH . In statistical physics, if ∆LD is large, then we call it Low Temperature Approximation. Note that ∆L D appears insides an exponential function in the theoretical analysis. People usually believe that, numerically, ∆LD > 6 can make a good approximation, for a similar reason of the 6-sigma rule in statistics. In the final training phase of deep networks, a common setting is η = 0.01 and B = 128. Thus, we may safely apply Assumption 3 to the loss valleys which satisfy the very mild condition ∆L H > 2.3 × 10 −4. Empirically, the condition ∆LH > 2.3 × 10 −4 holds well in SGD dynamics. It also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain barrier heights."},{"heading":"C THE STOCHASTIC GRADIENT NOISE ANALYSIS","text":"Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet with B = 50 on CIFAR-10. We also note that the SGN on ResNet seems less Gaussian than SGN on\nfully-connected networks with the same batch size. Panigrahi et al. (2019) presented more results on the Gaussianity of SGN under various conditions.\nBy Figure 8, we validate C = HB in the original coordinates on MNIST. By Figure 9, we also validate C = HB on another dataset, Avila, in the space spanned by the eigenvectors of Hessian. The relation C = HB can still be observed in these two cases.\nData Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on MNIST. We leave the preprocessing of Avila in D. Model: Fully-connected networks."},{"heading":"D MAIN EXPERIMENTS","text":"Figure 10, 11, and 12 respectively validate that the exponential relation of the escape rate with the Hessian, the batch size and the learning rate."},{"heading":"D.1 EXPERIMENTAL SETTINGS","text":"Datasets: a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis.\nData Precessing: We perform per-pixel zero-mean and unit-variance normalization on input data. For simplicity, we also transform multi-class problems into binary-class problems by grouping labels, although this is unnecessary.\nModel: Two-layer fully-connected networks with one hidden layer and 10 neurons per hidden layer.\nInitializations: To ensure the initialized models are near minima, we first pretrain models with 200-1000 epochs to fit each data set as well as possible. We set the pretrained models’ parameters as the initialized θt=0.\nValleys’ Boundary: In principle, any small neighborhood around θt=0 can be regarded as the inside of the start valleys. In our experiments, we set each dimension’s distance from θt=0 should be less than 0.05, namely |∆θi| ≤ 0.05 for each dimension i. If we rescale the landscape by a factor k, the neighborhood will also be rescaled by k. Although we don’t know which loss valleys exist inside the neighborhood, we know the landscape of the neighborhood is invariant in each simulation.\nHyperparameters: In Figure 10: (a) η = 0.001, B = 1, (b) η = 0.015, B = 1, (c) η = 0.005, B = 1, (d) η = 0.0005, B = 1. In Figure 11: (a) η = 0.02, (b) η = 0.6, (c) η = 0.18, (d) η = 0.01. In Figure 12: (a) B = 1, (b) B = 1, (c) B = 1, (d) B = 1. In Figure 13: (a) η = 0.0002, B = 100, (b) η = 0.001, B = 100, (c) η = 0.0002, B = 100, (d) η = 0.0001, B = 100. In Figure 14: (a) η = 0.0002, B = 100, D = 0.0002, (b) η = 0.001, B = 100, D = 0.0001, (c) η = 0.0002, B = 100, D = 0.0005, (d) η = 0.0001, B = 100, D = 0.0003. We note that the hyperparameters need be tuned for each initialized pretrained models, due to the stochastic property of deep learning.\nAccording to our experience, we can always find the hyperparameters to discover the quantitative relations as long as the pretrained model fits the data set well enough. The fined-tuned requirement can be avoided in Section E, because the models in Section E are artificially initialized.\nObservation: we observe the number of iterations from the initialized position to the terminated position. We repeat experiments 100 times to estimate the escape rate γ and the mean escape time τ . As the escape time is a random variable obeying an exponential distribution, t ∼ Exponential(γ), the estimated escape rate can be written as\nγ̂ = 100− 2∑100 i=1 ti . (84)\nThe 95% confidence interval of this estimator is\nγ̂(1− 1.96√ 100 ) ≤ γ̂ ≤ γ̂(1 + 1.96√ 100 ). (85)"},{"heading":"D.2 EXPERIMENTS ON SGLD","text":"Experimental Results: Figure 13 shows a highly precise exponential relation of the escape rate and the diffusion coefficient in the figure. Figure 14 shows a proportional relation of the escape rate and the Hessian determinant in the figure. Overall, the empirical results support the density diffusion theory in the dynamics of white noise. In experiments on SGLD, we carefully adjust the injected gradient noise scale in experiment to ensure that D is significantly smaller than the loss barrier’ height and large enough to dominate SGN scale. If D is too large, learning dynamics will be reduced to Free Brownian Motion."},{"heading":"E EXPERIMENTS ON MORE MODELS","text":"We supply experiments of training three models on artificial Gaussian datasets. In these experiments, we can analytically know the locations of the minima, Hessians and loss barriers, as each input feature is Gaussian noise."},{"heading":"E.1 EXPERIMENTS SETTINGS","text":"Data Set: We generate 50000 Gaussian samples and random two-class labels as the training data set, {(x(i), y(i))|x(i) ∼ N (0, I), y(i) ∈ {0, 1}, i ∈ {1, 2, · · · , 50000}}. Hyperparameters: In Figure 15: (a) η = 0.0001, B = 100, (b) η = 0.001, B = 100, (c) η = 0.0003, B = 100. In Figure 16: (a) η = 0.0001, B = 50, D = 0.2, (b) η = 0.001, B = 50, D = 0.0005, (c) η = 0.0003, B = 1, D = 0.0003. In Figure 17: (a) η = 0.006, B = 50, (b) η = 0.05, B = 50, (c) η = 0.005, B = 1. In Figure 18: (a) η = 0.006, (b) η = 0.06, (c) η = 0.1. In Figure 19: (a) B = 1, (b) B = 1, (c) B = 1. We note that the hyperparameters are recommended and needn’t be fine tuned again. The artificially initialized parameters avoids the stochastic property of the initial states.\nExperiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex optimization, written as\nf(θ) = 1\n2 n∑ i=1 (θ4i − 16θ2i + 5θi).\nWe use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as training data.\nL(θ) = f(θ − x),\nwhere data samples x ∼ N (0, I). The one-dimensional Styblinski-Tang Function has one global minimum located at a = −2.903534, one local minimum located at d, and one saddle point b = 0.156731 as the boundary separating Valley a1 and Valley a2. For a n-dimensional Styblinski-Tang Function, we initialize parameters θt=0 = 1√k (−2.903534, · · · ,−2.903534), and set the valley’s boundary as θi < 1√k0.156731, where i is the dimension index. We record the number of iterations required to escape from the valley to the outside of valley. The setting 1 does not need labels.\nExperiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initialization: θt=0 = (0, · · · , 0). Valley Boundary: −0.1 < θi < 0.1. Due to the randomness of training data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin valley group, and study the mean escape time from the group of valleys.\nExperiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization: θt=0 = (0.1, · · · , 0.1) with a small Gaussian noise = (0, 0.01I). Valley Boundary: 0.05 < θi < 0.15. To prevent the gradient disappearance problem of deep learning, we move the starting point from the origin. For symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value. Due to the complex loss landscape of deep networks, we can hardly know the exact information about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether or not the propositions can hold with very-small-batch gradient noise in practice."},{"heading":"E.2 EXPERIMENTS RESULTS","text":"Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. Figure 16 shows the relation of the escape rate and the Hessian determinant in the dynamics of white noise. Figure 17 shows the relation of the escape rate and the second order directional derivative in the dynamics of SGD. Figure 18 shows the relation of the escape rate and the batch size in the dynamics of SGD. Figure 19 shows the relation of the escape rate and the learning rate in the dynamics of SGD."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various application areas. Due to the over-parametrization and the highly complex loss landscape of deep networks, optimizing deep networks is a difficult task. Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks. Empirically, SGD can usually find flat minima among a large number of sharp minima and local minima (Hochreiter & Schmidhuber, 1995; 1997). More papers reported that learning flat minima closely relate to generalization (Hardt et al., 2016; Zhang et al., 2017a; Arpit et al., 2017; Hoffer et al., 2017; Dinh et al., 2017; Neyshabur et al., 2017; Wu et al., 2017; Dziugaite & Roy, 2017; Kleinberg et al., 2018). Some researchers specifically study flatness itself. They try to measure flatness (Hochreiter & Schmidhuber, 1997; Keskar et al., 2017; Sagun et al., 2017; Yao et al., 2018), rescale flatness (Tsuzuku et al., 2019; Xie et al., 2020b), and find flatter minima (Hoffer et al., 2017; Chaudhari et al., 2017; He et al., 2019b; Xie et al., 2020a). However, we still lack a quantitative theory that answers why deep learning dynamics selects a flat minimum.\\nThe diffusion theory is an important theoretical tool to understand how deep learning dynamics works. It helps us model the diffusion process of probability densities of parameters instead of model parameters themselves. The density diffusion process of Stochastic Gradient Langevin Dynamics (SGLD) under injected isotropic noise has been discussed by (Sato & Nakagawa, 2014; Raginsky et al., 2017; Zhang et al., 2017b; Xu et al., 2018). Zhu et al. (2019) revealed that anisotropic diffusion of SGD often leads to flatter minima than isotropic diffusion. A few papers has quantitatively studied the diffusion process of SGD under the isotropic gradient noise assumption. Jastrzębski et al. (2017) first studied the minima selection probability of SGD. Smith & Le (2018) presented a Beyesian perspective on generalization of SGD. Wu et al. (2018) studied the escape problems of\\nSGD from a dynamical perspective, and obtained the qualitative conclusion on the effects of batch size, learning rate, and sharpness. Hu et al. (2019) quantitatively showed that the mean escape time of SGD exponentially depends on the inverse learning rate. Achille & Soatto (2019) also obtained a related proposition that describes the mean escape time in terms of a free energy that depends on the Fisher Information. Li et al. (2017) analyzed Stochastic Differential Equation (SDE) of adaptive gradient methods. Nguyen et al. (2019) mainly contributed to closing the theoretical gap between continuous-time dynamics and discrete-time dynamics under isotropic heavy-tailed noise.\\nHowever, the related papers mainly analyzed the diffusion process under parameter-independent and isotropic gradient noise, while stochastic gradient noise (SGN) is highly parameter-dependent and anisotropic in deep learning dynamics. Thus, they failed to quantitatively formulate how SGD selects flat minima, which closely depends on the Hessian-dependent structure of SGN. We try to bridge the gap between the qualitative knowledge and the quantitative theory for SGD in the presence of parameter-dependent and anisotropic SGN. Mainly based on Theorem 3.2 , we have four contributions:\\n• The proposed theory formulates the fundamental roles of gradient noise, batch size, the learning rate, and the Hessian in minima selection. • The SGN covariance is approximately proportional to the Hessian and inverse to batch size. • Either a small learning rate or large-batch training requires exponentially many iterations to\\nescape minima in terms of ratio of batch size and learning rate. • To the best of our knowledge, we are the first to theoretically and empirically reveal that\\nSGD favors flat minima exponentially more than sharp minima.\"\n },\n {\n \"heading\": \"2 STOCHASTIC GRADIENT NOISE AND SGD DYNAMICS\",\n \"text\": \"We mainly introduce the necessary foundation for the proposed diffusion theory in this section. We denote the data samples as {xj}mj=1, the model parameters as θ and the loss function over data samples x as L(θ, x). For simplicity, we denote the training loss as L(θ). Following Mandt et al. (2017), we may write SGD dynamics as\\nθt+1 = θt − η ∂L̂(θt)\\n∂θt = θt − η\\n∂L(θt)\\n∂θt + ηC(θt)\\n1 2 ζt, (1)\\nwhere L̂(θ) is the loss of one minibatch, ζt ∼ N (0, I), and C(θ) represents the gradient noise covariance matrix. The classic approach is to model SGN by Gaussian noise, N (0, C(θ)) (Mandt et al., 2017; Smith & Le, 2018; Chaudhari & Soatto, 2018).\\nStochastic Gradient Noise Analysis. We first note that the SGN we study is introduced by minibatch training, C(θt) 1 2 ζt =\\n∂L(θt) ∂θt − ∂L̂(θt)∂θt , which is the difference between gradient descent and stochastic\\ngradient descent. According to Generalized Central Limit Theorem (Gnedenko et al., 1954), the mean of many infinite-variance random variables converges to a stable distribution, while the mean of many finite-variance random variables converges to a Gaussian distribution. As SGN is finite in practice, we believe the Gaussian approximation of SGN is reasonable.\\nSimsekli et al. (2019) argued that SGN is Lévy noise (stable variables), rather than Gaussian noise. They presented empirical evidence showing that SGN seems heavy-tailed, and the heavy-tailed distribution looks closer to a stable distribution than a Gaussian distribution. However, this research line (Simsekli et al., 2019; Nguyen et al., 2019) relies on a hidden strict assumption that SGN must be isotropic and obey the same distribution across dimensions. Simsekli et al. (2019) computed “SGN” across n model parameters and regarded “SGN\\\" as n samples drawn from a single-variant distribution. This is why one tail-index for all parameters was studied in Simsekli et al. (2019). The arguments in Simsekli et al. (2019) did not necessarily hold for parameter-dependent and anisotropic Gaussian noise. In our paper, SGN computed over different minibatches obeys a n-variant Gaussian distribution, which can be parameter-dependent and anisotropic.\\nIn Figure 1, we empirically verify that SGN is highly similar to Gaussian noise instead of heavy-tailed Lévy noise. We recover the experiment of Simsekli et al. (2019) to show that gradient noise is approximately Lévy noise only if it is computed across parameters. Figure 1 actually suggests that\\nthe contradicted observations are from the different formulations of gradient noise. Simsekli et al. (2019) studied the distribution of SGN as a single-variant distribution, while we relax it as a n-variant distribution. Our empirical analysis in Figure 1 holds well at least when the batch size B is larger than 16, which is common in practice. Similar empirical evidence can be observed for training ResNet18 (He et al., 2016) on CIFAR-10 (Krizhevsky et al., 2009), seen in Appendix C.\\nPanigrahi et al. (2019) also observed that for batch sizes 256 and above, the distribution of SGN is best described as Gaussian at-least in the early phases of training. Comparing our results with Panigrahi et al. (2019), we noticed that the Gaussianity of SGN may depend on more unknown factors. First, SGN on random models is more Gaussian than well-trained models. Second, the layer/network matters. Because SGN on some layers/networks is more Gaussian than other layers/networks.\\nThe isotropic gradient noise assumption is too rough to capture the Hessian-dependent covariance structure of SGN, which we will study in Figure 2 later. Our theory that focuses on parameterdependent and anisotropic SGN brings a large improvement over existing parameter-independent and isotropic noise, although Simsekli et al. (2019) brought an improvement over more conventional parameter-independent and isotropic Gaussian noise. A more sophisticated theory is interesting under parameter-independent anisotropic heavy-tailed noise, when the batch size is too small (B ∼ 1) to apply Central Limit Theorem. We will leave it as future work.\\nSGD Dynamics. Let us replace η by dt as unit time. Then the continuous-time dynamics of SGD (Coffey & Kalmykov, 2012) is written as\\ndθ = −∂L(θ) ∂θ dt+ [2D(θ)] 1 2 dWt, (2)\\nwhere dWt ∼ N (0, Idt) and D(θ) = η2C(θ). We note that the dynamical time t in the continuoustime dynamics is equal to the product of the number of iterations T and the learning rate η: t = ηT . The associated Fokker-Planck Equation is written as\\n∂P (θ, t)\\n∂t =∇ · [P (θ, t)∇L(θ)] +∇ · ∇D(θ)P (θ, t) (3)\\n= ∑ i ∂ ∂θi [ P (θ, t) ∂L(θ) ∂θi ] + ∑ i ∑ j ∂2 ∂θi∂θj Dij(θ)P (θ, t), (4)\\nwhere∇ is a nabla operator, and Dij is the element in the ith row and jth column of D. In standard SGLD, the injected gradient noise is fixed and isotropic Gaussian, D = I .\\nThe next question is how to formulate the SGN covariance C(θ) for SGD? Based on Smith & Le (2018), we can express the SGN covariance as\\nC(θ) = 1\\nB 1 m m∑ j=1 ∇L(θ, xj)∇L(θ, xj)> −∇L(θ)∇L(θ)> ≈ 1 Bm m∑ j=1 ∇L(θ, xj)∇L(θ, xj)>.\\n(5)\\nThe approximation is true near critical points, due to the fact that the gradient noise variance dominates the gradient mean near critical points. We know the observed fisher information matrix satisfies FIM(θ) ≈ H(θ) near minima, referring to Chapter 8 of (Pawitan, 2001). Following Jastrzębski et al. (2017); Zhu et al. (2019), we obtain\\nC(θ) ≈ 1 Bm m∑ j=1 ∇L(θ, xj)∇L(θ, xj)> = 1 B FIM(θ) ≈ 1 B H(θ), (6)\\nwhich approximately gives\\nD(θ) = η\\n2 C(θ) =\\nη\\n2B H(θ) (7)\\nnear minima. It indicates that the SGN covariance C(θ) is approximately proportional to the Hessian H(θ) and inverse to the batch size B. Obviously, we can generalize Equation 7 by D(θ) = ηC(θ)2 = η 2B [H(θ)]\\n+ near critical points, when there exist negative eigenvalues in H along some directions. We use [·]+ to denote the positive semidefinite transformation of a symmetric matrix: if we have the eigendecomposation H = U diag(H1, · · · , Hn−1, Hn)U>, then [H]+ = U diag(|H1|, · · · , |Hn−1|, |Hn|)U>. We empirically verify this relation in Figure 2 for pretrained fully-connected networks, and a followup paper Xie et al. (2020c) first verified this relation for randomly initialized fully-connected networks on real-world datasets. The Pearson Correlation is up to 0.999 for pretrained networks. We note that, the relation still approximately holds for even the randomly network, which is far from critical points. The correlation is especially high along the flat directions with small-magnitude eigenvalues of the Hessian (Xie et al., 2020c). We emphasize that previous papers with the isotropic Lévy or Gaussian noise approximation all failed to capture this core relation in deep learning dynamics.\"\n },\n {\n \"heading\": \"3 SGD DIFFUSION THEORY\",\n \"text\": \"We start the theoretical analysis from the classical Kramers Escape Problem (Kramers, 1940). We assume there are two valleys, Sharp Valley a1 and Flat Valley a2, seen in Figure 3. Also Col b is the\\nboundary between two valleys. What is the mean escape time for a particle governed by Equation 2 to escape from Sharp Valley a1 to Flat Valley a2? The mean escape time is widely used in related statistical physics and stochastic process (Van Kampen, 1992; Nguyen et al., 2019).\\nGauss’s Divergence Theorem (Arfken & Weber, 1999; Lipschutz et al., 2009) states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. We respectively denote the mean escape time as τ , the escape rate as γ, and the probability current as J . We apply Gauss’s Divergence Theorem to the Fokker-Planck Equation resulting in\\n∇ · [P (θ, t)∇L(θ)] +∇ · ∇D(θ)P (θ, t) = ∂P (θ, t) ∂t = −∇ · J(θ, t). (8)\\nThe mean escape time is expressed (Van Kampen, 1992) as\\nτ = 1 γ = P (θ ∈ Va)∫ Sa J · dS , (9)\\nwhere P (θ ∈ Va) = ∫ Va P (θ)dV is the current probability inside Valley a, J is the probability\\ncurrent produced by the probability source P (θ ∈ Va), j = ∫ Sa J · dS is the probability flux (surface integrals of probability current), Sa is the surface (boundary) surrounding Valley a, and Va is the volume surrounded by Sa. We have j = J in the case of one-dimensional escape.\\nClassical Assumptions. We state three classical assumptions first for the density diffusion theory. Assumption 1 is the common second order Taylor approximation, which was also used by (Mandt et al., 2017; Zhang et al., 2019). Assumptions 2 and 3 are widely used in many fields’ Kramers Escape Problems, including statistical physics (Kramers, 1940; Hanggi, 1986), chemistry (Eyring, 1935; Hänggi et al., 1990), biology (Zhou, 2010), electrical engineering (Coffey & Kalmykov, 2012), and stochastic process (Van Kampen, 1992; Berglund, 2013). Related machine learning papers (Jastrzębski et al., 2017) usually used Assumptions 2 and 3 as the background of Kramers Escape Problems.\\nAssumption 1 (The Second Order Taylor Approximation). The loss function around critical points θ? can be approximately written as\\nL(θ) = L(θ?) + g(θ?)(θ − θ?) + 1 2 (θ − θ?)>H(θ?)(θ − θ?).\\nAssumption 2 (Quasi-Equilibrium Approximation). The system is in quasi-equilibrium near minima. Assumption 3 (Low Temperature Approximation). The gradient noise is small (low temperature).\\nWe will dive into these two assumptions deeper than previous papers for SGD dynamics. Assumptions 2 and 3 both mean that our diffusion theory can better describe the escape processes that cost more iterations. As this class of “slow” escape processes takes main computational time compared with “fast” escape processes, this class of “slow” escape process is more interesting for training of deep neural networks. Our empirical analysis in Section 4 supports that the escape processes in the\\nwide range of iterations (50 to 100,000 iterations) can be modeled by our theory very well. Thus, Assumption 2 and 3 are reasonable in practice. More discussion can be found in Appendix B.\\nEscape paths. We generalize the concept of critical points into critical paths as the path where 1) the gradient perpendicular to the path direction must be zero, and 2) the second order directional derivatives perpendicular to the path direction must be nonnegative. The Most Possible Paths (MPPs) for escaping must be critical paths. The most possible escape direction at one point must be the direction of one eigenvector of the Hessian at the point. Under Assumption 3, the probability density far from critical points and MPPs is very small. Thus, the density diffusion will concentrate around MPPs. Draxler et al. (2018) reported that minima in the loss landscape of deep networks are connected by Minimum Energy Paths (MEPs) that are essentially flat and Local MEPs that have high-loss saddle points. Obviously, MPPs in our paper correspond to Local MEPs. The density diffusion along MEPs, which are strictly flat, is ignorable according to our following analysis.\\nThe boundary between Sharp Valley a1 and Flat Valley a2 is the saddle point b. The Hessian at b, Hb, must have only one negative eigenvalue and the corresponding eigenvector is the escape direction. Without losing generality, we first assume that there is only one most possible path through Col b existing between Sharp Valley a1 and Flat Valley a2.\\nSGLD diffusion. We first analyze a simple case: how does SGLD escape sharp minima? Researchers are interested in SGLD, when the injected noise dominates SGN as η → 0 in final epochs. Because SGLD may work as a Bayesian inference method in this limit (Welling & Teh, 2011). SGLD is usually simplified as Gradient Descent with injected white noise, whose behavior is identical to Kramers Escape Problem with thermo noise in statistical physics. We present Theorem 3.1. We leave the proof in Appendix A.1. We also note that more precise SGLD diffusion analysis should study a mixture of injected white noise and SGN. Theorem 3.1 (SGLD Escapes Minima). The loss function L(θ) is of class C2 and n-dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGLD, then the mean escape time from Valley a to the outside of Valley a is\\nτ = 1\\nγ = 2π √ −det(Hb) det(Ha) 1 |Hbe| exp ( ∆L D ) .\\nWe denote that Ha and Hb are the Hessians of the loss function at the minimum a and the saddle point b, ∆L = L(b)− L(a) is the loss barrier height, e indicates the escape direction, and Hbe is the eigenvalue of the Hessian Hb corresponding to the escape direction. The diffusion coefficient D is usually set to 1 in SGLD.\\nSGD diffusion. However, SGD diffusion is essentially different from SGLD diffusion in several aspects: 1) anisotropic noise, 2) parameter-dependent noise, and 3) the stationary distribution of SGD is far from the Gibs-Boltzmann distribution, P (θ) = 1Z exp ( −L(θ)D ) . These different characteristics make SGD diffusion behave differently from known physical dynamical systems and much less studied than SGLD diffusion. We formulate Theorem 3.2 for SGD. We leave the proof in Appendix A.2.The theoretical analysis of SGD can be easily generalized to the dynamics with a mixture of SGN and injected white noise, as long as the eigenvectors of D(θ) are closely aligned with the eigenvectors of H(θ). Theorem 3.2 (SGD Escapes Minima). The loss function L(θ) is of class C2 and n-dimensional. Only one most possible path exists between Valley a and the outside of Valley a. If Assumption 1, 2, and 3 hold, and the dynamics is governed by SGD, then the mean escape time from Valley a to the outside of Valley a is\\nτ = 2π 1\\n|Hbe| exp\\n[ 2B∆L\\nη\\n( s\\nHae + (1− s) |Hbe|\\n)] ,\\nwhere s ∈ (0, 1) is a path-dependent parameter, and Hae and Hbe are, respectively, the eigenvalues of the Hessians at the minimum a and the saddle point b corresponding to the escape direction e.\\nMultiple-path escape. Each escape path contributes to the total escape rate. Multiple paths combined together have a total escape rate. If there are multiple parallel from the start valley to the end valley, we can compute the total escape rate easily based on the following computation rule. The computation\\nrule is based on the fact that the probability flux integrals are additive. We can easily generalize the mean escape time analysis into the cases that there are multiple parallel escape paths indexed by p. As for multiple-valley escape problems, we can always reduce a multiple-valley escape problem into multiple two-valley escape problems. We also note that, while Theorem A.2 does not depend the dimensionality directly, higher dimensionality may increase the number of escape paths and loss valleys, and change the spectrum of the Hessians.\\nRule 1. If there are multiple MPPs between the start valley and the end valley, then γtotal = ∑ p γp.\\nThus, we only need to find the saddle points that connect two valleys as we analyzed in the paper and analyze the escape rates.\\nMinima selection. Now, we may formulate the probability of minima selection as Proposition 1. We leave the proof in Appendix A.3. In deep learning, one loss valley represents one mode and the landscape contain many good modes and bad modes. SGD transits from one mode to another mode during training. The mean escape time of one mode corresponds to the number of iterations which SGD spends on this mode during training, which is naturally proportional to the probability of selecting this mode after training.\\nProposition 1. Suppose there are two valleys connected by an escape path. If all assumptions of Theorem 3.2 hold, then the stationary distribution of locating these valleys is given by\\nP (θ ∈ Va) = τa∑ v τv ,\\nwhere v is the index of valleys, and τv is the mean escape time from Valley v to the outside of Valley v.\"\n },\n {\n \"heading\": \"4 EMPIRICAL ANALYSIS\",\n \"text\": \"In this section, we try to directly validate the escape formulas on real-world datasets. Each escape process, from the inside of loss valleys to the outside of loss valleys, are repeatedly simulated for 100 times under various gradient noise scales, batch sizes, learning rates, and sharpness.\\nHow to compare the escape rates under the same settings with various minima sharpness? Our method is to multiply a rescaling factor √ k to each parameter, and the Hessian will be proportionally rescaled by a factor k. If we let L(θ) = f(θ)→ L(θ) = f( √ kθ), then H(θ) = ∇2f(θ)→ H(θ) = k∇2f(θ). Thus, we can use k to indicate the minima sharpness. The theoretical relations of SGD we try to validate can be formulated as: (1) − log(γ) = O( 1k ), (2) − log(γ) = O(B), and (3) − log(γ) = O( 1η ).\\nThe mean escape time analysis of SGD. Styblinski-Tang Function, which has multiple minima and saddle points, is a common test function for nonconvex optimization. We conduct an intuitional 10-dimensional experiment, where the simulations start from a given minimum and terminate when reaching the boundary of the loss valley. The number of iterations is recorded for calculating\\nthe escape rate. We also train fully connected networks on four real-world datasets, including a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis (De Stefano et al., 2018; Dua & Graff, 2017). Figure 4 and Figure 5 clearly verifies that the escape rate exponentially depends on the minima sharpness (reflected by k), the batch size, and the learning rate on both test functions and real-world training, which fully supports our theoretical results.\\nModel architecture and details: We used fully-connected networks with the depth 2 and the width 10 in Figure 5. The experiments using Logistic Regression and Fully-connected networks with the depth 3 are presented in Appendix E. We leave more experimental details and results in Appendix D.1 and Appendix E.\\nThe mean escape time analysis of SGLD. We try to validate γ = O(k) and − log(γ) = O( 1D ) for SGLD (dominated by injected Gaussian noise). Figure 6 shows that SGLD only favors flat minima polynomially more than sharp minima as Theorem 3.1 indicates. Figure 6 also verifies that the injected gradient noise scale exponentially affects flat minima selection.\"\n },\n {\n \"heading\": \"5 DISCUSSION\",\n \"text\": \"SGD favors flat minima exponentially more than sharp minima. We can discover a few interesting insights about SGD by Theorem 3.2. Most importantly, the mean escape time exponentially depends on the eigenvalue of the Hessian at minima along the escape direction, Hae. Thus, SGD favors flat minima exponentially more than sharp minima. We claim one main advantage of SGD comes from the exponential relation of the mean escape time and the minima sharpness. The measure of “sharpness” has reformed in contexts of SGLD and SGD. In the context of SGLD, the “sharpness” is quantified by the determinant of the Hessian. In the context of SGD, the “sharpness” is quantified by the top eigenvalues of the Hessian along the escape direction. Based on the proposed diffusion theory, recent work (Xie et al., 2020c) successfully proved that SGD favors flat minima significantly more than Adam.\\nThe ratio of the batch size and the learning rate exponentially matters. Theorem 3.2 explains why large-batch training can easily get trapped near sharp minima, and increasing the learning rate proportionally is helpful for large-batch training (Krizhevsky, 2014; Keskar et al., 2017; Sagun et al., 2017; Smith et al., 2018; Yao et al., 2018; He et al., 2019a). We argue that the main cause is large-batch training expects exponentially longer time to escape minima. Note that, as the mean escape time in the theorems is equivalent to the product of the learning rate and the number of iterations, both the number of iterations and dynamical time exponentially depend on the ratio of the batch size and the learning rate. The practical computational time in large-batch training is usually too short to search many enough flat minima. We conjecture that exponentially increasing training iterations may be helpful for large batch training, while this is often too expensive in practice.\\nLow dimensional diffusion. Most eigenvalues of the Hessian at the loss landscape of overparametrized deep networks are close to zero, while only a small number of eigenvalues are large (Sagun et al., 2017; Li et al., 2018). Zero eigenvalues indicate zero diffusion along the corresponding directions. Thus, we may theoretically ignore these zero-eigenvalue directions. This also indicates that the density diffusion is ignorable along an essentially flat MEP in Draxler et al. (2018).\\nAs the escape rate exponentially depends the corresponding eigenvalues, a small number of large eigenvalues means that the process of minima selection mainly happens in the relatively low dimensional subspace corresponding to top eigenvalues of the Hessian. Gur-Ari et al. (2018) also reported a similar finding. Although the parameter space is very high-dimensional, SGD dynamics hardly depends on those “meaningless” dimensions with small second order directional derivatives. This novel characteristic of SGD significantly reduces the explorable parameter space around one minimum into a much lower dimensional space.\\nHigh-order effects. As we have applied the second-order Taylor approximation near critical points, our SGD diffusion theory actually excludes the third-order and higher-order effect. The asymmetric valley in He et al. (2019b), which only appears in high-order analysis, is beyond the scope of this paper. However, we also argue that the third-order effect is much smaller than the second-order effect under the low temperature assumption in Kramers Escape Problems. We will leave the more refined high-order theory as future work.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"In this paper, we demonstrate that one essential advantage of SGD is selecting flat minima with an exponentially higher probability than sharp minima. To the best of our knowledge, we are the first to formulate the exponential relation of minima selection to the minima sharpness, the batch size, and the learning rate. Our work bridges the gap between the qualitative knowledge and the quantitative theoretical knowledge on the minima selection mechanism of SGD. We believe the proposed theory not only helps us understand how SGD selects flat minima, but also will provide researchers a powerful theoretical tool to analyze more learning behaviors and design better optimizers in future.\"\n },\n {\n \"heading\": \"ACKNOWLEDGEMENT\",\n \"text\": \"We thanks Dr. Yuanqian Tang for helpful discussion. MS was supported by the International Research Center for Neurointelligence (WPI-IRCN) at The University of Tokyo Institutes for Advanced Study.\"\n },\n {\n \"heading\": \"A PROOFS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A.1 PROOF OF THEOREM 3.1\",\n \"text\": \"Proof. This proposition is a well known conclusion in statistical physics under Assumption 1, 2 and 3. We still provide an intuitional proof here, and the following proof of SGD Diffusion will closely relate to this proof. We decompose the proof into two steps: 1) compute the probability of locating in valley a, P (θ ∈ Va), and 2) compute the probability flux j = ∫ Sa J · dS.\\nWithout losing generality, we first prove the one-dimensional case.\\nStep 1: Under Assumption 1, the stationary distribution around minimum a is P (θ) = P (a) exp[−L(θ)−L(a)T ], where T = D. Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points. We use the T notation as the temperature parameter in the stationary distribution, and use the D notation as the diffusion coefficient in the dynamics, for their different roles.\\nP (θ ∈ Va) (10)\\n= ∫ θ∈Va P (θ)dV (11)\\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) T ] dθ (12)\\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) T ] dθ (13)\\n=P (a) (2πT )\\n1 2\\nH 1 2 a\\n. (14)\\nStep 2:\\nJ =P (θ)∇L(θ) + P (θ)∇D +D∇P (θ) (15)\\nJ =P (θ) ( ∇L(θ) +∇D − D\\nT ∇L(θ)\\n) (16)\\n∇D = ( D T − 1 ) ∇L (17)\\nApply this result to the Fokker-Planck Equation 4, we have\\n∇ · ∇[D(θ)P (θ, t)] (18) =∇ ·D∇P (θ, t) +∇ · [( D T − 1 ) ∇L(θ) ] P (θ, t) (19)\\nAnd thus we obtain the Smoluchowski equation and a new form of J\\n∂P (θ, t)\\n∂t = ∇ ·\\n[ D ( 1\\nT ∇L(θ) +∇\\n) P (θ, t) ] = −∇ · J(θ, t), (20)\\nJ(θ) = D exp ( −L(θ) T ) ∇ [ exp ( L(θ) T ) P (θ) ] . (21)\\nWe note that the probability density outside Valley a must be zero, P (c) = 0. As we want to compute the probability flux escaping from Valley a in the proof, the probability flux escaping from other valleys into Valley a should be ignored. Under Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path∫ c\\na\\n∂\\n∂θ\\n[ exp ( L(θ)\\nT\\n) P (θ) ] dθ = ∫ c a − J D exp ( L(θ) T ) dθ (22)\\nexp\\n( L(θ)\\nT\\n) P (θ)|ca =− J\\nD ∫ c a exp ( L(θ) T ) dθ (23)\\n0− exp ( L(a)\\nT\\n) P (a) =− J\\nD ∫ c a exp ( L(θ) T ) dθ (24)\\nJ = D exp\\n( L(a) T ) P (a)∫ c\\na exp ( L(θ) T ) dθ . (25)\\nWe move J to the outside of integral based on Gauss’s Divergence Theorem, because J is fixed on the escape path from one minimum to another. As there is no field source on the escape path,∫ V ∇ · J(θ)dV = 0. Then ∇J(θ) = 0. Obviously, only minima are probability sources in deep learning. Under Assumption 3 and the second-order Taylor approximation, we have∫ c a exp ( L(θ) T ) dθ (26)\\n= ∫ c a exp [ L(b) + 12 (θ − b) >Hb(θ − b) +O(∆θ3) T ] dθ (27)\\n≈ exp ( L(b)\\nT )∫ +∞ −∞ exp [ 1 2 (θ − b) >Hb(θ − b) T ] dθ (28)\\n= exp\\n( L(b)\\nT\\n)√ 2πT\\n|Hb| . (29)\\nBased on the results of Step 1 and Step 2, we obtain\\nγ =\\n∫ Sa J · dS P (θ ∈ Va) =\\nJ\\nP (θ ∈ Va) (30)\\n= DP (a) exp\\n( L(a) T ) exp ( L(b) T )√ 2πT |Hb|\\n1 P (a) √\\n2πT Ha\\n(31)\\n= D √ Ha|Hb|\\n2πT exp\\n( −∆Lab\\nT\\n) (32)\\n= √ Ha|Hb| 2π exp ( −∆Lab D ) (33)\\nWe generalize the proof of one-dimensional diffusion to high-dimensional diffusion\\nStep 1: P (θ ∈ Va) (34)\\n= ∫ θ∈Va P (θ)dV (35)\\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) T ] dV (36)\\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) T ] dV (37)\\n=P (a) (2πT )\\nn 2\\ndet(Ha) 1 2\\n(38)\\nStep 2: Based on the formula of the one-dimensional probability current and flux, we obtain∫ Sb J · dS (39)\\n=Jb ∫ Sb exp [ − 1 2 (θ − b) >H+b (θ − b) T ] dS (40)\\n=Jb (2πT )\\nn−1 2 ( ∏n−1 i=1 Hbi) 1 2\\n(41)\\nSo we have\\nτ =2π\\n√ ∏n−1 i=1 Hbi\\ndet(Ha)|Hbe| exp\\n( ∆L\\nT\\n) (42)\\n=2π √ −det(Hb) det(Ha) 1 |Hbe| exp ( ∆L D ) . (43)\"\n },\n {\n \"heading\": \"A.2 PROOF OF THEOREM 3.2\",\n \"text\": \"Proof. We decompose the proof into two steps and analyze the one-dimensional case like before. The following proof is similar to the proof of SGLD except that we make Ta the temperature near the minimum a and Tb the temperature near the saddle point b.\\nOne-dimensional SGD Diffusion:\\nStep 1: Under Assumption 3, we may only consider the second order Taylor approximation of the density function around critical points.\\nP (θ ∈ Va) (44)\\n= ∫ θ∈Va P (θ)dV (45)\\n= ∫ θ∈Va P (a) exp [ −L(θ)− L(a) Ta ] dV (46)\\n=P (a) ∫ θ∈Va exp [ − 1 2 (θ − a) >Ha(θ − a) +O(∆θ3) Ta ] dθ (47)\\n=P (a) (2πTa)\\n1 2\\nH 1 2 a\\n(48)\\nStep 2: J =P (θ)∇L(θ) + P (θ)∇D +D∇P (θ) (49)\\nJ =P (θ) [ ∇L(θ) +∇D − D\\nT ∇L(θ)−DL(θ)∇\\n( 1\\nT\\n)] (50)\\nAccording to Equation 7, ∇ (\\n1 T\\n) is ignorable near the minimum a and the col b, thus\\n∇D = ( D T − 1 ) ∇L. (51)\\nApply this result to the Fokker-Planck Equation 4, we have\\n∇ · ∇[D(θ)P (θ, t)] (52) =∇ ·D∇P (θ, t) +∇ · [( D T − 1 ) ∇L(θ) ] P (θ, t) (53)\\nAnd thus we obtain the Smoluchowski equation and a new form of J\\n∂P (θ, t)\\n∂t = ∇ ·\\n[ D ( 1\\nT ∇L(θ) +∇\\n) P (θ, t) ] = −∇ · J, (54)\\nJ = D exp ( −L(θ) T ) ∇ [ exp ( L(θ) T ) P (θ) ] . (55)\\nWe note that the Smoluchowski equation is true only near critical points. We assume the point s is the midpoint on the most possible path between a and b, where L(s) = (1− s)L(a) + sL(b). The temperature Ta dominates the path a→ s, while temperature Tb dominates the path s→ b. So we have\\n∇ [ exp ( L(θ)− L(s)\\nT\\n) P (θ) ] = JD−1 exp ( L(θ)− L(s)\\nT\\n) . (56)\\nUnder Assumption 2, we integrate the equation from Valley a to the outside of Valley a along the most possible escape path\\nLeft = ∫ c a ∂ ∂θ [exp ( L(θ)− L(s) T ) P (θ)]dθ (57)\\n= ∫ s a ∂ ∂θ [ exp ( L(θ)− L(s) Ta ) P (θ) ] dθ (58)\\n+ ∫ c s ∂ ∂θ [ exp ( L(θ)− L(s) Tb ) P (θ) ] dθ (59)\\n=[P (s)− exp ( L(a)− L(s)\\nTa\\n) P (a)] + [0− P (s)] (60)\\n=− exp ( L(a)− L(s)\\nTa\\n) P (a) (61)\\nRight =− J ∫ c a D−1 exp ( L(θ)− L(s) T ) dθ (62)\\nWe move J to the outside of integral based on Gauss’s Divergence Theorem, because J is fixed on the escape path from one minimum to another. As there is no field source on the escape path,∫ V ∇·J(θ)dV = 0 and∇J(θ) = 0. Obviously, only minima are probability sources in deep learning. So we obtain\\nJ = exp\\n( L(a)−L(s)\\nTa ) P (a)∫ c\\na D−1 exp\\n( L(θ)−L(s)\\nT\\n) dθ . (63)\\nUnder Assumption 3, we have∫ c a D−1 exp ( L(θ)− L(s) T ) dθ (64)\\n≈ ∫ c a D−1 exp [ L(b)− L(s) + 12 (θ − b) >Hb(θ − b) Tb ] dθ (65)\\n≈D−1b ∫ +∞ −∞ exp [ L(b)− L(s) + 12 (θ − b) >Hb(θ − b) Tb ] dθ (66)\\n=D−1b exp\\n( L(b)− L(s)\\nTb\\n)√ 2πTb |Hb| . (67)\\nBased on the results of Step 1 and Step 2, we have\\nγ =\\n∫ Sa J · dS P (θ ∈ Va) =\\nJ\\nP (θ ∈ Va) (68)\\n= P (a) exp\\n( L(a)−L(s)\\nTa ) D−1b exp ( L(b)−L(s)\\nTb\\n)√ 2πTb |Hb|\\n1 P (a) √\\n2πTa Ha\\n(69)\\n= √ TbHa|Hb| 2π √ Ta exp ( −L(s)− L(a) Ta − L(b)− L(s) Tb ) (70)\\n= √ TbHa|Hb| 2π √ Ta exp ( −s∆L Ta − (1− s)∆L Tb ) (71)\\nSo we have\\nτ = 1\\nγ = 2π\\n√ Ta\\nTbHa|Hb| exp\\n( s∆L\\nTa + (1− s)∆L Tb\\n) . (72)\\nIn the case of pure SGN, Ta = η2BHa and Tb = − η 2BHb gives\\nτ = 1\\nγ = 2π\\n1\\n|Hb| exp\\n[ 2B∆L\\nη ( s Ha + (1− s) |Hb| )\\n] . (73)\\nWe generalize the proof above into the high-dimensional SGD diffusion.\\nStep 1: P (θ ∈ Va) (74)\\n= ∫ θ∈Va P (θ)dV (75)\\n=P (a) ∫ θ∈Va exp [ −1 2 (θ − a)>(D− 1 2 a HaD − 12 a )(θ − a) ] dV (76)\\n=P (a) (2π)\\nn 2\\ndet(D−1a Ha) 1 2\\n(77)\\nStep 2: Based on the formula of the one-dimensional probability current and flux, we obtain the high-dimensional flux escaping through Col b:∫\\nSb\\nJ · dS (78)\\n=J1d ∫ Sb exp [ −1 2 (θ − b)>[D− 1 2 b HbD − 12 b ] ⊥e(θ − b) ] dS (79)\\n=J1d (2π)\\nn−1 2 ( ∏ i6=e(D −1 bi Hbi)) 1 2 , (80)\\nwhere [·]⊥e indicates the directions perpendicular to the escape direction e. So we have\\nγ = 1\\n2π\\n√ det(HaD −1 a )\\n−det(HbD−1b ) |Hbe| exp\\n( −s∆L\\nTa − (1− s)∆L Tb\\n) (81)\\nTa and Tb are the eigenvalues of H−1a Da and H −1 b Db corresponding to the escape direction. We know Da = η2BHa and Db = η 2B [Hb] +. As D must be positive semidefinite, we replace Hb = U>b diag(Hb1, · · · , Hb(n−1), Hbe)Ub by its positive semidefinite analog [Hb]+ = U>b diag(Hb1, · · · , Hb(n−1), |Hbe|)Ub. Thus, we have\\nτ = 1\\nγ = 2π\\n1\\n|Hbe| exp\\n[ 2B∆L\\nη\\n( s\\nHae + (1− s) |Hbe|\\n)] . (82)\"\n },\n {\n \"heading\": \"A.3 PROOF OF PROPOSITION 1\",\n \"text\": \"Proof. A stationary distribution must have a balanced probability flux between valleys. So the probability flux of each valley must be equivalent,\\nP (θ ∈ V1)γ12 = P (θ ∈ V2)γ21 (83)\\nAs τ = γ−1, it leads to P (θ ∈ Vv) ∝ τv . We normalize the total probability to 1, then we obtain the result.\"\n },\n {\n \"heading\": \"B ASSUMPTIONS\",\n \"text\": \"Assumption 2 indicates that the dynamical system is in equilibrium near minima but not necessarily near saddle points. It means that ∂P (θ,t)∂t = −∇ · J(θ, t) ≈ 0 holds near minima a1 and a2, but not necessarily holds near saddle point b. Quasi-Equilibrium Assumption is actually weaker but more useful than the conventional stationary assumption for deep learning (Welling & Teh, 2011; Mandt et al., 2017). Under Assumption 2, the probability density P can behave like a stationary distribution only inside valleys, but density transportation through saddle points can be busy. Quasi-Equilibrium is more like: stable lakes (loss valleys) is connected by rapid Rivers (escape paths). In contrast, the stationary assumption requires strictly zero flux between lakes (loss valleys). Little knowledge about density motion can be obtained under the stationary assumption.\\nLow Temperature Assumption is common (Van Kampen, 1992; Zhou, 2010; Berglund, 2013; Jastrzębski et al., 2017), and is always justified when ηB is small. Under Assumption 3, the probability densities will concentrate around minima and MPPs. Numerically, the 6-sigma rule may often provide good approximation for a Gaussian distribution. Assumption 3 will make the second order Taylor approximation, Assumption 1, even more reasonable in SGD diffusion.\\nHere, we try to provide a more intuitive explanation about Low Temperature Assumption in the domain of deep learning. Without loss of generality, we discuss it in one-dimensional dynamics. The temperature can be interpreted as a real number D. In SGD, we have the temperature as D = η2BH . In statistical physics, if ∆LD is large, then we call it Low Temperature Approximation. Note that ∆L D appears insides an exponential function in the theoretical analysis. People usually believe that, numerically, ∆LD > 6 can make a good approximation, for a similar reason of the 6-sigma rule in statistics. In the final training phase of deep networks, a common setting is η = 0.01 and B = 128. Thus, we may safely apply Assumption 3 to the loss valleys which satisfy the very mild condition ∆L H > 2.3 × 10 −4. Empirically, the condition ∆LH > 2.3 × 10 −4 holds well in SGD dynamics. It also suggests that, we can adjust the learning rate to let SGD search among loss valleys with certain barrier heights.\"\n },\n {\n \"heading\": \"C THE STOCHASTIC GRADIENT NOISE ANALYSIS\",\n \"text\": \"Figure 7 demonstrates that the SGN is also approximately Gaussian on a randomly initialized ResNet with B = 50 on CIFAR-10. We also note that the SGN on ResNet seems less Gaussian than SGN on\\nfully-connected networks with the same batch size. Panigrahi et al. (2019) presented more results on the Gaussianity of SGN under various conditions.\\nBy Figure 8, we validate C = HB in the original coordinates on MNIST. By Figure 9, we also validate C = HB on another dataset, Avila, in the space spanned by the eigenvectors of Hessian. The relation C = HB can still be observed in these two cases.\\nData Precessing: We perform the usual per-pixel zero-mean and unit-variance normalization on MNIST. We leave the preprocessing of Avila in D. Model: Fully-connected networks.\"\n },\n {\n \"heading\": \"D MAIN EXPERIMENTS\",\n \"text\": \"Figure 10, 11, and 12 respectively validate that the exponential relation of the escape rate with the Hessian, the batch size and the learning rate.\"\n },\n {\n \"heading\": \"D.1 EXPERIMENTAL SETTINGS\",\n \"text\": \"Datasets: a) Avila, b) Banknote Authentication, c) Cardiotocography, d) Dataset for Sensorless Drive Diagnosis.\\nData Precessing: We perform per-pixel zero-mean and unit-variance normalization on input data. For simplicity, we also transform multi-class problems into binary-class problems by grouping labels, although this is unnecessary.\\nModel: Two-layer fully-connected networks with one hidden layer and 10 neurons per hidden layer.\\nInitializations: To ensure the initialized models are near minima, we first pretrain models with 200-1000 epochs to fit each data set as well as possible. We set the pretrained models’ parameters as the initialized θt=0.\\nValleys’ Boundary: In principle, any small neighborhood around θt=0 can be regarded as the inside of the start valleys. In our experiments, we set each dimension’s distance from θt=0 should be less than 0.05, namely |∆θi| ≤ 0.05 for each dimension i. If we rescale the landscape by a factor k, the neighborhood will also be rescaled by k. Although we don’t know which loss valleys exist inside the neighborhood, we know the landscape of the neighborhood is invariant in each simulation.\\nHyperparameters: In Figure 10: (a) η = 0.001, B = 1, (b) η = 0.015, B = 1, (c) η = 0.005, B = 1, (d) η = 0.0005, B = 1. In Figure 11: (a) η = 0.02, (b) η = 0.6, (c) η = 0.18, (d) η = 0.01. In Figure 12: (a) B = 1, (b) B = 1, (c) B = 1, (d) B = 1. In Figure 13: (a) η = 0.0002, B = 100, (b) η = 0.001, B = 100, (c) η = 0.0002, B = 100, (d) η = 0.0001, B = 100. In Figure 14: (a) η = 0.0002, B = 100, D = 0.0002, (b) η = 0.001, B = 100, D = 0.0001, (c) η = 0.0002, B = 100, D = 0.0005, (d) η = 0.0001, B = 100, D = 0.0003. We note that the hyperparameters need be tuned for each initialized pretrained models, due to the stochastic property of deep learning.\\nAccording to our experience, we can always find the hyperparameters to discover the quantitative relations as long as the pretrained model fits the data set well enough. The fined-tuned requirement can be avoided in Section E, because the models in Section E are artificially initialized.\\nObservation: we observe the number of iterations from the initialized position to the terminated position. We repeat experiments 100 times to estimate the escape rate γ and the mean escape time τ . As the escape time is a random variable obeying an exponential distribution, t ∼ Exponential(γ), the estimated escape rate can be written as\\nγ̂ = 100− 2∑100 i=1 ti . (84)\\nThe 95% confidence interval of this estimator is\\nγ̂(1− 1.96√ 100 ) ≤ γ̂ ≤ γ̂(1 + 1.96√ 100 ). (85)\"\n },\n {\n \"heading\": \"D.2 EXPERIMENTS ON SGLD\",\n \"text\": \"Experimental Results: Figure 13 shows a highly precise exponential relation of the escape rate and the diffusion coefficient in the figure. Figure 14 shows a proportional relation of the escape rate and the Hessian determinant in the figure. Overall, the empirical results support the density diffusion theory in the dynamics of white noise. In experiments on SGLD, we carefully adjust the injected gradient noise scale in experiment to ensure that D is significantly smaller than the loss barrier’ height and large enough to dominate SGN scale. If D is too large, learning dynamics will be reduced to Free Brownian Motion.\"\n },\n {\n \"heading\": \"E EXPERIMENTS ON MORE MODELS\",\n \"text\": \"We supply experiments of training three models on artificial Gaussian datasets. In these experiments, we can analytically know the locations of the minima, Hessians and loss barriers, as each input feature is Gaussian noise.\"\n },\n {\n \"heading\": \"E.1 EXPERIMENTS SETTINGS\",\n \"text\": \"Data Set: We generate 50000 Gaussian samples and random two-class labels as the training data set, {(x(i), y(i))|x(i) ∼ N (0, I), y(i) ∈ {0, 1}, i ∈ {1, 2, · · · , 50000}}. Hyperparameters: In Figure 15: (a) η = 0.0001, B = 100, (b) η = 0.001, B = 100, (c) η = 0.0003, B = 100. In Figure 16: (a) η = 0.0001, B = 50, D = 0.2, (b) η = 0.001, B = 50, D = 0.0005, (c) η = 0.0003, B = 1, D = 0.0003. In Figure 17: (a) η = 0.006, B = 50, (b) η = 0.05, B = 50, (c) η = 0.005, B = 1. In Figure 18: (a) η = 0.006, (b) η = 0.06, (c) η = 0.1. In Figure 19: (a) B = 1, (b) B = 1, (c) B = 1. We note that the hyperparameters are recommended and needn’t be fine tuned again. The artificially initialized parameters avoids the stochastic property of the initial states.\\nExperiment Setting 1: Styblinski-Tang Function is a commonly used function in nonconvex optimization, written as\\nf(θ) = 1\\n2 n∑ i=1 (θ4i − 16θ2i + 5θi).\\nWe use high-dimensional Styblinski-Tang Function as the test function, and Gaussian samples as training data.\\nL(θ) = f(θ − x),\\nwhere data samples x ∼ N (0, I). The one-dimensional Styblinski-Tang Function has one global minimum located at a = −2.903534, one local minimum located at d, and one saddle point b = 0.156731 as the boundary separating Valley a1 and Valley a2. For a n-dimensional Styblinski-Tang Function, we initialize parameters θt=0 = 1√k (−2.903534, · · · ,−2.903534), and set the valley’s boundary as θi < 1√k0.156731, where i is the dimension index. We record the number of iterations required to escape from the valley to the outside of valley. The setting 1 does not need labels.\\nExperiment Setting 2: We study the learning dynamics of Logistic Regression. Parameters Initialization: θt=0 = (0, · · · , 0). Valley Boundary: −0.1 < θi < 0.1. Due to the randomness of training data and the symmetry of dimension, the origin must be a minimum and there are a lot unknown valleys neighboring the origin valley. And we can set an arbitrary boundary surrounding the origin valley group, and study the mean escape time from the group of valleys.\\nExperiment Setting 3: We study the learning dynamics of MLP with ReLu activations, cross entropy losses, depth as 3, and hidden layers’ width as 10. Parameters Initialization: θt=0 = (0.1, · · · , 0.1) with a small Gaussian noise = (0, 0.01I). Valley Boundary: 0.05 < θi < 0.15. To prevent the gradient disappearance problem of deep learning, we move the starting point from the origin. For symmetry breaking of deep learning, we add a small Gaussian noise to each parameter’s initial value. Due to the complex loss landscape of deep networks, we can hardly know the exact information about valleys and cols. However, the escape formula can still approximately hold even if an arbitrary boundary surrounding an arbitrary group of valleys. We set the batch size as 1 in this setting. When the batch size is small, the gradient noise is more like a heavy-tailed noise. We can validate whether or not the propositions can hold with very-small-batch gradient noise in practice.\"\n },\n {\n \"heading\": \"E.2 EXPERIMENTS RESULTS\",\n \"text\": \"Figure 15 shows the relation of the escape rate and the isotropic diffusion coefficient D. Figure 16 shows the relation of the escape rate and the Hessian determinant in the dynamics of white noise. Figure 17 shows the relation of the escape rate and the second order directional derivative in the dynamics of SGD. Figure 18 shows the relation of the escape rate and the batch size in the dynamics of SGD. Figure 19 shows the relation of the escape rate and the learning rate in the dynamics of SGD.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2021,"string":"2,021"},"title":{"kind":"string","value":"A DIFFUSION THEORY FOR DEEP LEARNING DYNAM-"}}},{"rowIdx":94,"cells":{"paper_id":{"kind":"string","value":"SP:d92fe94e29672783f906710a2ecb7a02aa4bd67d"},"summaries":{"kind":"list like","value":["The value of the optimal objective as a function of the cost vector $c$ can be written as $z^*(c) = c^T u^*(c)$ where the optimal solution $u^*$ also depends on $c$. The function $u^*(c)$ is piecewise constant -- there are finitely (resp. countably) many feasible solutions; candidates for $u^*$ -- and so the function $z^*(c)$ is a piecewise linear function of $c$, with gradient $u^*(c)$, wherever it exists (otherwise there is analogous subgradient). Obviously, all it takes for computing $u^*(c)$ is solving -- anyhow -- the combinatorial problem. This is all trivial and well-known, yet the authors do precisely that."],"string":"[\n \"The value of the optimal objective as a function of the cost vector $c$ can be written as $z^*(c) = c^T u^*(c)$ where the optimal solution $u^*$ also depends on $c$. The function $u^*(c)$ is piecewise constant -- there are finitely (resp. countably) many feasible solutions; candidates for $u^*$ -- and so the function $z^*(c)$ is a piecewise linear function of $c$, with gradient $u^*(c)$, wherever it exists (otherwise there is analogous subgradient). Obviously, all it takes for computing $u^*(c)$ is solving -- anyhow -- the combinatorial problem. This is all trivial and well-known, yet the authors do precisely that.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Combinatorial problems with linear objective function play a central role in many computer science applications, and efficient algorithms for solving them are well known. However, the solutions to these problems are not differentiable with respect to the parameters specifying the problem instance – for example, shortest distance between two nodes in a graph is not a differentiable function of graph edge weights. Recently, attempts to integrate combinatorial and, more broadly, convex optimization solvers into gradient-trained models resulted in several approaches for differentiating over the solution vector to the optimization problem. However, in many cases, the interest is in differentiating over only the objective value, not the solution vector, and using existing approaches introduces unnecessary overhead. Here, we show how to perform gradient descent directly over the objective value of the solution to combinatorial problems. We demonstrate advantage of the approach in examples involving sequence-to-sequence modeling using differentiable encoder-decoder architecture with softmax or Gumbel-softmax, and in weakly supervised learning involving a convolutional, residual feed-forward network for image classification."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Akshay Agrawal","Brandon Amos","Shane Barratt","Stephen Boyd","Steven Diamond","J Zico Kolter"],"title":"Differentiable convex optimization layers","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Brandon Amos","J Zico Kolter"],"title":"Optnet: Differentiable optimization as a layer in neural networks","venue":"In Proceedings of the 34th International Conference on Machine Learning,","year":2017},{"authors":["Yuri Boykov","Olga Veksler","Ramin Zabih"],"title":"Fast approximate energy minimization via graph cuts","venue":"IEEE Transactions on Pattern Analysis and Machine Intelligence,","year":2001},{"authors":["Chien-Yi Chang","De-An Huang","Yanan Sui","Li Fei-Fei","Juan Carlos Niebles"],"title":"D3TW: Discriminative differentiable dynamic time warping for weakly supervised action alignment and segmentation","venue":"In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR),","year":2019},{"authors":["Liqun Chen","Yizhe Zhang","Ruiyi Zhang","Chenyang Tao","Zhe Gan","Haichao Zhang","Bai Li","Dinghan Shen","Changyou Chen","Lawrence Carin"],"title":"Improving sequence-to-sequence learning via optimal transport","venue":"In International Conference on Learning Representations, pp","year":2019},{"authors":["Frank H Clarke"],"title":"Generalized gradients and applications","venue":"Transactions of the American Mathematical Society,","year":1975},{"authors":["Thomas H. Cormen","Charles E. Leiserson","Ronald L. Rivest","Clifford Stein"],"title":"Introduction to Algorithms, Third Edition","venue":null,"year":2009},{"authors":["Daniel De Wolf","Yves Smeers"],"title":"Generalized derivatives of the optimal value of a linear program with respect to matrix coefficients","venue":"Technical report. Université Catholique de Louvain,","year":2000},{"authors":["Josip Djolonga","Andreas Krause"],"title":"Differentiable learning of submodular models","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["LawrenceCraig Evans"],"title":"Measure theory and fine properties of functions","venue":null,"year":1992},{"authors":["Aaron Ferber","Bryan Wilder","Bistra Dilina","Milind Tambe"],"title":"MIPaaL: Mixed integer program as a layer","venue":"arXiv preprint arXiv:1907.05912,","year":2019},{"authors":["Robert M Freund"],"title":"Postoptimal analysis of a linear program under simultaneous changes in matrix coefficients","venue":"In Mathematical Programming Essays in Honor of George B. Dantzig Part I,","year":1985},{"authors":["Tomas Gal"],"title":"Rim multiparametric linear programming","venue":"Management Science,","year":1975},{"authors":["Zhiting Hu","Haoran Shi","Bowen Tan","Wentao Wang","Zichao Yang","Tiancheng Zhao","Junxian He","Lianhui Qin","Di Wang"],"title":"Texar: A modularized, versatile, and extensible toolkit for text generation","venue":"arXiv preprint arXiv:1809.00794,","year":2018},{"authors":["Eric Jang","Shixiang Gu","Ben Poole"],"title":"Categorical reparameterization with Gumbel-softmax","venue":"In International Conference on Learning Representations ICLR’17","year":2016},{"authors":["Stefan Lendl","Ante Ćustić","Abraham P Punnen"],"title":"Combinatorial optimization with interaction costs: Complexity and solvable cases","venue":"Discrete Optimization,","year":2019},{"authors":["Chin-Yew Lin"],"title":"Rouge: A package for automatic evaluation of summaries","venue":"In Text summarization branches out,","year":2004},{"authors":["Di Lin","Jifeng Dai","Jiaya Jia","Kaiming He","Jian Sun"],"title":"Scribblesup: Scribble-supervised convolutional networks for semantic segmentation","venue":"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,","year":2016},{"authors":["Huidong Liu","Xianfeng Gu","Dimitris Samaras"],"title":"A two-step computation of the exact GAN Wasserstein distance","venue":"In International Conference on Machine Learning,","year":2018},{"authors":["Minh-Thang Luong","Hieu Pham","Christopher D Manning"],"title":"Effective approaches to attention-based neural machine translation","venue":"In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing,","year":2015},{"authors":["Dmitrii Marin","Meng Tang","Ismail Ben Ayed","Yuri Boykov"],"title":"Beyond gradient descent for regularized segmentation losses","venue":"In IEEE Conference on Computer Vision and Pattern Recognition","year":2019},{"authors":["R Kipp Martin"],"title":"Using separation algorithms to generate mixed integer model reformulations","venue":"Operations Research Letters,","year":1991},{"authors":["Arthur Mensch","Mathieu Blondel"],"title":"Differentiable dynamic programming for structured prediction and attention","venue":"In Proceedings of the 35th International Conference on Machine Learning,","year":2018},{"authors":["Saul B Needleman","Christian D Wunsch"],"title":"A general method applicable to the search for similarities in the amino acid sequence of two proteins","venue":"Journal of molecular biology,","year":1970},{"authors":["Nicholas J Redding","Tom Downs"],"title":"Learning in feedforward networks with nonsmooth functions","venue":"In Advances in Neural Information Processing Systems,","year":1992},{"authors":["Michal Rolı́nek","Paul Swoboda","Dominik Zietlow","Anselm Paulus","Vı́t Musil","Georg Martius"],"title":"Deep graph matching via blackbox differentiation of combinatorial solvers","venue":"arXiv preprint arXiv:2003.11657,","year":2020},{"authors":["Alexander Schrijver"],"title":"Combinatorial optimization: polyhedra and efficiency, volume 24","venue":"Springer Science & Business Media,","year":2003},{"authors":["Sebastian Tschiatschek","Aytunc Sahin","Andreas Krause"],"title":"Differentiable submodular maximization","venue":"In Proceedings of the 27th International Joint Conference on Artificial Intelligence,","year":2411},{"authors":["Marin Vlastelica Pogančić","Anselm Paulus","Vit Musil","Georg Martius","Michal Rolinek"],"title":"Differentiation of blackbox combinatorial solvers","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Bryan Wilder","Bistra Dilkina","Milind Tambe"],"title":"Melding the data-decisions pipeline: Decisionfocused learning for combinatorial optimization","venue":"In The Thirty-Third Conference on Artificial Intelligence (AAAI),","year":2019},{"authors":["Laurence Wolsey"],"title":"Strong formulations for mixed integer programming: a survey","venue":"Mathematical Programming,","year":1989},{"authors":["Sergey Zagoruyko","Nikos Komodakis"],"title":"Wide residual networks","venue":"In International Conference on Learning Representations ICLR’17","year":2016},{"authors":["Shuai Zheng","Sadeep Jayasumana","Bernardino Romera-Paredes","Vibhav Vineet","Zhizhong Su","Dalong Du","Chang Huang","Philip HS Torr"],"title":"Conditional random fields as recurrent neural networks","venue":"In Proceedings of the IEEE International Conference on Computer Vision, pp. 1529–1537,","year":2015}],"string":"[\n {\n \"authors\": [\n \"Akshay Agrawal\",\n \"Brandon Amos\",\n \"Shane Barratt\",\n \"Stephen Boyd\",\n \"Steven Diamond\",\n \"J Zico Kolter\"\n ],\n \"title\": \"Differentiable convex optimization layers\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Brandon Amos\",\n \"J Zico Kolter\"\n ],\n \"title\": \"Optnet: Differentiable optimization as a layer in neural networks\",\n \"venue\": \"In Proceedings of the 34th International Conference on Machine Learning,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yuri Boykov\",\n \"Olga Veksler\",\n \"Ramin Zabih\"\n ],\n \"title\": \"Fast approximate energy minimization via graph cuts\",\n \"venue\": \"IEEE Transactions on Pattern Analysis and Machine Intelligence,\",\n \"year\": 2001\n },\n {\n \"authors\": [\n \"Chien-Yi Chang\",\n \"De-An Huang\",\n \"Yanan Sui\",\n \"Li Fei-Fei\",\n \"Juan Carlos Niebles\"\n ],\n \"title\": \"D3TW: Discriminative differentiable dynamic time warping for weakly supervised action alignment and segmentation\",\n \"venue\": \"In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR),\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Liqun Chen\",\n \"Yizhe Zhang\",\n \"Ruiyi Zhang\",\n \"Chenyang Tao\",\n \"Zhe Gan\",\n \"Haichao Zhang\",\n \"Bai Li\",\n \"Dinghan Shen\",\n \"Changyou Chen\",\n \"Lawrence Carin\"\n ],\n \"title\": \"Improving sequence-to-sequence learning via optimal transport\",\n \"venue\": \"In International Conference on Learning Representations, pp\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Frank H Clarke\"\n ],\n \"title\": \"Generalized gradients and applications\",\n \"venue\": \"Transactions of the American Mathematical Society,\",\n \"year\": 1975\n },\n {\n \"authors\": [\n \"Thomas H. Cormen\",\n \"Charles E. Leiserson\",\n \"Ronald L. Rivest\",\n \"Clifford Stein\"\n ],\n \"title\": \"Introduction to Algorithms, Third Edition\",\n \"venue\": null,\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Daniel De Wolf\",\n \"Yves Smeers\"\n ],\n \"title\": \"Generalized derivatives of the optimal value of a linear program with respect to matrix coefficients\",\n \"venue\": \"Technical report. Université Catholique de Louvain,\",\n \"year\": 2000\n },\n {\n \"authors\": [\n \"Josip Djolonga\",\n \"Andreas Krause\"\n ],\n \"title\": \"Differentiable learning of submodular models\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"LawrenceCraig Evans\"\n ],\n \"title\": \"Measure theory and fine properties of functions\",\n \"venue\": null,\n \"year\": 1992\n },\n {\n \"authors\": [\n \"Aaron Ferber\",\n \"Bryan Wilder\",\n \"Bistra Dilina\",\n \"Milind Tambe\"\n ],\n \"title\": \"MIPaaL: Mixed integer program as a layer\",\n \"venue\": \"arXiv preprint arXiv:1907.05912,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Robert M Freund\"\n ],\n \"title\": \"Postoptimal analysis of a linear program under simultaneous changes in matrix coefficients\",\n \"venue\": \"In Mathematical Programming Essays in Honor of George B. Dantzig Part I,\",\n \"year\": 1985\n },\n {\n \"authors\": [\n \"Tomas Gal\"\n ],\n \"title\": \"Rim multiparametric linear programming\",\n \"venue\": \"Management Science,\",\n \"year\": 1975\n },\n {\n \"authors\": [\n \"Zhiting Hu\",\n \"Haoran Shi\",\n \"Bowen Tan\",\n \"Wentao Wang\",\n \"Zichao Yang\",\n \"Tiancheng Zhao\",\n \"Junxian He\",\n \"Lianhui Qin\",\n \"Di Wang\"\n ],\n \"title\": \"Texar: A modularized, versatile, and extensible toolkit for text generation\",\n \"venue\": \"arXiv preprint arXiv:1809.00794,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Eric Jang\",\n \"Shixiang Gu\",\n \"Ben Poole\"\n ],\n \"title\": \"Categorical reparameterization with Gumbel-softmax\",\n \"venue\": \"In International Conference on Learning Representations ICLR’17\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Stefan Lendl\",\n \"Ante Ćustić\",\n \"Abraham P Punnen\"\n ],\n \"title\": \"Combinatorial optimization with interaction costs: Complexity and solvable cases\",\n \"venue\": \"Discrete Optimization,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Chin-Yew Lin\"\n ],\n \"title\": \"Rouge: A package for automatic evaluation of summaries\",\n \"venue\": \"In Text summarization branches out,\",\n \"year\": 2004\n },\n {\n \"authors\": [\n \"Di Lin\",\n \"Jifeng Dai\",\n \"Jiaya Jia\",\n \"Kaiming He\",\n \"Jian Sun\"\n ],\n \"title\": \"Scribblesup: Scribble-supervised convolutional networks for semantic segmentation\",\n \"venue\": \"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Huidong Liu\",\n \"Xianfeng Gu\",\n \"Dimitris Samaras\"\n ],\n \"title\": \"A two-step computation of the exact GAN Wasserstein distance\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Minh-Thang Luong\",\n \"Hieu Pham\",\n \"Christopher D Manning\"\n ],\n \"title\": \"Effective approaches to attention-based neural machine translation\",\n \"venue\": \"In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Dmitrii Marin\",\n \"Meng Tang\",\n \"Ismail Ben Ayed\",\n \"Yuri Boykov\"\n ],\n \"title\": \"Beyond gradient descent for regularized segmentation losses\",\n \"venue\": \"In IEEE Conference on Computer Vision and Pattern Recognition\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"R Kipp Martin\"\n ],\n \"title\": \"Using separation algorithms to generate mixed integer model reformulations\",\n \"venue\": \"Operations Research Letters,\",\n \"year\": 1991\n },\n {\n \"authors\": [\n \"Arthur Mensch\",\n \"Mathieu Blondel\"\n ],\n \"title\": \"Differentiable dynamic programming for structured prediction and attention\",\n \"venue\": \"In Proceedings of the 35th International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Saul B Needleman\",\n \"Christian D Wunsch\"\n ],\n \"title\": \"A general method applicable to the search for similarities in the amino acid sequence of two proteins\",\n \"venue\": \"Journal of molecular biology,\",\n \"year\": 1970\n },\n {\n \"authors\": [\n \"Nicholas J Redding\",\n \"Tom Downs\"\n ],\n \"title\": \"Learning in feedforward networks with nonsmooth functions\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 1992\n },\n {\n \"authors\": [\n \"Michal Rolı́nek\",\n \"Paul Swoboda\",\n \"Dominik Zietlow\",\n \"Anselm Paulus\",\n \"Vı́t Musil\",\n \"Georg Martius\"\n ],\n \"title\": \"Deep graph matching via blackbox differentiation of combinatorial solvers\",\n \"venue\": \"arXiv preprint arXiv:2003.11657,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Alexander Schrijver\"\n ],\n \"title\": \"Combinatorial optimization: polyhedra and efficiency, volume 24\",\n \"venue\": \"Springer Science & Business Media,\",\n \"year\": 2003\n },\n {\n \"authors\": [\n \"Sebastian Tschiatschek\",\n \"Aytunc Sahin\",\n \"Andreas Krause\"\n ],\n \"title\": \"Differentiable submodular maximization\",\n \"venue\": \"In Proceedings of the 27th International Joint Conference on Artificial Intelligence,\",\n \"year\": 2411\n },\n {\n \"authors\": [\n \"Marin Vlastelica Pogančić\",\n \"Anselm Paulus\",\n \"Vit Musil\",\n \"Georg Martius\",\n \"Michal Rolinek\"\n ],\n \"title\": \"Differentiation of blackbox combinatorial solvers\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Bryan Wilder\",\n \"Bistra Dilkina\",\n \"Milind Tambe\"\n ],\n \"title\": \"Melding the data-decisions pipeline: Decisionfocused learning for combinatorial optimization\",\n \"venue\": \"In The Thirty-Third Conference on Artificial Intelligence (AAAI),\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Laurence Wolsey\"\n ],\n \"title\": \"Strong formulations for mixed integer programming: a survey\",\n \"venue\": \"Mathematical Programming,\",\n \"year\": 1989\n },\n {\n \"authors\": [\n \"Sergey Zagoruyko\",\n \"Nikos Komodakis\"\n ],\n \"title\": \"Wide residual networks\",\n \"venue\": \"In International Conference on Learning Representations ICLR’17\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Shuai Zheng\",\n \"Sadeep Jayasumana\",\n \"Bernardino Romera-Paredes\",\n \"Vibhav Vineet\",\n \"Zhizhong Su\",\n \"Dalong Du\",\n \"Chang Huang\",\n \"Philip HS Torr\"\n ],\n \"title\": \"Conditional random fields as recurrent neural networks\",\n \"venue\": \"In Proceedings of the IEEE International Conference on Computer Vision, pp. 1529–1537,\",\n \"year\": 2015\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Combinatorial optimization problems, such as shortest path in a weighted directed graph, minimum spanning tree in a weighted undirected graph, or optimal assignment of tasks to workers, play a central role in many computer science applications. We have highly refined, efficient algorithms for solving these fundamental problems (Cormen et al., 2009; Schrijver, 2003). However, while we can easily find, for example, the minimal spanning tree in a graph, the total weight of the tree as function of graph edge weights is not differentiable. This problem hinders using solutions to combinatorial problems as criteria in training models that rely on differentiability of the objective function with respect to the model parameters.\nLosses that are defined by objective value of some feasible solution to a combinatorial problem, not the optimal one, have been recently proposed for image segmentation using deep models (Zheng et al., 2015; Lin et al., 2016). These focus on a problem where some pixels in the image have segmentation labels, and the goal is to train a convolutional network that predicts segmentation labels for all pixels. For pixels with labels, a classification loss can be used. For the remaining pixels, a criterion based on a combinatorial problem – for example the maximum flow / minimal cut problem in a regular, lattice graph connecting all pixels (Boykov et al., 2001) or derived, higher-level super-pixels (Lin et al., 2016) – is often used as a loss, in an iterative process of improving discrete segmentation labels (Zheng et al., 2015; Marin et al., 2019). In this approach, the instance of the combinatorial problem is either fixed, or depends only on the input to the network; for example, similarity of neighboring pixel colors defines edge weights. The output of the neural network gives rise to a feasible, but rarely optimal, solution to that fixed instance a combinatorial problem, and its quality is used as a loss. For example, pixel labeling proposed by the network is interpreted as a cut in a pre-defined graph connecting then pixels. Training the network should result in improved cuts, but no attempt to use a solver to find an optimal cut is made.\nHere, we are considering a different setup, in which each new output of the neural network gives rise to a new instance of a combinatorial problem. A combinatorial algorithm is then used to find the optimal solution to the problem defined by the output, and the value of the objective function of\nthe optimal solution is used as a loss. After each gradient update, the network will produce a new combinatorial problem instance, even for the same input sample. Iteratively, the network is expected to learn to produce combinatorial problem instances that have low optimal objective function value. For example, in sequence-to-sequence modeling, the network will output a new sentence that is supposed to closely match the desired sentence, leading to a new optimal sequence alignment problem to be solved. Initially, the optimal alignment will be poor, but as the network improves and the quality of the output sentences get higher, the optimal alignment scores will be lower.\nRecently, progress in integrating combinatorial problems into differentiable models have been made by modifying combinatorial algorithms to use only differentiable elements (Tschiatschek et al., 2018; Mensch & Blondel, 2018; Chang et al., 2019), for example smoothed max instead of max in dynamic programming. Another approach involves executing two runs of a non-differentiable, black-box combinatorial algorithm and uses the two solutions to define a differentiable interpolation (Vlastelica Pogančić et al., 2020; Rolı́nek et al., 2020). Finally, differentiable linear programming and quadratic programming layers, which can be used to model many combinatorial problems, have been proposed recently (Amos & Kolter, 2017; Agrawal et al., 2019; Wilder et al., 2019; Ferber et al., 2019).\nThe approaches above allow for differentiating through optimal solution vectors. In many cases, we are interested only in the optimal objective value, not the solution vector, and the approaches above introduce unnecessary overhead. We propose an approach for gradient-descent based training of a network f(x;β) for supervised learning problems involving samples (x, y) with the objective criterion involving a loss term of the form L(β) = h(OptSolutionObjectiveValue(Π(F (x;β), y)), where h : R → R is some differentiable function, and Π is a combinatorial solver for a problem instance defined by the output of the β-parameterized network F for feature vector x and by the true label y. We show that a broad class of combinatorial problems can be integrated into models trained using variants of gradient descent. Specifically, we show that for an efficiently solvable combinatorial problem that can be efficiently expressed as an integer linear program, generalized gradients of the problem’s objective value with respect to real-valued parameters defining the problem exist and can be efficiently computed from a single run of a black-box combinatorial algorithm. Using the above result, we show how generalized gradients of combinatorial problems can provide sentence-level loss for text summarization using differentiable encoder-decoder models that involve softmax or Gumbel softmax (Jang et al., 2016), and a multi-element loss for training classification models when only weakly supervised, bagged training data is available."},{"heading":"2 DIFFERENTIABLE COMBINATORIAL LOSSES","text":""},{"heading":"2.1 BACKGROUND ON GENERALIZED GRADIENTS","text":"A function f : X → R defined over a convex, bounded open set X ∈ Rp is Lipschitz continuous on an open set B ∈ X if there is a finite K ∈ R such that ∀x, y ∈ B |f(x)− f(y)| ≤ K||x− y||. A function is locally Lipschitz-continuous if for every point x0 in its domain, there is a neighborhood B0, an open ball centered at x0, on which the function is Lipschitz-continuous. For such functions, a generalized gradient can be defined.\nDefinition 1. (Clarke, 1975) Let f : X → R be Lipschitz-continuous in the neighborhood of x ∈ X . Then, the Clarke subdifferential ∂f(x) of f at x is defined as\n∂f(x) = conv { lim xk→x ∇f(xk) } ,\nwhere the limit is over all convergent sequences involving those xk for which gradient exists, and conv denotes convex hull, that is, the smallest polyhedron that contains all vectors from a given set. Each element of the set ∂f(x) is called a generalized gradient of f at x.\nThe Rademacher theorem (see e.g. (Evans, 1992)) states that for any locally Lipschitz-continuous function the gradient exists almost everywhere; convergent sequences can be found.\nIn optimization algorithms, generalized gradients can be used in the same way as subgradients (Redding & Downs, 1992), that is, nondifferentiability may affect convergence in certain cases."},{"heading":"2.2 GRADIENT DESCENT OVER COMBINATORIAL OPTIMIZATION","text":"Many combinatorial problems have linear objective function and can be intuitively expressed as integer linear programs (ILP), that is, linear programs with additional constraint that the solution vector involves only integers. Any ILP can be reduced to a linear program. Consider an ILP\nz∗ = ILP (c, A′, b′) := minu c Tu s.t. A′u = b′, u ≥ 0, u ∈ Zp,\nwith an optimal solution vector u∗ and optimal objective value z∗. Then, there exists a corresponding linear program LP (c, A, b)\nz∗ = LP (c, A, b) := minu c Tu s.t. Au = b, u ≥ 0,\ncalled ideal formulation (Wolsey, 1989), for which u∗ is also an optimal solution vector, with the same objective value z∗. For a feasible, bounded p-dimensional integer program, we can view the pair (A′, b′) as a convex polyhedron A′, the set of all feasible solutions. Then, the pair (A, b) in the ideal formulation LP is defined as the set of constraints specifying the feasible set A = conv {A′ ∩ Zp}. Convex hull of a subset of a convex set A′ cannot extend beyond A′, thus, A is convex, contains all integer solutions from A′, and no other integer solutions. The number of linear constraints in the ideal formulation may be exponential in p, and/or in m, the number of the original constraints in A′. Thus, the existence of the ideal formulation LP for an ILP may not have practical utility for solving the ILP.\nFor a combinatorial problem and its corresponding ILP, we use the ideal formulation of the ILP as a conceptual tool to define generalized gradient of the objective value of the optimal solution to the combinatorial problem with respect to the parameters defining the combinatorial problem. Specifically, our approach first uses a single run of an efficient, black-box combinatorial algorithm to produce the optimal solution vector and the associated objective value. Then, the combinatorial problem is conceptually viewed as an instance of an ILP. A possibly exponentially large linear program (LP) equivalent to the ILP is then used, without actually being spelled out or solved, to derive generalized gradients based on the solution vector returned by the combinatorial algorithm.\nFirst, we introduce several notions of efficiency of transforming a combinatorial problem into a linear integer program that will be convenient in defining the generalized gradients of combinatorial problems. Definition 2. Let P (w) be a combinatorial problem that is parameterized by a continuous vector w ∈ W ⊆ Rn, whereW is simply connected and n is the problem size, and let k ∈ Z be a constant that may depend on the problem type but not on its size. Then, a combinatorial problem is\n• primal-dual ∂-efficient if it can be phrased as an integer linear program involving n variables, with kn constraints in an LP formulation equivalent to the ILP, and the parameters (A, b, c) of the LP formulation depend on w through (sub)differentiable functions, c = c(w), A = A(w), b = b(w).\n• primal ∂-efficient if it can be phrased as an integer linear program involving n variables, the parameters w of the problem influence the cost vector c through a (sub)differentiable function c = c(w), and do not influence the constraints A, b.\n• dual ∂-efficient if it can be phrased as an integer linear program in which the number of constraints in the equivalent LP formulation is kn, the parametersw of the problem influence b through a (sub)differentiable function b = b(w), and do no influence the constraint matrix A nor the cost vector c.\nThe class of ∂-efficient problems includes polynomially solvable combinatorial problems with objective function that is linear in terms of problem parameters. Typically, the functions c = c(w), b = b(w) and A = A(w) are either identity mapping or are constant; for example, in the LP for maximum network flow, the cost vector c is composed directly of edge capacities, and A an b are constant for a given flow network topology, and do not depend on capacities.\nFor any polynomially solvable combinatorial problem, we can construct a poly(n)-sized Boolean circuit for the algorithm solving it. For each poly(n)-sized circuit, there is a linear program with poly(n) variables and constraints that gives the same solution (see (Dasgupta et al., 2008), Chap.\n7). For example, for MST in a graph with V vertices and E edges, the Martin’s ILP formulation (Martin, 1991) has only poly(V + E) constraints, but it is an extended formulation that involves V E additional variables on top of the typical E variables used in the standard ILP formulations for MST. Thus, we cannot use it to construct an ILP formulation that would make MST primal-dual ∂-efficient. Alternatively, there is an ILP for MST with one binary variable per edge, and the weight of the edge only influences the cost vector c, but to prohibit cycles in the solution there is a constraint for each cycle in the graph, thus the number of constraints is not poly(n) for arbitrary graphs. These constraints are specified fully by the topology of the graph, not by the edge weights, so w does not influence A nor b, meeting the conditions for primal ∂-efficiency. The MST example shows that there are problems that are primal ∂-efficient and not primal-dual ∂-efficient.\nSome polynomially solvable combinatorial problems are not ∂-efficient in any of the above sense. For example, fixed-rank combinatorial problems with interaction costs (Lendl et al., 2019) can be phrased succinctly as a bilinear program, but lead to prohibitively large linear programs both in terms of the number of variables and the number of constraints.\nFor ∂-efficient problems, we can efficiently obtain generalized gradients of the objective value. Theorem 1. Consider a combinatorial problem P (w) of size n, a parameter vector w from the interior of the parameter domainW , and an algorithm Π(w) for solving it in time poly(n). Let z∗ be the optimal objective value returned by Π. Then,\n• if P is primal ∂-efficient, then the generalized gradients ∂z∗(w) exist, and can be efficiently computed from U∗, the set of primal solution of the ideal formulation of integer program corresponding to P ;\n• if P is dual ∂-efficient, then the generalized gradients of ∂z∗(w) exist, and can be efficiently computed from V ∗, the set of all dual solution to the ideal formulation of the integer program corresponding to P ;\n• if P is primal-dual ∂-efficient, then the generalized gradients of A over w exist, and can be efficiently computed from U∗ and V ∗, as defined above.\nProof. A series of results (Gal, 1975; Freund, 1985; De Wolf & Smeers, 2000) shows that if the optimal objective value z∗ = LP (c, A, b) for a linear program is finite at (c, A, b) and in some neighborhood of (c, A, b), then generalized gradients of z∗ with respect to c, b, and A exist and are\n∂z∗(c) = U∗, ∂z∗(b) = V ∗, ∂z∗(A) = { −vuT : (u, v) ∈ V ∗ × U∗ } .\nWe build on these results to obtain generalized gradients of the linear program corresponding to the combinatorial problem. For the first case in the theorem, definition 2 states that in the linear program corresponding to P , only the cost vector c depends on w, through a (sub)differentiable function c = c(w). Since w is in the interior of the parameter domainW , the objective value is finite over some neighborhood of w. Then,\n∂z∗(w) = ∂z∗(c) ∂c\n∂w =\n∂c ∂w U∗,\nwhere the generalized gradient z∗(c) exists and is equal to U∗. For the second case, the ideal formulation LP exists. Then, from definition 2 we have that\n∂z∗(w) = ∂z∗(b) ∂b\n∂w =\n∂b ∂w V ∗.\nThe third case is a direct extension of the first two cases.\nTheorem 1 indicates that black-box combinatorial algorithms can be used to expand the range of transformations that can be efficiently utilized in neural networks. One immediate area of application is using them to specify a loss function. Consider a network F (x;β) parameterized by a vector of tunable parameters β. The network transforms a batch of input samples x into a batch of outputs χ = F (x;β). Then, in the broadest primal-dual ∂-efficient case, χ is used, possibly with the true classes y, to formulate parameters (c, A, b) = g(χ, y) of a linear program corresponding to the combinatorial problem, through some (sub)differentiable function g. For\nAlgorithm 1 Minimization of a combinatorial loss Input: batch x ⊂ X , y ⊂ Y , network F (x;β), functions g, h, combinatorial algorithm Π Output: Loss and its generalized gradient, L(β), ∂L(β)\n1: procedure COMBLOSSMIN(x, y, β, F, g, h,Π) 2: forward pass χ = F (x;β) 3: forward pass (c, A, b) = g(χ, y) 4: run combinatorial solver to find optimal objective value z∗ = Π(c, A, b) and optimal primal and/or dual solution vectors u∗, v∗ 5: forward pass L(β) = h(z∗) 6: backward pass through h: ∂L/∂z∗ 7: backward pass through Π: ∂z∗(c) = u∗, ∂z∗(b) = v∗, ∂z∗(A) = −v∗u∗T 8: backward pass through g and F 9: ∂L(β) = ∂L∂z ( u∗ ∂c∂β − v ∗u∗T ∂A∂β + v ∗ ∂b ∂β\n) 10: return L(β), ∂L(β) 11: end procedure\na given β and given batch samples (x, y), we can then define loss as a function of the optimal objective value of the linear program corresponding to the combinatorial problem resulting from g(F (x;β), y), L(β) = h(z∗(c, A, b)). This approach, summarized in Algorithm 1, allows us to obtain the generalized gradient of the loss with respect to β as long as functions g and h are differentiable. For clarity, in Algorithm 1, we did not consider functions h depending not just on z but also on x or y, but the extension is straightforward."},{"heading":"3 EXAMPLE USE CASES AND EXPERIMENTAL VALIDATION","text":""},{"heading":"3.1 DIFFERENTIATING OVER BIPARTITE MATCHING FOR WEAKLY-SUPERVISED LEARNING","text":"To illustrate gradient descent over a combinatorial loss, we first focus on a simple image recognition problem. Consider a photo of a group of people with a caption listing each of the persons in the picture, but missing the ”from left to right” part. Given a collection of such labeled photos, can a model learn to recognize individual faces? Similarly, consider a shopping cart and a printout from the register. Given a collection of unordered shopping carts together with matching receipts, can a model learn to recognize individual shopping items? These are example of a weakly-supervised learning where the goal is to learn to classify previously unseen feature vectors, but a training sample is a bag of feature vectors accompanied by a bag of correct labels, instead of a feature-vector and a correct label. We are not told which class belongs to which sample, which prevents us from directly using the standard cross-entropy loss.\nMore formally, consider a d-class classification problem, and a model F (xj ;β) that for sample xj returns a d-dimensional vector of class probabilities, pj , with pcj denoting the predicted conditional probability of class c given feature vector xj . Let yj denote a d-dimensional, one-hot representation of the true class label of sample xj , with ycj = 1 if sample j is of class c, and zero otherwise. In weakly supervised learning involving bags of size b, we are given a tuple of b feature vectors, X = (xj) b j=1,\nand a tuple of permuted labels Y = ( yσ(i) )b i=1\nas one-hot-vectors, for some permutation σ; we will refer to the j-th element of the tuple Y as Yj . The permutation σ is unknown, thus using a loss `(pj , Yj) = `(pj , yσ(i)) to compare predicted distribution over classes for sample j with one-hot representation of j-th element in the randomly ordered set of true classes Yj makes no sense, since most likely i 6= j; Yj = yσ(i) is the class for some other sample i, not for sample j. While the permutation is unknown, with repeated presentation of bags of samples and bags of corresponding labels, we do have some information connecting the feature vector to classes. Intuitively, we can try to match model’s outputs for feature vectors in the bag to the class labels using the information in the probability distribution pj over classes provided by the model for each feature vector xj . That is, we can aim to find permutation σ̂ optimal in the average loss sense minσ̂ ∑b j=1 `(pj , σ̂(Y )j). If the class conditional probabilities pj resulting from the model perfectly match the one-hot vectors, the optimal σ̂ will be the inverse of the permutation σ, that is, σ̂(Y )j = yj .\nAlgorithm 2 Loss based on bipartite matching for weakly-supervised image classification\nInput: X = (xj) b j=1 – bag of b input images; Y = (Yk) b k=1 – a set of b sample classes to match, in\none-hot representation, in arbitrary order; β – ResNet18 network weights. Output: Loss (optimal matching cost) and its generalized gradient, L(β), ∂L(β)\n1: procedure MATCHBAG(X,Y, β) 2: forward pass, class probabilities pj = softmax(ResNet18(xj ;β)) for j = 1, ..., b 3: forward pass, cross-entropy for all image-label pairs Cjk = 〈log pj , Yk〉 for j, k = 1, ..., b 4: optimal matching cost and matching matrix: z∗,M∗ = OptMatching(C), i.e., M∗ = arg minM 〈C,M〉F , z∗ = 〈C,M∗〉F 5: final loss: cost of optimal matching L(β) = z∗ 6: backward pass through bipartite matching ∂z∗(C) = M∗ 7: backward pass through cross-entropy, softmax and ResNet18: ∂L(β) = M∗ ∂C∂β 8: return L(β), ∂L(β) 9: end procedure\nA b-element permutation can be represented by a b× b permutation matrix M . To find M , we define a b× b matrix C with Cjk = `(pj , Yk), where ` represents cross-entropy loss `(p, y) = −〈log p, y〉, with the logarithm applied element-wise. The elements Cjk correspond to edge weight in a bipartite graph with the feature vectors x processed by the neural network on one side, and labels y on the other side. We use a combinatorial solver, for example the Hungarian method with computational complexity O ( b3 ) , to find the the permutation matrix M∗ = arg minM 〈C,M〉F minimizing the Frobenius inner product of C and M . The procedure is outlined in Algorithm 2.\nTo test the approach, we used the CIFAR100 benchmark image dataset. As a baseline, we trained 5 independent fully supervised models with ResNet18 architecture (Zagoruyko & Komodakis, 2016) (see Supplementary Material for details), that is, models where each image is a separate sample with its true class available for loss calculation. To evaluate the ability of our method to provide gradients of a combinatorial loss defined by weighted matching, during training we explored image bags of samples consisting of b=4, 8, 12, 16, 24, or 32 images, and including correct but shuffled image labels. We trained 5 independent models for each bag size with the loss and its gradient provided using Algorithm 2. To avoid situations where the combinatorial loss is superficially aided by bags with mostly one class, we ignored any bag that has less than 75% of different classes, that is, for bag of size 8, we only consider bags that consist of at least 6 different classes. During testing, same as in the baseline model experiments, each image had the matching label available for test error calculations. For comparison, we trained a model with the same setup of image bags using cvxpylayers (Agrawal et al., 2019), a recently proposed methods for differentiable layers defined by conic programs. In contrast to our approach, which uses a combinatorial algorithm and relies on the LP formulation of the weighted bipartite matching only conceptually, for the definition of gradients, cvxpylayers solve the linear program in order to obtain gradients. We also trained the same model using a recently proposed approach to approximate gradients of the optimal solution vector, not the optimal objective value, of a combinatorial problem (Vlastelica Pogančić et al., 2020); we used the same combinatorial solver as in the experiments with our method.\nTest error for CIFAR100 of the training set reshuffled into bags after each epoch (Fig. 1, left) shows that for bag sizes up to twelve elements, weak supervision through weighted bipartite graph matching is almost as effective as supervised learning with true label available for each individual image, that is, bag of size one. Training using the bipartite matching loss was implemented in three different ways: through interpolated combinatorial gradients proposed in (Vlastelica Pogančić et al., 2020), through differentiable LP approach (cvxpylayers), and through the proposed approach for obtaining gradients of the objective value. All three approaches lead to very similar error rates (Fig. 1, left), indicating these three ways of obtaining gradients provide similar training signal to the network. The two methods that use combinatorial solvers are much more efficient than LP solver-based cvxpylayers (Fig. 1, right). The performance of the LP-based method decreases for very small bag sizes, where each epoch has large number of individual problems to solve, as well as for large bag sizes, where each problem to be solved involves more computation. Among the two methods using the same combinatorial solver, our proposed method is twice as fast as the interpolation method of (Vlastelica Pogančić et al., 2020), which requires solving a combinatorial problem not only in the\nforward pass, but also in the backwards pass in order to obtain gradients of the solution vector. These results show that the generalized gradient over combinatorial optimization is effective in providing training signal to train a large neural network, and can do it much faster than the state-of-the-art alternative approaches."},{"heading":"3.2 DIFFERENTIATING OVER GLOBAL SEQUENCE ALIGNMENT FOR SENTENCE-LEVEL LOSS","text":"IN SEQUENCE-TO-SEQUENCE MODELS\nAnother use case where a combinatorial loss is advantageous occurs in to sequence-to-sequence natural language models. We used a standard encoder-decoder architecture for the model (see Supplementary Material for details). The encoder takes the source sequence on input and prepares a context vector capturing the source sequence. The decoder is a recurrent network that outputs the predicted sequence one token at a time, based on the context vector and the output of the previous step. The output of the decoder at a step t is a vector of probabilities pt over the set of all possible output tokens.\nExisting encoder-decoder models use cross-entropy loss to compare predicted probabilities pt to the target word at position t, encoded as one-hot vector yt. Instead of a sequence-level optimization, position-specific cross entropy loss results in an averaged token-level optimization. We hypothesize this has detrimental effect on the training process of differentiable sequence-to-sequence models that involve softmax or Gumbel-softmax (Jang et al., 2016) as the mechanism for feeding the output of the previous step of the decoder as input for the next step. For example, a recurrent model that learned to output almost all of the target sentence correctly but is still making the mistake of missing one word early in the sentence will have very high loss at all the words following the missing word – correcting the mistake should involve keeping most of the model and focusing on the missing word, but with position-specific loss, all the outputs are considered wrong and in need of correction.\nGaps or spurious words in the output sequence can be treated naturally if we consider global sequence alignment (GSA) as the loss. Global sequence alignment (Needleman & Wunsch, 1970) is a combinatorial problem in which two sequences are aligned by choosing, at each position, to either match a token from one sequence to a token from the other, or to introduce a gap in one or the other sequence; each choice has a cost (see Fig. 2). In sequence-to-sequence modeling, the cost of matching the decoder’s output from position i to the target sequence token as position k will be given by 〈− log pi, yk〉. The cost of a gap, that is, of a horizontal or a vertical move in Fig. 2, is specified in a way that promotes closing of the gap; we use the cost of diagonal move from that position as the cost of the gap, multiplied by a scalar γ > 1 to prioritize closing the gaps over improving the matchings. In our experiments, we used γ = 1.5. The GSA problem can stated as a linear program\nwith p variables and m+ 1 constraints, with the costs of the moves forming the right-hand side of the constraints. Thus, by Theorem 1, the generalized gradient of the minimum global sequence alignment with respect to matching and gap costs is efficiently available.\nIn experiments involving global sequence alignment in sequence-to-sequence models, we used an encoder-decoder sequence-to-sequence architecture with bidirectional forward-backward RNN encoder and an attention-based RNN decoder (Luong et al., 2015), as implemented in PyTorch-Texar (Hu et al., 2018). While this architecture is no longer the top performer in terms of ROUGE metric – currently, large pre-trained self-attention models are the state-of-the-art – it is much more efficient in training, allowing for experimenting with different loss functions. During inference, we used beam search. During training, to have a differentiable decoder, we use two alternative approaches. First, we feed the probabilities resulting from the softmax layer applied to the outputs of the RNN directly as the recursive inputs to the RNN. Second, inputs to the RNN are provided by the straight-through Gumbel-softmax distribution (Jang et al., 2016) based on the outputs of the RNN, which is an approximation of the categorical distribution from which one-hot, single-token outputs are sampled. In both cases, as a baseline for comparisons with the GSA-based loss, we use word-level maximum likelihood, that is, cross-entropy between the probability vector on output of the softmax layer of the RNN and the desired target word at that position. In evaluating the combinatorial GSA loss, we used text summarization task involving the GIGAWORD dataset (Graff & Cieri, 2003) as an example of a sequence-to-sequence problem. We used test set ROUGE 1, 2, and L scores (Lin, 2004) as the measure of quality of the summarizations.\nThe results in Table 1 show that the GSA-based loss leads to improved text summarization results in all three ROUGE metrics compared to position-specific cross-entropy maximum likelihood training, both for the softmax and the Gumbel-softmax approach for providing the recursive input to the RNN in a differentiable way. The increase in accuracy comes at the cost of doubling the training time when our method is used to provide gradients of the optimal alignment score. A similar increased accuracy can be observed when the interpolation approach (Vlastelica Pogančić et al., 2020) for gradients of optimal alignment path is used instead, but the interpolation method further increases the training time, by a factor of two compared to our method. The proposed combinatorial approach is much more accurate and efficient than the recently proposed cvxpylayers method. The running time for the cvxpylayers approach is orders of magnitude slower. The cvxpylayers solver managed to reduce the training loss for several initial epochs, after which solver errors start to occur and the learning process diverges. In order to confirm this behavior, we performed 3 additional runs of the cvxpylayers-based training for the softmax model. In all cases, the loss dropped from the initial value in the 90-95 range to above 50, after which it increased to 500 or more. For comparison, the proposed combinatorial loss approach and the standard cross-entropy approach reach loss in the 30-32 range by epoch 10."},{"heading":"4 RELATED WORK","text":"Recently, (Tschiatschek et al., 2018) proposed an approximate solver for submodular function maximization that uses differentiable elements and allows for differentiating through the solver. Differentiable solvers are also considered in (Mensch & Blondel, 2018), where dynamic programming solver is re-implemented with the maximum operation replaced by smoothed max. Similar approach is used in differentiable dynamic time warping (Chang et al., 2019). Several authors used a differential approximation to linear program solutions instead of introducing differentiable operations into combinatorial algorithms. WGAN-TS (Liu et al., 2018) solves an LP to obtain the exact empirical Wasserstein distance. Then, to circumvent lack of differentiability of linear programs, WGAN-TS proceeds by training a neural network to approximate the LP solution in order to obtain gradients. In seq2seq-OT (Chen et al., 2019), an approximation is used to model optimal transport between word embeddings serving as a regularizer in training sequence-to-sequence models. These approximation approaches are limited to specific problems and preclude using off-the-shelf combinatorial solvers.\nRecently, an approach that relies on interpolation to obtain gradients of the optimal solution vector – not optimal objective value as in our method – produced by combinatorial solvers has been proposed (Vlastelica Pogančić et al., 2020; Rolı́nek et al., 2020). Similar to our approach, it allows for using off-the-shelf, black-box implementations of combinatorial algorithms. However, unlike our approach, it requires two executions of the solver, one in the forward phase, and a second execution for a slightly perturbed problem for the backward phase. As can be seen in our experiments, this results in doubling the performance overhead compared to our approach.\nAn alternative approach is to use mathematical programming solvers in gradient-trained neural networks. OptNet (Amos & Kolter, 2017) provides differentiable quadratic programming layers, and an efficient GPU-based batch solver, qpth. Cvxpylayers (Agrawal et al., 2019) generalizes this approach to a broad class of convex optimization problems expressed as cone programs, which include QP and LP as special cases, using conic solver based on ADMM, providing a general-purpose package based on the easy-to-use interface of cvxpy, with speed comparable to qpth for QP problems. Other authors (Wilder et al., 2019; Ferber et al., 2019) focus on LP problems, regularize them by adding the quadratic term, and use a QP solver as in OptNet to obtain the optimal solution vector and its gradient. Quadratic smoothing is also used in (Djolonga & Krause, 2017) in submodular set function minimization. While these methods can handle broader class of problems than our method, the reliance on quadratic or linear programming solvers translates to increased solving time. In the approach proposed here, linear programming is used only as a theoretical tool that allows for defining a mapping from the solution to a combinatorial problem to the gradient of its objective value. The solution is obtained by a single run of a combinatorial algorithm, which, as our experiments confirm, is faster than using mathematical programming and not affected by numerical instability and convergence problems."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Combinatorial optimization problems, such as shortest path in a weighted directed graph, minimum spanning tree in a weighted undirected graph, or optimal assignment of tasks to workers, play a central role in many computer science applications. We have highly refined, efficient algorithms for solving these fundamental problems (Cormen et al., 2009; Schrijver, 2003). However, while we can easily find, for example, the minimal spanning tree in a graph, the total weight of the tree as function of graph edge weights is not differentiable. This problem hinders using solutions to combinatorial problems as criteria in training models that rely on differentiability of the objective function with respect to the model parameters.\\nLosses that are defined by objective value of some feasible solution to a combinatorial problem, not the optimal one, have been recently proposed for image segmentation using deep models (Zheng et al., 2015; Lin et al., 2016). These focus on a problem where some pixels in the image have segmentation labels, and the goal is to train a convolutional network that predicts segmentation labels for all pixels. For pixels with labels, a classification loss can be used. For the remaining pixels, a criterion based on a combinatorial problem – for example the maximum flow / minimal cut problem in a regular, lattice graph connecting all pixels (Boykov et al., 2001) or derived, higher-level super-pixels (Lin et al., 2016) – is often used as a loss, in an iterative process of improving discrete segmentation labels (Zheng et al., 2015; Marin et al., 2019). In this approach, the instance of the combinatorial problem is either fixed, or depends only on the input to the network; for example, similarity of neighboring pixel colors defines edge weights. The output of the neural network gives rise to a feasible, but rarely optimal, solution to that fixed instance a combinatorial problem, and its quality is used as a loss. For example, pixel labeling proposed by the network is interpreted as a cut in a pre-defined graph connecting then pixels. Training the network should result in improved cuts, but no attempt to use a solver to find an optimal cut is made.\\nHere, we are considering a different setup, in which each new output of the neural network gives rise to a new instance of a combinatorial problem. A combinatorial algorithm is then used to find the optimal solution to the problem defined by the output, and the value of the objective function of\\nthe optimal solution is used as a loss. After each gradient update, the network will produce a new combinatorial problem instance, even for the same input sample. Iteratively, the network is expected to learn to produce combinatorial problem instances that have low optimal objective function value. For example, in sequence-to-sequence modeling, the network will output a new sentence that is supposed to closely match the desired sentence, leading to a new optimal sequence alignment problem to be solved. Initially, the optimal alignment will be poor, but as the network improves and the quality of the output sentences get higher, the optimal alignment scores will be lower.\\nRecently, progress in integrating combinatorial problems into differentiable models have been made by modifying combinatorial algorithms to use only differentiable elements (Tschiatschek et al., 2018; Mensch & Blondel, 2018; Chang et al., 2019), for example smoothed max instead of max in dynamic programming. Another approach involves executing two runs of a non-differentiable, black-box combinatorial algorithm and uses the two solutions to define a differentiable interpolation (Vlastelica Pogančić et al., 2020; Rolı́nek et al., 2020). Finally, differentiable linear programming and quadratic programming layers, which can be used to model many combinatorial problems, have been proposed recently (Amos & Kolter, 2017; Agrawal et al., 2019; Wilder et al., 2019; Ferber et al., 2019).\\nThe approaches above allow for differentiating through optimal solution vectors. In many cases, we are interested only in the optimal objective value, not the solution vector, and the approaches above introduce unnecessary overhead. We propose an approach for gradient-descent based training of a network f(x;β) for supervised learning problems involving samples (x, y) with the objective criterion involving a loss term of the form L(β) = h(OptSolutionObjectiveValue(Π(F (x;β), y)), where h : R → R is some differentiable function, and Π is a combinatorial solver for a problem instance defined by the output of the β-parameterized network F for feature vector x and by the true label y. We show that a broad class of combinatorial problems can be integrated into models trained using variants of gradient descent. Specifically, we show that for an efficiently solvable combinatorial problem that can be efficiently expressed as an integer linear program, generalized gradients of the problem’s objective value with respect to real-valued parameters defining the problem exist and can be efficiently computed from a single run of a black-box combinatorial algorithm. Using the above result, we show how generalized gradients of combinatorial problems can provide sentence-level loss for text summarization using differentiable encoder-decoder models that involve softmax or Gumbel softmax (Jang et al., 2016), and a multi-element loss for training classification models when only weakly supervised, bagged training data is available.\"\n },\n {\n \"heading\": \"2 DIFFERENTIABLE COMBINATORIAL LOSSES\",\n \"text\": \"\"\n },\n {\n \"heading\": \"2.1 BACKGROUND ON GENERALIZED GRADIENTS\",\n \"text\": \"A function f : X → R defined over a convex, bounded open set X ∈ Rp is Lipschitz continuous on an open set B ∈ X if there is a finite K ∈ R such that ∀x, y ∈ B |f(x)− f(y)| ≤ K||x− y||. A function is locally Lipschitz-continuous if for every point x0 in its domain, there is a neighborhood B0, an open ball centered at x0, on which the function is Lipschitz-continuous. For such functions, a generalized gradient can be defined.\\nDefinition 1. (Clarke, 1975) Let f : X → R be Lipschitz-continuous in the neighborhood of x ∈ X . Then, the Clarke subdifferential ∂f(x) of f at x is defined as\\n∂f(x) = conv { lim xk→x ∇f(xk) } ,\\nwhere the limit is over all convergent sequences involving those xk for which gradient exists, and conv denotes convex hull, that is, the smallest polyhedron that contains all vectors from a given set. Each element of the set ∂f(x) is called a generalized gradient of f at x.\\nThe Rademacher theorem (see e.g. (Evans, 1992)) states that for any locally Lipschitz-continuous function the gradient exists almost everywhere; convergent sequences can be found.\\nIn optimization algorithms, generalized gradients can be used in the same way as subgradients (Redding & Downs, 1992), that is, nondifferentiability may affect convergence in certain cases.\"\n },\n {\n \"heading\": \"2.2 GRADIENT DESCENT OVER COMBINATORIAL OPTIMIZATION\",\n \"text\": \"Many combinatorial problems have linear objective function and can be intuitively expressed as integer linear programs (ILP), that is, linear programs with additional constraint that the solution vector involves only integers. Any ILP can be reduced to a linear program. Consider an ILP\\nz∗ = ILP (c, A′, b′) := minu c Tu s.t. A′u = b′, u ≥ 0, u ∈ Zp,\\nwith an optimal solution vector u∗ and optimal objective value z∗. Then, there exists a corresponding linear program LP (c, A, b)\\nz∗ = LP (c, A, b) := minu c Tu s.t. Au = b, u ≥ 0,\\ncalled ideal formulation (Wolsey, 1989), for which u∗ is also an optimal solution vector, with the same objective value z∗. For a feasible, bounded p-dimensional integer program, we can view the pair (A′, b′) as a convex polyhedron A′, the set of all feasible solutions. Then, the pair (A, b) in the ideal formulation LP is defined as the set of constraints specifying the feasible set A = conv {A′ ∩ Zp}. Convex hull of a subset of a convex set A′ cannot extend beyond A′, thus, A is convex, contains all integer solutions from A′, and no other integer solutions. The number of linear constraints in the ideal formulation may be exponential in p, and/or in m, the number of the original constraints in A′. Thus, the existence of the ideal formulation LP for an ILP may not have practical utility for solving the ILP.\\nFor a combinatorial problem and its corresponding ILP, we use the ideal formulation of the ILP as a conceptual tool to define generalized gradient of the objective value of the optimal solution to the combinatorial problem with respect to the parameters defining the combinatorial problem. Specifically, our approach first uses a single run of an efficient, black-box combinatorial algorithm to produce the optimal solution vector and the associated objective value. Then, the combinatorial problem is conceptually viewed as an instance of an ILP. A possibly exponentially large linear program (LP) equivalent to the ILP is then used, without actually being spelled out or solved, to derive generalized gradients based on the solution vector returned by the combinatorial algorithm.\\nFirst, we introduce several notions of efficiency of transforming a combinatorial problem into a linear integer program that will be convenient in defining the generalized gradients of combinatorial problems. Definition 2. Let P (w) be a combinatorial problem that is parameterized by a continuous vector w ∈ W ⊆ Rn, whereW is simply connected and n is the problem size, and let k ∈ Z be a constant that may depend on the problem type but not on its size. Then, a combinatorial problem is\\n• primal-dual ∂-efficient if it can be phrased as an integer linear program involving n variables, with kn constraints in an LP formulation equivalent to the ILP, and the parameters (A, b, c) of the LP formulation depend on w through (sub)differentiable functions, c = c(w), A = A(w), b = b(w).\\n• primal ∂-efficient if it can be phrased as an integer linear program involving n variables, the parameters w of the problem influence the cost vector c through a (sub)differentiable function c = c(w), and do not influence the constraints A, b.\\n• dual ∂-efficient if it can be phrased as an integer linear program in which the number of constraints in the equivalent LP formulation is kn, the parametersw of the problem influence b through a (sub)differentiable function b = b(w), and do no influence the constraint matrix A nor the cost vector c.\\nThe class of ∂-efficient problems includes polynomially solvable combinatorial problems with objective function that is linear in terms of problem parameters. Typically, the functions c = c(w), b = b(w) and A = A(w) are either identity mapping or are constant; for example, in the LP for maximum network flow, the cost vector c is composed directly of edge capacities, and A an b are constant for a given flow network topology, and do not depend on capacities.\\nFor any polynomially solvable combinatorial problem, we can construct a poly(n)-sized Boolean circuit for the algorithm solving it. For each poly(n)-sized circuit, there is a linear program with poly(n) variables and constraints that gives the same solution (see (Dasgupta et al., 2008), Chap.\\n7). For example, for MST in a graph with V vertices and E edges, the Martin’s ILP formulation (Martin, 1991) has only poly(V + E) constraints, but it is an extended formulation that involves V E additional variables on top of the typical E variables used in the standard ILP formulations for MST. Thus, we cannot use it to construct an ILP formulation that would make MST primal-dual ∂-efficient. Alternatively, there is an ILP for MST with one binary variable per edge, and the weight of the edge only influences the cost vector c, but to prohibit cycles in the solution there is a constraint for each cycle in the graph, thus the number of constraints is not poly(n) for arbitrary graphs. These constraints are specified fully by the topology of the graph, not by the edge weights, so w does not influence A nor b, meeting the conditions for primal ∂-efficiency. The MST example shows that there are problems that are primal ∂-efficient and not primal-dual ∂-efficient.\\nSome polynomially solvable combinatorial problems are not ∂-efficient in any of the above sense. For example, fixed-rank combinatorial problems with interaction costs (Lendl et al., 2019) can be phrased succinctly as a bilinear program, but lead to prohibitively large linear programs both in terms of the number of variables and the number of constraints.\\nFor ∂-efficient problems, we can efficiently obtain generalized gradients of the objective value. Theorem 1. Consider a combinatorial problem P (w) of size n, a parameter vector w from the interior of the parameter domainW , and an algorithm Π(w) for solving it in time poly(n). Let z∗ be the optimal objective value returned by Π. Then,\\n• if P is primal ∂-efficient, then the generalized gradients ∂z∗(w) exist, and can be efficiently computed from U∗, the set of primal solution of the ideal formulation of integer program corresponding to P ;\\n• if P is dual ∂-efficient, then the generalized gradients of ∂z∗(w) exist, and can be efficiently computed from V ∗, the set of all dual solution to the ideal formulation of the integer program corresponding to P ;\\n• if P is primal-dual ∂-efficient, then the generalized gradients of A over w exist, and can be efficiently computed from U∗ and V ∗, as defined above.\\nProof. A series of results (Gal, 1975; Freund, 1985; De Wolf & Smeers, 2000) shows that if the optimal objective value z∗ = LP (c, A, b) for a linear program is finite at (c, A, b) and in some neighborhood of (c, A, b), then generalized gradients of z∗ with respect to c, b, and A exist and are\\n∂z∗(c) = U∗, ∂z∗(b) = V ∗, ∂z∗(A) = { −vuT : (u, v) ∈ V ∗ × U∗ } .\\nWe build on these results to obtain generalized gradients of the linear program corresponding to the combinatorial problem. For the first case in the theorem, definition 2 states that in the linear program corresponding to P , only the cost vector c depends on w, through a (sub)differentiable function c = c(w). Since w is in the interior of the parameter domainW , the objective value is finite over some neighborhood of w. Then,\\n∂z∗(w) = ∂z∗(c) ∂c\\n∂w =\\n∂c ∂w U∗,\\nwhere the generalized gradient z∗(c) exists and is equal to U∗. For the second case, the ideal formulation LP exists. Then, from definition 2 we have that\\n∂z∗(w) = ∂z∗(b) ∂b\\n∂w =\\n∂b ∂w V ∗.\\nThe third case is a direct extension of the first two cases.\\nTheorem 1 indicates that black-box combinatorial algorithms can be used to expand the range of transformations that can be efficiently utilized in neural networks. One immediate area of application is using them to specify a loss function. Consider a network F (x;β) parameterized by a vector of tunable parameters β. The network transforms a batch of input samples x into a batch of outputs χ = F (x;β). Then, in the broadest primal-dual ∂-efficient case, χ is used, possibly with the true classes y, to formulate parameters (c, A, b) = g(χ, y) of a linear program corresponding to the combinatorial problem, through some (sub)differentiable function g. For\\nAlgorithm 1 Minimization of a combinatorial loss Input: batch x ⊂ X , y ⊂ Y , network F (x;β), functions g, h, combinatorial algorithm Π Output: Loss and its generalized gradient, L(β), ∂L(β)\\n1: procedure COMBLOSSMIN(x, y, β, F, g, h,Π) 2: forward pass χ = F (x;β) 3: forward pass (c, A, b) = g(χ, y) 4: run combinatorial solver to find optimal objective value z∗ = Π(c, A, b) and optimal primal and/or dual solution vectors u∗, v∗ 5: forward pass L(β) = h(z∗) 6: backward pass through h: ∂L/∂z∗ 7: backward pass through Π: ∂z∗(c) = u∗, ∂z∗(b) = v∗, ∂z∗(A) = −v∗u∗T 8: backward pass through g and F 9: ∂L(β) = ∂L∂z ( u∗ ∂c∂β − v ∗u∗T ∂A∂β + v ∗ ∂b ∂β\\n) 10: return L(β), ∂L(β) 11: end procedure\\na given β and given batch samples (x, y), we can then define loss as a function of the optimal objective value of the linear program corresponding to the combinatorial problem resulting from g(F (x;β), y), L(β) = h(z∗(c, A, b)). This approach, summarized in Algorithm 1, allows us to obtain the generalized gradient of the loss with respect to β as long as functions g and h are differentiable. For clarity, in Algorithm 1, we did not consider functions h depending not just on z but also on x or y, but the extension is straightforward.\"\n },\n {\n \"heading\": \"3 EXAMPLE USE CASES AND EXPERIMENTAL VALIDATION\",\n \"text\": \"\"\n },\n {\n \"heading\": \"3.1 DIFFERENTIATING OVER BIPARTITE MATCHING FOR WEAKLY-SUPERVISED LEARNING\",\n \"text\": \"To illustrate gradient descent over a combinatorial loss, we first focus on a simple image recognition problem. Consider a photo of a group of people with a caption listing each of the persons in the picture, but missing the ”from left to right” part. Given a collection of such labeled photos, can a model learn to recognize individual faces? Similarly, consider a shopping cart and a printout from the register. Given a collection of unordered shopping carts together with matching receipts, can a model learn to recognize individual shopping items? These are example of a weakly-supervised learning where the goal is to learn to classify previously unseen feature vectors, but a training sample is a bag of feature vectors accompanied by a bag of correct labels, instead of a feature-vector and a correct label. We are not told which class belongs to which sample, which prevents us from directly using the standard cross-entropy loss.\\nMore formally, consider a d-class classification problem, and a model F (xj ;β) that for sample xj returns a d-dimensional vector of class probabilities, pj , with pcj denoting the predicted conditional probability of class c given feature vector xj . Let yj denote a d-dimensional, one-hot representation of the true class label of sample xj , with ycj = 1 if sample j is of class c, and zero otherwise. In weakly supervised learning involving bags of size b, we are given a tuple of b feature vectors, X = (xj) b j=1,\\nand a tuple of permuted labels Y = ( yσ(i) )b i=1\\nas one-hot-vectors, for some permutation σ; we will refer to the j-th element of the tuple Y as Yj . The permutation σ is unknown, thus using a loss `(pj , Yj) = `(pj , yσ(i)) to compare predicted distribution over classes for sample j with one-hot representation of j-th element in the randomly ordered set of true classes Yj makes no sense, since most likely i 6= j; Yj = yσ(i) is the class for some other sample i, not for sample j. While the permutation is unknown, with repeated presentation of bags of samples and bags of corresponding labels, we do have some information connecting the feature vector to classes. Intuitively, we can try to match model’s outputs for feature vectors in the bag to the class labels using the information in the probability distribution pj over classes provided by the model for each feature vector xj . That is, we can aim to find permutation σ̂ optimal in the average loss sense minσ̂ ∑b j=1 `(pj , σ̂(Y )j). If the class conditional probabilities pj resulting from the model perfectly match the one-hot vectors, the optimal σ̂ will be the inverse of the permutation σ, that is, σ̂(Y )j = yj .\\nAlgorithm 2 Loss based on bipartite matching for weakly-supervised image classification\\nInput: X = (xj) b j=1 – bag of b input images; Y = (Yk) b k=1 – a set of b sample classes to match, in\\none-hot representation, in arbitrary order; β – ResNet18 network weights. Output: Loss (optimal matching cost) and its generalized gradient, L(β), ∂L(β)\\n1: procedure MATCHBAG(X,Y, β) 2: forward pass, class probabilities pj = softmax(ResNet18(xj ;β)) for j = 1, ..., b 3: forward pass, cross-entropy for all image-label pairs Cjk = 〈log pj , Yk〉 for j, k = 1, ..., b 4: optimal matching cost and matching matrix: z∗,M∗ = OptMatching(C), i.e., M∗ = arg minM 〈C,M〉F , z∗ = 〈C,M∗〉F 5: final loss: cost of optimal matching L(β) = z∗ 6: backward pass through bipartite matching ∂z∗(C) = M∗ 7: backward pass through cross-entropy, softmax and ResNet18: ∂L(β) = M∗ ∂C∂β 8: return L(β), ∂L(β) 9: end procedure\\nA b-element permutation can be represented by a b× b permutation matrix M . To find M , we define a b× b matrix C with Cjk = `(pj , Yk), where ` represents cross-entropy loss `(p, y) = −〈log p, y〉, with the logarithm applied element-wise. The elements Cjk correspond to edge weight in a bipartite graph with the feature vectors x processed by the neural network on one side, and labels y on the other side. We use a combinatorial solver, for example the Hungarian method with computational complexity O ( b3 ) , to find the the permutation matrix M∗ = arg minM 〈C,M〉F minimizing the Frobenius inner product of C and M . The procedure is outlined in Algorithm 2.\\nTo test the approach, we used the CIFAR100 benchmark image dataset. As a baseline, we trained 5 independent fully supervised models with ResNet18 architecture (Zagoruyko & Komodakis, 2016) (see Supplementary Material for details), that is, models where each image is a separate sample with its true class available for loss calculation. To evaluate the ability of our method to provide gradients of a combinatorial loss defined by weighted matching, during training we explored image bags of samples consisting of b=4, 8, 12, 16, 24, or 32 images, and including correct but shuffled image labels. We trained 5 independent models for each bag size with the loss and its gradient provided using Algorithm 2. To avoid situations where the combinatorial loss is superficially aided by bags with mostly one class, we ignored any bag that has less than 75% of different classes, that is, for bag of size 8, we only consider bags that consist of at least 6 different classes. During testing, same as in the baseline model experiments, each image had the matching label available for test error calculations. For comparison, we trained a model with the same setup of image bags using cvxpylayers (Agrawal et al., 2019), a recently proposed methods for differentiable layers defined by conic programs. In contrast to our approach, which uses a combinatorial algorithm and relies on the LP formulation of the weighted bipartite matching only conceptually, for the definition of gradients, cvxpylayers solve the linear program in order to obtain gradients. We also trained the same model using a recently proposed approach to approximate gradients of the optimal solution vector, not the optimal objective value, of a combinatorial problem (Vlastelica Pogančić et al., 2020); we used the same combinatorial solver as in the experiments with our method.\\nTest error for CIFAR100 of the training set reshuffled into bags after each epoch (Fig. 1, left) shows that for bag sizes up to twelve elements, weak supervision through weighted bipartite graph matching is almost as effective as supervised learning with true label available for each individual image, that is, bag of size one. Training using the bipartite matching loss was implemented in three different ways: through interpolated combinatorial gradients proposed in (Vlastelica Pogančić et al., 2020), through differentiable LP approach (cvxpylayers), and through the proposed approach for obtaining gradients of the objective value. All three approaches lead to very similar error rates (Fig. 1, left), indicating these three ways of obtaining gradients provide similar training signal to the network. The two methods that use combinatorial solvers are much more efficient than LP solver-based cvxpylayers (Fig. 1, right). The performance of the LP-based method decreases for very small bag sizes, where each epoch has large number of individual problems to solve, as well as for large bag sizes, where each problem to be solved involves more computation. Among the two methods using the same combinatorial solver, our proposed method is twice as fast as the interpolation method of (Vlastelica Pogančić et al., 2020), which requires solving a combinatorial problem not only in the\\nforward pass, but also in the backwards pass in order to obtain gradients of the solution vector. These results show that the generalized gradient over combinatorial optimization is effective in providing training signal to train a large neural network, and can do it much faster than the state-of-the-art alternative approaches.\"\n },\n {\n \"heading\": \"3.2 DIFFERENTIATING OVER GLOBAL SEQUENCE ALIGNMENT FOR SENTENCE-LEVEL LOSS\",\n \"text\": \"IN SEQUENCE-TO-SEQUENCE MODELS\\nAnother use case where a combinatorial loss is advantageous occurs in to sequence-to-sequence natural language models. We used a standard encoder-decoder architecture for the model (see Supplementary Material for details). The encoder takes the source sequence on input and prepares a context vector capturing the source sequence. The decoder is a recurrent network that outputs the predicted sequence one token at a time, based on the context vector and the output of the previous step. The output of the decoder at a step t is a vector of probabilities pt over the set of all possible output tokens.\\nExisting encoder-decoder models use cross-entropy loss to compare predicted probabilities pt to the target word at position t, encoded as one-hot vector yt. Instead of a sequence-level optimization, position-specific cross entropy loss results in an averaged token-level optimization. We hypothesize this has detrimental effect on the training process of differentiable sequence-to-sequence models that involve softmax or Gumbel-softmax (Jang et al., 2016) as the mechanism for feeding the output of the previous step of the decoder as input for the next step. For example, a recurrent model that learned to output almost all of the target sentence correctly but is still making the mistake of missing one word early in the sentence will have very high loss at all the words following the missing word – correcting the mistake should involve keeping most of the model and focusing on the missing word, but with position-specific loss, all the outputs are considered wrong and in need of correction.\\nGaps or spurious words in the output sequence can be treated naturally if we consider global sequence alignment (GSA) as the loss. Global sequence alignment (Needleman & Wunsch, 1970) is a combinatorial problem in which two sequences are aligned by choosing, at each position, to either match a token from one sequence to a token from the other, or to introduce a gap in one or the other sequence; each choice has a cost (see Fig. 2). In sequence-to-sequence modeling, the cost of matching the decoder’s output from position i to the target sequence token as position k will be given by 〈− log pi, yk〉. The cost of a gap, that is, of a horizontal or a vertical move in Fig. 2, is specified in a way that promotes closing of the gap; we use the cost of diagonal move from that position as the cost of the gap, multiplied by a scalar γ > 1 to prioritize closing the gaps over improving the matchings. In our experiments, we used γ = 1.5. The GSA problem can stated as a linear program\\nwith p variables and m+ 1 constraints, with the costs of the moves forming the right-hand side of the constraints. Thus, by Theorem 1, the generalized gradient of the minimum global sequence alignment with respect to matching and gap costs is efficiently available.\\nIn experiments involving global sequence alignment in sequence-to-sequence models, we used an encoder-decoder sequence-to-sequence architecture with bidirectional forward-backward RNN encoder and an attention-based RNN decoder (Luong et al., 2015), as implemented in PyTorch-Texar (Hu et al., 2018). While this architecture is no longer the top performer in terms of ROUGE metric – currently, large pre-trained self-attention models are the state-of-the-art – it is much more efficient in training, allowing for experimenting with different loss functions. During inference, we used beam search. During training, to have a differentiable decoder, we use two alternative approaches. First, we feed the probabilities resulting from the softmax layer applied to the outputs of the RNN directly as the recursive inputs to the RNN. Second, inputs to the RNN are provided by the straight-through Gumbel-softmax distribution (Jang et al., 2016) based on the outputs of the RNN, which is an approximation of the categorical distribution from which one-hot, single-token outputs are sampled. In both cases, as a baseline for comparisons with the GSA-based loss, we use word-level maximum likelihood, that is, cross-entropy between the probability vector on output of the softmax layer of the RNN and the desired target word at that position. In evaluating the combinatorial GSA loss, we used text summarization task involving the GIGAWORD dataset (Graff & Cieri, 2003) as an example of a sequence-to-sequence problem. We used test set ROUGE 1, 2, and L scores (Lin, 2004) as the measure of quality of the summarizations.\\nThe results in Table 1 show that the GSA-based loss leads to improved text summarization results in all three ROUGE metrics compared to position-specific cross-entropy maximum likelihood training, both for the softmax and the Gumbel-softmax approach for providing the recursive input to the RNN in a differentiable way. The increase in accuracy comes at the cost of doubling the training time when our method is used to provide gradients of the optimal alignment score. A similar increased accuracy can be observed when the interpolation approach (Vlastelica Pogančić et al., 2020) for gradients of optimal alignment path is used instead, but the interpolation method further increases the training time, by a factor of two compared to our method. The proposed combinatorial approach is much more accurate and efficient than the recently proposed cvxpylayers method. The running time for the cvxpylayers approach is orders of magnitude slower. The cvxpylayers solver managed to reduce the training loss for several initial epochs, after which solver errors start to occur and the learning process diverges. In order to confirm this behavior, we performed 3 additional runs of the cvxpylayers-based training for the softmax model. In all cases, the loss dropped from the initial value in the 90-95 range to above 50, after which it increased to 500 or more. For comparison, the proposed combinatorial loss approach and the standard cross-entropy approach reach loss in the 30-32 range by epoch 10.\"\n },\n {\n \"heading\": \"4 RELATED WORK\",\n \"text\": \"Recently, (Tschiatschek et al., 2018) proposed an approximate solver for submodular function maximization that uses differentiable elements and allows for differentiating through the solver. Differentiable solvers are also considered in (Mensch & Blondel, 2018), where dynamic programming solver is re-implemented with the maximum operation replaced by smoothed max. Similar approach is used in differentiable dynamic time warping (Chang et al., 2019). Several authors used a differential approximation to linear program solutions instead of introducing differentiable operations into combinatorial algorithms. WGAN-TS (Liu et al., 2018) solves an LP to obtain the exact empirical Wasserstein distance. Then, to circumvent lack of differentiability of linear programs, WGAN-TS proceeds by training a neural network to approximate the LP solution in order to obtain gradients. In seq2seq-OT (Chen et al., 2019), an approximation is used to model optimal transport between word embeddings serving as a regularizer in training sequence-to-sequence models. These approximation approaches are limited to specific problems and preclude using off-the-shelf combinatorial solvers.\\nRecently, an approach that relies on interpolation to obtain gradients of the optimal solution vector – not optimal objective value as in our method – produced by combinatorial solvers has been proposed (Vlastelica Pogančić et al., 2020; Rolı́nek et al., 2020). Similar to our approach, it allows for using off-the-shelf, black-box implementations of combinatorial algorithms. However, unlike our approach, it requires two executions of the solver, one in the forward phase, and a second execution for a slightly perturbed problem for the backward phase. As can be seen in our experiments, this results in doubling the performance overhead compared to our approach.\\nAn alternative approach is to use mathematical programming solvers in gradient-trained neural networks. OptNet (Amos & Kolter, 2017) provides differentiable quadratic programming layers, and an efficient GPU-based batch solver, qpth. Cvxpylayers (Agrawal et al., 2019) generalizes this approach to a broad class of convex optimization problems expressed as cone programs, which include QP and LP as special cases, using conic solver based on ADMM, providing a general-purpose package based on the easy-to-use interface of cvxpy, with speed comparable to qpth for QP problems. Other authors (Wilder et al., 2019; Ferber et al., 2019) focus on LP problems, regularize them by adding the quadratic term, and use a QP solver as in OptNet to obtain the optimal solution vector and its gradient. Quadratic smoothing is also used in (Djolonga & Krause, 2017) in submodular set function minimization. While these methods can handle broader class of problems than our method, the reliance on quadratic or linear programming solvers translates to increased solving time. In the approach proposed here, linear programming is used only as a theoretical tool that allows for defining a mapping from the solution to a combinatorial problem to the gradient of its objective value. The solution is obtained by a single run of a combinatorial algorithm, which, as our experiments confirm, is faster than using mathematical programming and not affected by numerical instability and convergence problems.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":95,"cells":{"paper_id":{"kind":"string","value":"SP:16d9ab54eb8e4f24314ceca6e0f86f4ca586d7f1"},"summaries":{"kind":"list like","value":["This paper provides the interesting method that leverages GPU memory resources more efficiently for supernet (meta-graph) of differentiable NAS. For this, this paper proposes binary neural architecture search and consecutive model parallel (CMP). CMP parallelizes one supernet with multiple GPUs, which allows NAS model to use larger batch size and search space. Additionally, this paper improves neural architecture search speed and hardware utilization with waiting cycles reduction by dividing forward/backward phases into several sub-tasks and executing the same type of sub-tasks. The proposed method shows 1.2x faster search time compared with other model parallel methods and the highest performance among differentiable NAS methods in the experiment section."],"string":"[\n \"This paper provides the interesting method that leverages GPU memory resources more efficiently for supernet (meta-graph) of differentiable NAS. For this, this paper proposes binary neural architecture search and consecutive model parallel (CMP). CMP parallelizes one supernet with multiple GPUs, which allows NAS model to use larger batch size and search space. Additionally, this paper improves neural architecture search speed and hardware utilization with waiting cycles reduction by dividing forward/backward phases into several sub-tasks and executing the same type of sub-tasks. The proposed method shows 1.2x faster search time compared with other model parallel methods and the highest performance among differentiable NAS methods in the experiment section.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Neural architecture search (NAS) automatically designs effective network architectures. Differentiable NAS with supernets that encompass all potential architectures in a large graph cuts down search overhead to few GPU days or less. However, these algorithms consume massive GPU memory, which will restrain NAS from large batch sizes and large search spaces (e.g., more candidate operations, diverse cell structures, and large depth of supernets). In this paper, we present binary neural architecture search (NASB) with consecutive model parallel (CMP) to tackle the problem of insufficient GPU memory. CMP aggregates memory from multiple GPUs for supernets. It divides forward/backward phases into several sub-tasks and executes the same type of sub-tasks together to reduce waiting cycles. This approach improves the hardware utilization of model parallel, but it utilizes large GPU memory. NASB is proposed to reduce memory footprint, which excludes inactive operations from computation graphs and computes those operations on the fly for inactive architectural gradients in backward phases. Experiments show that NASB-CMP runs 1.2× faster than other model parallel approaches and outperforms state-of-the-art differentiable NAS. NASB can also save twice GPU memory more than PC-DARTS. Finally, we apply NASB-CMP to complicated supernet architectures. Although deep supernets with diverse cell structures do not improve NAS performance, NASB-CMP shows its potential to explore supernet architecture design in large search space 1."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Han Cai","Ligeng Zhu","Song Han"],"title":"ProxylessNAS: Direct neural architecture search on target task and hardware","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Tianqi Chen","Bing Xu","Chiyuan Zhang","Carlos Guestrin"],"title":"Training deep nets with sublinear memory cost","venue":"arXiv preprint arXiv:1604.06174,","year":2016},{"authors":["Xin Chen","Lingxi Xie","Jun Wu","Qi Tian"],"title":"Progressive differentiable architecture search: Bridging the depth gap between search and evaluation","venue":"In Proceedings of the IEEE International Conference on Computer Vision,","year":2019},{"authors":["Terrance DeVries","Graham W Taylor"],"title":"Improved regularization of convolutional neural networks with cutout","venue":"arXiv preprint arXiv:1708.04552,","year":2017},{"authors":["Aaron Harlap","Deepak Narayanan","Amar Phanishayee","Vivek Seshadri","Nikhil Devanur","Greg Ganger","Phil Gibbons"],"title":"Pipedream: Fast and efficient pipeline parallel dnn training","venue":"arXiv preprint arXiv:1806.03377,","year":2018},{"authors":["Gao Huang","Zhuang Liu","Laurens Van Der Maaten","Kilian Q Weinberger"],"title":"Densely connected convolutional networks","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2017},{"authors":["Yanping Huang","Youlong Cheng","Ankur Bapna","Orhan Firat","Dehao Chen","Mia Chen","HyoukJoong Lee","Jiquan Ngiam","Quoc V Le","Yonghui Wu"],"title":"Gpipe: Efficient training of giant neural networks using pipeline parallelism","venue":"In Advances in neural information processing systems,","year":2019},{"authors":["Chiheon Kim","Heungsub Lee","Myungryong Jeong","Woonhyuk Baek","Boogeon Yoon","Ildoo Kim","Sungbin Lim","Sungwoong Kim"],"title":"torchgpipe: On-the-fly pipeline parallelism for training giant models","venue":null,"year":2004},{"authors":["Alex Krizhevsky","Geoffrey Hinton"],"title":"Learning multiple layers of features from tiny images","venue":"Technical report, Citeseer,","year":2009},{"authors":["Seunghak Lee","Jin Kyu Kim","Xun Zheng","Qirong Ho","Garth A Gibson","Eric P Xing"],"title":"On model parallelization and scheduling strategies for distributed machine learning","venue":"In Advances in neural information processing systems,","year":2014},{"authors":["Chenxi Liu","Liang-Chieh Chen","Florian Schroff","Hartwig Adam","Wei Hua","Alan L Yuille","Li FeiFei"],"title":"Auto-deeplab: Hierarchical neural architecture search for semantic image segmentation","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2019},{"authors":["Hanxiao Liu","Karen Simonyan","Yiming Yang"],"title":"DARTS: Differentiable architecture search","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Renqian Luo","Fei Tian","Tao Qin","Enhong Chen","Tie-Yan Liu"],"title":"Neural architecture optimization","venue":"In Advances in neural information processing systems,","year":2018},{"authors":["Jieru Mei","Yingwei Li","Xiaochen Lian","Xiaojie Jin","Linjie Yang","Alan Yuille","Jianchao Yang"],"title":"Atomnas: Fine-grained end-to-end neural architecture search","venue":null,"year":1912},{"authors":["Adam Paszke","Sam Gross","Soumith Chintala","Gregory Chanan","Edward Yang","Zachary DeVito","Zeming Lin","Alban Desmaison","Luca Antiga","Adam Lerer"],"title":"Automatic differentiation in pytorch","venue":null,"year":2017},{"authors":["Hieu Pham","Melody Guan","Barret Zoph","Quoc Le","Jeff Dean"],"title":"Efficient neural architecture search via parameters sharing","venue":"In Proceedings of the 35th International Conference on Machine Learning,","year":2018},{"authors":["Esteban Real","Sherry Moore","Andrew Selle","Saurabh Saxena","Yutaka Leon Suematsu","Jie Tan","Quoc Le","Alex Kurakin"],"title":"Large-scale evolution of image classifiers","venue":"arXiv preprint arXiv:1703.01041,","year":2017},{"authors":["Esteban Real","Alok Aggarwal","Yanping Huang","Quoc V Le"],"title":"Regularized evolution for image classifier architecture search","venue":"In Proceedings of the AAAI conference on artificial intelligence,","year":2019},{"authors":["Alexander Sergeev","Mike Del"],"title":"Balso. Horovod: fast and easy distributed deep learning in tensorflow","venue":"arXiv preprint arXiv:1802.05799,","year":2018},{"authors":["Linnan Wang","Yiyang Zhao","Yuu Jinnai","Yuandong Tian","Rodrigo Fonseca"],"title":"Alphax: exploring neural architectures with deep neural networks and monte carlo tree search","venue":null,"year":1903},{"authors":["Sirui Xie","Hehui Zheng","Chunxiao Liu","Liang Lin"],"title":"SNAS: stochastic neural architecture search","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Yuhui Xu","Lingxi Xie","Xiaopeng Zhang","Xin Chen","Guo-Jun Qi","Qi Tian","Hongkai Xiong"],"title":"PCDARTS: Partial channel connections for memory-efficient architecture search","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Quanming Yao","Ju Xu","Wei-Wei Tu","Zhanxing Zhu"],"title":"Efficient neural architecture search via proximal iterations","venue":"In AAAI,","year":2020},{"authors":["Barret Zoph","Quoc V Le"],"title":"Neural architecture search with reinforcement learning","venue":"arXiv preprint arXiv:1611.01578,","year":2016},{"authors":["Barret Zoph","Vijay Vasudevan","Jonathon Shlens","Quoc V Le"],"title":"Learning transferable architectures for scalable image recognition","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2018}],"string":"[\n {\n \"authors\": [\n \"Han Cai\",\n \"Ligeng Zhu\",\n \"Song Han\"\n ],\n \"title\": \"ProxylessNAS: Direct neural architecture search on target task and hardware\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Tianqi Chen\",\n \"Bing Xu\",\n \"Chiyuan Zhang\",\n \"Carlos Guestrin\"\n ],\n \"title\": \"Training deep nets with sublinear memory cost\",\n \"venue\": \"arXiv preprint arXiv:1604.06174,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Xin Chen\",\n \"Lingxi Xie\",\n \"Jun Wu\",\n \"Qi Tian\"\n ],\n \"title\": \"Progressive differentiable architecture search: Bridging the depth gap between search and evaluation\",\n \"venue\": \"In Proceedings of the IEEE International Conference on Computer Vision,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Terrance DeVries\",\n \"Graham W Taylor\"\n ],\n \"title\": \"Improved regularization of convolutional neural networks with cutout\",\n \"venue\": \"arXiv preprint arXiv:1708.04552,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Aaron Harlap\",\n \"Deepak Narayanan\",\n \"Amar Phanishayee\",\n \"Vivek Seshadri\",\n \"Nikhil Devanur\",\n \"Greg Ganger\",\n \"Phil Gibbons\"\n ],\n \"title\": \"Pipedream: Fast and efficient pipeline parallel dnn training\",\n \"venue\": \"arXiv preprint arXiv:1806.03377,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Gao Huang\",\n \"Zhuang Liu\",\n \"Laurens Van Der Maaten\",\n \"Kilian Q Weinberger\"\n ],\n \"title\": \"Densely connected convolutional networks\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yanping Huang\",\n \"Youlong Cheng\",\n \"Ankur Bapna\",\n \"Orhan Firat\",\n \"Dehao Chen\",\n \"Mia Chen\",\n \"HyoukJoong Lee\",\n \"Jiquan Ngiam\",\n \"Quoc V Le\",\n \"Yonghui Wu\"\n ],\n \"title\": \"Gpipe: Efficient training of giant neural networks using pipeline parallelism\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Chiheon Kim\",\n \"Heungsub Lee\",\n \"Myungryong Jeong\",\n \"Woonhyuk Baek\",\n \"Boogeon Yoon\",\n \"Ildoo Kim\",\n \"Sungbin Lim\",\n \"Sungwoong Kim\"\n ],\n \"title\": \"torchgpipe: On-the-fly pipeline parallelism for training giant models\",\n \"venue\": null,\n \"year\": 2004\n },\n {\n \"authors\": [\n \"Alex Krizhevsky\",\n \"Geoffrey Hinton\"\n ],\n \"title\": \"Learning multiple layers of features from tiny images\",\n \"venue\": \"Technical report, Citeseer,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Seunghak Lee\",\n \"Jin Kyu Kim\",\n \"Xun Zheng\",\n \"Qirong Ho\",\n \"Garth A Gibson\",\n \"Eric P Xing\"\n ],\n \"title\": \"On model parallelization and scheduling strategies for distributed machine learning\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Chenxi Liu\",\n \"Liang-Chieh Chen\",\n \"Florian Schroff\",\n \"Hartwig Adam\",\n \"Wei Hua\",\n \"Alan L Yuille\",\n \"Li FeiFei\"\n ],\n \"title\": \"Auto-deeplab: Hierarchical neural architecture search for semantic image segmentation\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Hanxiao Liu\",\n \"Karen Simonyan\",\n \"Yiming Yang\"\n ],\n \"title\": \"DARTS: Differentiable architecture search\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Renqian Luo\",\n \"Fei Tian\",\n \"Tao Qin\",\n \"Enhong Chen\",\n \"Tie-Yan Liu\"\n ],\n \"title\": \"Neural architecture optimization\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Jieru Mei\",\n \"Yingwei Li\",\n \"Xiaochen Lian\",\n \"Xiaojie Jin\",\n \"Linjie Yang\",\n \"Alan Yuille\",\n \"Jianchao Yang\"\n ],\n \"title\": \"Atomnas: Fine-grained end-to-end neural architecture search\",\n \"venue\": null,\n \"year\": 1912\n },\n {\n \"authors\": [\n \"Adam Paszke\",\n \"Sam Gross\",\n \"Soumith Chintala\",\n \"Gregory Chanan\",\n \"Edward Yang\",\n \"Zachary DeVito\",\n \"Zeming Lin\",\n \"Alban Desmaison\",\n \"Luca Antiga\",\n \"Adam Lerer\"\n ],\n \"title\": \"Automatic differentiation in pytorch\",\n \"venue\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Hieu Pham\",\n \"Melody Guan\",\n \"Barret Zoph\",\n \"Quoc Le\",\n \"Jeff Dean\"\n ],\n \"title\": \"Efficient neural architecture search via parameters sharing\",\n \"venue\": \"In Proceedings of the 35th International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Esteban Real\",\n \"Sherry Moore\",\n \"Andrew Selle\",\n \"Saurabh Saxena\",\n \"Yutaka Leon Suematsu\",\n \"Jie Tan\",\n \"Quoc Le\",\n \"Alex Kurakin\"\n ],\n \"title\": \"Large-scale evolution of image classifiers\",\n \"venue\": \"arXiv preprint arXiv:1703.01041,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Esteban Real\",\n \"Alok Aggarwal\",\n \"Yanping Huang\",\n \"Quoc V Le\"\n ],\n \"title\": \"Regularized evolution for image classifier architecture search\",\n \"venue\": \"In Proceedings of the AAAI conference on artificial intelligence,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Alexander Sergeev\",\n \"Mike Del\"\n ],\n \"title\": \"Balso. Horovod: fast and easy distributed deep learning in tensorflow\",\n \"venue\": \"arXiv preprint arXiv:1802.05799,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Linnan Wang\",\n \"Yiyang Zhao\",\n \"Yuu Jinnai\",\n \"Yuandong Tian\",\n \"Rodrigo Fonseca\"\n ],\n \"title\": \"Alphax: exploring neural architectures with deep neural networks and monte carlo tree search\",\n \"venue\": null,\n \"year\": 1903\n },\n {\n \"authors\": [\n \"Sirui Xie\",\n \"Hehui Zheng\",\n \"Chunxiao Liu\",\n \"Liang Lin\"\n ],\n \"title\": \"SNAS: stochastic neural architecture search\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yuhui Xu\",\n \"Lingxi Xie\",\n \"Xiaopeng Zhang\",\n \"Xin Chen\",\n \"Guo-Jun Qi\",\n \"Qi Tian\",\n \"Hongkai Xiong\"\n ],\n \"title\": \"PCDARTS: Partial channel connections for memory-efficient architecture search\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Quanming Yao\",\n \"Ju Xu\",\n \"Wei-Wei Tu\",\n \"Zhanxing Zhu\"\n ],\n \"title\": \"Efficient neural architecture search via proximal iterations\",\n \"venue\": \"In AAAI,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Barret Zoph\",\n \"Quoc V Le\"\n ],\n \"title\": \"Neural architecture search with reinforcement learning\",\n \"venue\": \"arXiv preprint arXiv:1611.01578,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Barret Zoph\",\n \"Vijay Vasudevan\",\n \"Jonathon Shlens\",\n \"Quoc V Le\"\n ],\n \"title\": \"Learning transferable architectures for scalable image recognition\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Neural architecture search (NAS) has revolutionized architecture designs of deep learning from manually to automatically in various applications, such as image classification (Zoph & Le, 2016) and semantic segmentation (Liu et al., 2019a). Reinforcement learning (Zoph & Le, 2016; Zoph et al., 2018; Pham et al., 2018), evolutionary algorithms (Real et al., 2017; 2019), and differentiable algorithms (Liu et al., 2019b; Cai et al., 2019) have been applied to discover the optimal architecture from a large search space of candidate network structures. Supernets (Zoph et al., 2018; Pham et al., 2018) comprising all possible networks reduce search spaces from complete network architectures to cell structures. Recent acceleration techniques of differentiable NAS (Xie et al., 2019; Yao et al., 2020; Chen et al., 2019; Xu et al., 2020) further diminish search costs to affordable computation overheads (e.g., half GPU day). Prior work (Xu et al., 2020) randomly samples partial channels of intermediate feature maps in the mixed operations.\nHowever, supernets of differentiable NAS consume gigantic GPU memory, which constrains NAS from using large batch sizes and imposes restrictions on supernet architectures’ complexity. For example, NAS determines networks in shallow supernets (e.g., 8 layers) for deep compact networks (e.g., 20 layers). The cell structures are also required to remain identical for the same type of cells. Data parallelism can increase the search efficiency of NAS by using large batch sizes, such as SNAS (Xie et al., 2019), but it requires supernet complexity low enough to fit in a single GPU. In contrast, model parallelism can parallelize complex supernets, which distributes partial models to multiple devices. Nevertheless, model parallelism suffers from low hardware utilization. Only one device executes its model partition, while other devices stay idle. How to take advantage of multiple GPUs for large supernets efficiently is an open problem.\n1Search and evaluation code are released at link\nIn this paper, we propose a simple and efficient solution, binary neural architecture search (NASB) using consecutive model parallel (CMP), to tackle the above limitations. Specifically, supernets have two forward and two backward phases to learn architecture parameters and network weights. CMP distributes several sub-tasks split from the four phases in multiple GPUs and executes the sub-tasks of all forward/backward phases together. Figure 1 illustrates that sub-tasks of forward/backward phases will be overlapped to reduce waiting cycles. Nevertheless, CMP consumes large GPU memory due to two computation graphs existing at the same time. Thus, we introduce NASB to declines GPU memory occupation. NASB utilizes binary and sparse architecture parameters (1 or 0) for mixed operations. It excludes inactive operations in the computation graph and computes feature maps of inactive operations for architecture gradients during the back-propagation. In this way, NASB-CMP can increase hardware utilization of model parallelism with efficient GPU memory in differentiable NAS.\nIn our experiments on CIFAR-10, NASB-CMP runs 1.2× faster than using model parallel and pipeline parallel, TorchGPipe (Kim et al., 2020) in a server with 4 GPUs 2. It can achieve the test error of 2.53 ± 0.06% by searching for only 1.48 hours. Our contribution can be summarized as follows:\n• NASB-CMP is the first NAS algorithm that can parallelize large supernets with large batch sizes. We analyze the acceleration ratio between CMP and traditional model parallelism. Even though complex supernets (e.g., large layers and different cell structures) will not boost NAS performance, NASB-CMP paves the way to explore the supernet architecture design in the future.\n• NASB utilizes binary architecture parameters and extra architecture gradients computation to reduce GPU usage. It can save memory consumption by accepting twice batch sizes larger than the other memory saving algorithm, PC-DARTS (Xu et al., 2020).\n• We fairly compare NASB-CMP with state-of-the-art differentiable NAS in the same hardware and search space. Extensive experiments show that NASB-CMP can achieve competitive test error in short search time.\n2NVIDIA GTX 1080 Ti."},{"heading":"2 METHODOLOGY","text":"We first describe the fundamental concepts of one-shot neural architecture search (NAS) in Section 2.1. We then portray the consecutive model parallel to enhance NAS search efficiency in multiple devices in Section 2.2. Finally, we explain how we binarize the architectural weights and compute their gradients to cut down the GPU memory consumption in Section 2.3."},{"heading":"2.1 ONE-SHOT NEURAL ARCHITECTURE SEARCH","text":"One-shot neural NAS (Zoph et al., 2018) is built on a supernet (a.k.a. meta graph) in which we stack normal cells and reduce cells sequentially in Figure 2 (a). Normal cells are analogous to convolutional layers to extract images features. Reduce cells are equivalent to pooling layers to reduce the spatial dimension of feature maps. All normal cells share the same structure, but each cell still has its network weights. So do all reduce cells. One-shot approaches are required to design two cell structures instead of complete neural networks. Figure 2 (b) illustrates one popular cell structure (Pham et al., 2018), an N -node directed-acyclic-graph (DAG) with total edges E, not counting the “concat” node. In the h-th cell, the first two nodes are the (h − 2)-th and (h − 1)-th cells having no inbound edges. The other nodes accept previous nodes whose index is lower than the current index. Total edges E (red lines of Figure 2 (b)) is (N +1)(N −2)/2. We denote the h-th cell’s output as yh = concat(nj), where 2 ≤ j ≤ N − 1 and nj is a DAG node signified in Eq. 1.\nnj = yh−2, if j = 0, yh−1, j = 1,∑\ni<j mO(ni), 2 ≤ j ≤ N − 1. (1)\nA mixed operation mO is the edge between node i and j in the DAG. Let O be a set of candidate operations (e.g., convolution, pooling, identity, zero) and A ∈ RE×|O| be a matrix of architecture parameters. Eq. 2 formulates the mixed operation mO from node i to j as the weighted sum of all operations ok (Liu et al., 2019b).\nmjO(ni) = |O|∑ k=1 Ae,kok(ni), j ≥ 2, i < j, (2)\nwhere e = (j+1)(j− 2)/2+ i is the edge index. The mixed operations transform the cell structure search to the problem of learning two matrices, AN and AR, for the normal and reduce cell.\nGiven that Lval and Ltrain is the loss function L beyond a training and validation dataset, respectively. Let A comprise AN and AR. Mathematically, one-shot NAS can be formulated in the following optimization problem,\nminA Lval(w ∗,A)\ns.t. w∗ = argmin w Ltrain(w,A). (3)\nNAS leverages the validation performance to choose well-trained networks that outperform others. After training A, we derive the compact network by pruning unused operations in the supernet. Since the whole paper follows the image classification setting (Liu et al., 2019b; Cai et al., 2019), we assume each node is assigned two inputs and two operations. And we prune node inputs of cells of the supernet by the largest two values of A associated with that node. For simplicity, we use A in replace of A in the following discussion."},{"heading":"2.2 CONSECUTIVE MODEL PARALLEL","text":"Data parallelism can scale up supernets with large batch sizes, but it cannot handle large supernets (e.g., deep supernets with different cell structures). Model parallelism (MP) is able to amortize such large supernets across multiple GPUs, but its hardware utilization is low. MP would generate unwanted waiting cycles across devices. Figure 1 displays that the first device becomes idle until the second device finishes its forward and backward phases. The parallelization gets worse as we use large available GPUs.\nMotivated by pipeline parallelism (Huang et al., 2019), we propose consecutive model parallel (CMP) to decrease GPU idle time. Let FA and BA signify the forward and backward phase to update A, and Fw and Bw be two phases to update w. CMP divides the four phases into several sub-tasks and performs sub-tasks of FA and Fw consecutively, followed by sub-tasks of Bw and BA. Figure 1 illustrates that the execution order change by CMP overlaps sub-tasks without waiting for others to finish. Given the number of available GPUs M , Eq. 4 reveals the ratio of execution time between CMP and MP in theory.\nTime of CMP Time of MP = 1 M [4M − 2(M − 1)] 4 = 1− M − 1 2M . (4)\nWe assume FA, BA, Fw, and Bw take the same time unit. MP will complete an iteration in 4 units. For CMP, the total sub-tasks is 4M , and 2(M − 1) sub-tasks can be overlapped. If a sub-task takes 1/M ideally, CMP will finish an iteration in 1/M(4M − 2(M − 1)) units. According to Eq. 4, CMP with two devices could reduce (2-1)/(2*2)=25% time from MP. In practice, Experiment 3.1 demonstrates that NASB-CMP runs 1.2× faster than model parallelism without sacrificing test error. The theoretical value for 4 GPU is 1.6 (or reduce 37.5% time). We believe communication overhead and uneven model balance cause the deviation. Communication overhead comes from the intermediate tensors transfer from one to another GPU when models are split into different GPUs. Moreover, the main thread is responsible for loading data and backward propagation. The GPU with the main thread always consumes the most GPU memory, which causes uneven model balance.\nCMP is a general model parallel approach for any existing differentiable NAS algorithm. However, runningBA andBw consecutively asks for two computation graphs, which doubles GPU utilization and deteriorates CMP efficiency. To address the problem of great GPU consumption, we introduce a memory-efficient NAS to CMP, called binary neural architecture search (NASB)."},{"heading":"2.3 BINARY NEURAL ARCHITECTURE SEARCH","text":"Binary neural architecture search (NASB) harnesses binary mixed operations mBO (Yao et al., 2020) that convert the real-valued A into sparse binary matrix G, as illustrated in Figure 2. Among rows Ae,: associate node j, mBO enforces the two largest elements to 1 (active) and the rest elements to 0 (inactive). The row indexes of active elements indicate selected edges to node j, while column indexes indicate chosen operations. Notice that NASB does not directly multiply G with candidate operations in Eq. 5. Instead, NASB constructs a set of active operations O(active) based on active elements in G. Only those active operations oa ∈ O(active) are included in the forward phase. This technique could stop inactive operations being stored in the computation graph and decrease roughly\nAlgorithm 1: NASB - Consecutive Model Parallel 1: Initialize architecture weights A and network weights w 2: while not stopped do 3: Gt = binarize(At) 4: Create mBO using Gt and Eq. 5 5: Compute Lvalid(wt,Gt) and Ltrain(wt,Gt) consecutively // model parallel 6: Compute∇wLtrain(wt,Gt) and∇ALvalid(wt,Gt) consecutively // model parallel 7: Update wt+1 by descending∇wLtrain(wt,Gt) 8: Update At+1 by descending∇ALvalid(wt,Gt) 9: end while\n|O| times GPU memory compared to using the multiplication by G.\nmBO(ni) = |O|∑ k=1 Ge,kok(ni) = oa(ni). (5)\nNASB computes gradients of network weights w using standard back-propagation in the supernet. For the gradients of A, NASB estimates ∂L/∂A approximately by ∂L/∂G:\n∂L ∂Ae,k = ∂L ∂mO ∂mO ∂Ae,k ≈ ∂L ∂mBO ∂mBO ∂Ge,k = ∂L ∂mBO × ok(n) = ∂L ∂Ge,k . (6)\nEq. 6 states that gradients of elements in A come from ∂L/∂mBO × ok(n). However, inactive operations are not in the computation graph. NASB saves inputs of inactive operations n in PyTorch Context that is used for backward computation. During the backward phase, NASB will compute inactive operations ok′(n) on the fly and multiply the results with the ∂L/∂mBO.\nApart from saving unneeded GPU FLOPS and memory, mBO can avoid performance bias between supernets and compact networks. Supernets using mO assume that the performance of supernets can represent derived compact networks, but non-linear operations (e.g., ReLU-Conv-BN) break the representation that causes performance bias (Xie et al., 2019). Instead, the sparse matrix of mBO activates one operation. The performance of supernets during the search is only for one compact network. Thus, NASB can mitigate the bias caused by non-linear operations.\nAlgorithm 1 describes how CMP works with NASB. Note that NASB-CMP does not update any parameter (including A and w) until FA, BA, Fw, and Bw complete. Ltrain will use the current binary architecture matrix Gt rather than updated Gt+1, which is the major difference from the alternate algorithm (See Appendix A). Experiment 3.2 demonstrates NASB could save substantial GPU memory than PC-DARTS (Xu et al., 2020), which reduces GPU memory by partial channels of feature maps in mixed operations.\nComparison with other methods. NASP (Yao et al., 2020) binarizes A based on A itself, while ProxylessNAS (Cai et al., 2019) binarizes A based on the softmax results of A. The two binarization approaches are equivalent, but how they handle binary mixed operations (Eq. 5) is different. NASP multiplies G with all operations (i.e., saving active and inactive operations in the computation graph). ProxylessNAS selects two sampled operations (paths) in the computation graph according to multinomial distribution. NASB utilizes the same binarization as NASP but only keeps one active operation in the computation graph according to G."},{"heading":"3 EXPERIMENTS","text":"We compare NASB-CMP with other parallelisms on the CIFAR-10 in Section 3.1. We then inspect the quality of NASB and compare NASB-CMP with state-of-the-art NAS in Section 3.2. Finally, we investigate the design of supernet architectures using large layers and different cell structures in Section 3.3, which cannot be conducted without saving GPU consumption or model parallel.\nDataset. CIFAR-10 Krizhevsky & Hinton (2009) is a color-image dataset for image classification, composed of 50,000 training images and 10,000 test images for 10 classes. The dataset preparation can be found in Append E.\nSearch Space. The DAG (See Section 2.1) has N = 6 intermediate nodes and E = 14 total edges. The set of candidate operations follows NASP (Yao et al., 2020), where normal operations |ON |=8 and reduce operations |OR|=5. Notice that our baselines also use the same operation sets rather than their original one (|ON | = |OR| = 8). All operations are included in Appendix F."},{"heading":"3.1 PARALLELISM COMPARISON ON CIFAR-10","text":"The performance of NASB-CMP is compared with other parallel approaches on CIFAR-10, including data parallelism, model parallelism, and GPipe (Huang et al., 2019), the state-of-the-art model parallel that pipelines chunks of data into several model partitions. The implementation of Parallel NASB is written in Appendix B, and the search and evaluation settings are in Appendix C and D.\nFigure 3 compares the performance of different parallelizations in NASB in varied GPUs. CMP runs 1.2× faster than model parallel (MP) and GPipe especially running in 3 and 4 GPUs. According to Eq. 4, four GPUs should run 1.6X faster (or reduce 37.5% search time) than MP. In practice, communication overhead and uneven model partitions reduce the ideal speedup ratio. Compared with all parallel approaches, CMP’s execution order change does not degrade the test error. Data parallel takes the lowest search cost, but it does not generate as low test error as other model parallel approaches. The reason might be that model replicas in data parallel utilize partial batches to compute architectural gradients, while model parallel can make use of the whole batches. Therefore, CMP is an efficient model parallel approach that helps NAS to utilize large batches.\nDespite the competitive performance, the scalability of CMP is inferior. CMP disallows batch sizes from linearly scaling up as large GPUs are involved. For example, 2 GPUs should use 448 (we used 416 instead) if 1 GPU uses 224. Besides, 1-GPU NASB can utilize batch size 448, but NASB-CMP needs 4 GPUs to double batch sizes. The main reason is that CMP keeps two computation graphs (for BA and Bw) simultaneously for overlapping computations, resulting in twice GPU consumption. We believe that a mixed parallel combining CMP and data parallel can mitigate the drawback by merging two advantages, accuracy of CMP and scalability of data parallel."},{"heading":"3.2 STATE-OF-THE-ART NAS COMPARISON ON CIFAR-10","text":"Following experiment settings in Appendix C and D, we compare NASB and NASB-CMP with several NAS algorithms on CIFAR-10. DARTS (Liu et al., 2019b), SNAS (Xie et al., 2019), NASP (Yao et al., 2020), and PC-DARTS (Xu et al., 2020) are selected as our baselines. DARTS is the pioneer of differentiable NAS. SNAS points out the performance bias between a supernet and derived networks in DARTS. Both NASP and PC-DARTS reduce GPU memory, which overlaps the scope of this paper. We should select ProxylessNAS (Cai et al., 2019) as a baseline, but their search code on CIFAR-10 is not released. We prefer not to ruin their performance with improper implementation.\nInstead of directly using their reported results, we re-run the baselines from scratch to ensure their hardware and search space are the same. So, we can fairly compare them in terms of test error and search cost.\nThe test error and search cost on CIFAR-10 are stated in Table 1, where “c/o” signifies Cutout (DeVries & Taylor, 2017) used in the evaluation phase. The first row (human designed networks) and the second group of rows are extracted from their papers. The third group compares NASB with differentiable NAS baselines. The fourth group compares NAS algorithms using large batch sizes. Notably, ProxylessNAS attains the outstanding test error, but its supernet structure and search space are different from what we use, which might bias the comparison.\nIn the third group, NASB significantly takes the cheapest search cost, roughly 4 hours, to reach a comparable test error of 2.64% with SNAS 2.58% and PC-DARTS 2.59%, not to mention the search cost is smaller than the second group. NASB and NASP use similar mixed binary operations, but NASB outperforms NASP in both search cost (3.92 versus 6.44) and test error (2.64 versus 2.76). The GPU memory utilization of NASB and NASP is 2,117 MB and 9,587 MB, respectively. These three comparisons indicate that the additional gradient computation for inactive operations is a useful technique. Notice that DARTS, SNAS, and PC-DARTS use different search space (See Appendix F), so the original test errors (second group) differ from what we report in the third group of Table 1. Especially, DARTS tends to overfit the validation set by selecting “skip connect” in our search space. Its results are not as good as their original search space. Even though NASP uses the same search space, the different batch sizes and random seeds for the search and retrain setting still lead to different results.\nThe fourth group points out that NASB can considerably reduce GPU memory by using twice batch sizes larger than PC-DARTS within 1.64 hours to attain a test error of 2.49. PC-DARTS, however, becomes worse when using large batch sizes. We consider that the Hessian approximation in PC-DARTS fluctuates greatly with large batch sizes, which misleads PC-DARTS to easily select “skip connect”. NASB-CMP using four GPUs enables twice batch sizes for NASB to finish its search in 1.48 hours without severe test error degradation. Its test error 2.53% also performs better than other differentiable NAS. The empirical results in the third and fourth groups demonstrate the high efficient NASB with significant memory saving and the strong performance of NASB-CMP."},{"heading":"3.3 LARGE SUPERNETS ON CIFAR-10","text":"One-shot NAS embraces two limitations, (1) searching 8-layer supernets for 20-layer compact networks and (2) same cell structures (Liu et al., 2019b; Pham et al., 2018). We hypothesize that\n20-layer supernets with different cell structures can build suitable compact networks. Thanks to NASB and NASB-CMP that reduce GPU utilization and exploit multiple GPUs, we can examine how supernet architectures affect NAS. The search and retrain setting follow Appendix C and D. Table 2 shows test errors on CIFAR-10 with various supernet architectures, where the 1st and 2nd rows indicate cell diversity and layers (cells) numbers. Since 8-layer supernets could have six varying normal cells, we magnify each normal cell three times to construct compact networks.\nFirst, supernets with large layers do not benefit NAS to discover high-quality compact networks. Test errors in NASB (3rd row) and NASB-CMP (4th row) show that most 8-layer supernets can generate lower test errors than 20-layer supernets. The reason is 20-layer supernets have numerous architecture parameters and network weights, and they should ask for more search epochs than 8- layer supernets to train. Insufficient search epochs for deep supernets do not help NAS reach strong compact networks. Furthermore, supernets with different cell structures are not beneficial for NAS as well. When we compare results in 2nd and 4th columns (or 3rd and 5th columns), most supernets using the same cell structures can generate similar or lower test errors than using different cell structures. The reason is close to the previous one. Different cell structures demand extra search epochs to train high-dimensional architecture parameters compared to homogeneous cell structures. Not enough epochs for different cell structures do not produce low test error. Although the results contradict the hypothesis, NASB-CMP shows its potential to explore complicated supernet architectures, which paves the way for designing supernet architectures."},{"heading":"4 RELATED WORK","text":"Parallelism has been applied to NAS for acceleration (Zoph & Le, 2016; Xie et al., 2019; Cai et al., 2019; Mei et al., 2019). Parameter servers in NAS (Zoph & Le, 2016) train several child networks in parallel to speed up the learning process of the controller. ProxylessNAS (Cai et al., 2019) speed up its retrain phase by a distributed framework, Horovod (Sergeev & Del Balso, 2018). SNAS (Xie et al., 2019) and AtomNAS (Mei et al., 2019) have accelerated the search phase by data parallelism. Data parallelism runs data partitions simultaneously across multiple devices, but it cannot parallelize large models exceeding the memory of a single device, especially complicated supernets with large batch sizes. In contrast, model parallelism (Lee et al., 2014; Harlap et al., 2018; Huang et al., 2019; Kim et al., 2020) excels at parallelizing large models. GPipe (Huang et al., 2019) splits mini-batches to micro-batches and execute micro-batches in the pipeline of model partitions. The pipeline manner mitigates low hardware utilization in model parallelism. Consecutive model parallel is motivated by pipeline parallelism to overlap sub-tasks of forward/backward phases. We found that batch splitting and re-materialization (Chen et al., 2016) of GPipe increase NAS search time because frequently updating A and w intensifies extra computation. To the best of our knowledge, CMP is the most efficient model parallelism for NAS.\nReducing GPU utilization to enlarge search batch sizes is another acceleration techniques (Xu et al., 2020; Chen et al., 2019; Xie et al., 2019; Yao et al., 2020; Cai et al., 2019). PC-DARTS (Xu et al., 2020) samples channels of feature maps in mixed operations. P-DARTS (Chen et al., 2019) reduce search space as it progressively increases layers of supernets in the search phase. ProxylessNAS (Cai et al., 2019) and NASP (Yao et al., 2020) binarize A to reduce all operations saved in GPU. NASB uses the same binarization as NASP but saves one active operation in the mixed operations. Thus, NASB can reduce GPU consumption substantially and give CMP more space to keep two computation graphs in GPUs."},{"heading":"5 CONCLUSION","text":"We proposed a simple and efficient model parallel approach, NASB-CMP, which overlaps sub-tasks of forward and backward phases to reduce idle time across GPUs and utilize binary architecture parameters to reduce GPU utilization for heavy supernets. Experiments on CIFAR-10 show NASBCMP runs 1.2× faster with a large batch size of 896 than other model parallel approaches in 4 GPUs and only took 1.48 hours to attain a test error of 2.53, surpassing state-of-the-art differentiable NAS. Moreover, NASB-CMP is able to accommodate high complicated supernets for search, which paves the way for supernet network architecture design. In the future, we will combine the data parallel with NASB-CMP to overcome its inferior scalability, investigate effective and complicated supernet architectures, and analyze the communication overhead of NASB-CMP in a multi-node GPU cluster."},{"heading":"A ALTERNATE ALGORITHM OF BINARY NEURAL ARCHITECTURE SEARCH","text":"Algorithm 2 displays the alternate fashion to update A and w in the NAS: updating A with fixed w and then updating w with fixed A. Note that Line 9 of Algorithm 2 computes the gradients ∇wLtrain with updated Gt+1, which is different from the consecutive algorithm (Algorithm 1).\nAlgorithm 2: NASB 1: Initialize architecture weights A and network weights w 2: while not stopped do 3: Gt = binarize(At) 4: Create mBO using Gt and Eq. 5 5: Compute∇ALvalid(wt,Gt) using Eq. 6 // handle the gradients of inactive elements 6: Update At+1 by descending∇ALvalid(wt,Gt) 7: Gt+1 = binarize(At+1) 8: Create mBO using Gt+1 and Eq 5 9: Compute∇wLtrain(wt,Gt+1) // standard back-propagation 10: Update wt+1 by descending ∇wLtrain(wt,Gt+1) 11: end while\nB IMPLEMENTATION OF PARALLEL NAS\nThe data parallel leverages PyTorch (Paszke et al., 2017) distributed module providing communication interfaces to update parameter tensors between multiple processes. Model parallel and CMP are implemented in multi-threading. Each GPU has a specialized thread responsible for its model partition. Those threads enable different model partitions to run simultaneously. Without multi-threading, only assigning model partitions to specific devices do not automatically overlap sub-tasks. For GPipe, we adopt the corresponding PyTorch package, torchgpipe (Kim et al., 2020), in replace of GPipe, since GPipe is written in Tensorflow. The chunk setting to split mini-batch size to micro-batch size is disabled in the experiment, because enabling the setting increases the search cost."},{"heading":"C SEARCH DETAILS ON CIFAR-10","text":"Our platform is a server with 4 GPUs of NVIDIA GTX 1080 Ti, in which all search experiments are executed. Supernets consist of 8 cells in which the 3rd and 6th cells are reduce cells, and others are normal cells with initial channels 16. The optimizer for network weights w is momentum SGD with moment 0.9, L2 penalty 3e − 4, and cosine anneal learning rate initialized by 0.025 and minimal 0.001. The optimizer for architecture parameter A is Adam with learning rate 3e − 4, L2 penalty 1e− 3, and (β1, β2) = (0.5, 0.999). PC-DARTS with large batch sizes (Xu et al., 2020) has unique configurations: initial learning rate 0.1 and minimal 0.0 for SGD optimizer and learning rate 6e− 4 for Adam optimizer.\nAll NAS algorithms will search networks for 50 epochs with varied batch sizes and random seeds. In Experiment 3.1, NABS-CMP is specified search batch size 224, 416, 512, 896 for 1, 2, 3, 4 GPUs, respectively It random seed is 2. In Experiment 3.2, The batch size is 60 determined by DARTS because DARTS consumes the largest GPU memory. We want all NAS algorithms to use the same batch size in order to compare each other fairly. Since PC-DARTS is proposed to reduce GPU memory consumption, we also compare the performance of PC-DARTS using a large batch size 224 with NASB and NASB-CMP. NASB is specified with its allowable maximal batch size 448 in a single GPU, and NASB-CMP uses a batch size of 896 in 4 GPUs. All NAS baselines and NASB use 2, 3, 4, 5, 6 as random seeds, and NASP-CMP uses 2, 3, 9, 11, 18 instead. In Experiment 3.3, NASB and NASB-CMP exploit batch size 160 and 256, respectively, for 50 epochs. We ran search experiments twice using random seed 2 and 3 and reported the average test error among the two searches in Table 2."},{"heading":"D EVALUATION DETAILS ON CIFAR-10","text":"The compact networks used in the retrain (evaluation) phase have 20 cells (layers), where the onethird and two-thirds of the depth are reduce cells and others are normal cells. We retrain the compact networks from scratch for 600 epochs with the batch size 96, dropout path of probability 0.2, and initial channels of 36. We also add the auxiliary layer in the network with a loss weight 0.4. During the evaluation phase, the cutout length 16 is additionally applied for image transformation. The optimizer setting for network weights w is the same as the searching setting. The retrain random seed is assigned to 0, which is different from the search seeds."},{"heading":"E PRE-PROCESSING SUMMARY ON CIFAR-10","text":"We preprocess the training images in the following techniques: padding 32×32 images with 4 pixels, and then randomly cropping them back to 32 × 32; randomly flipping images in the horizontal direction; normalizing image pixels by the channel mean and standard deviation. The processed training set is split evenly: the first half serves as the final training set, and the other serves as the validation set. SNAS merely relies on the training set to search, so its training set is not split."},{"heading":"F CANDIDATE OPERATIONS ON CIFAR-10","text":"Table 3 summarizes candidate operations for mixed operations used in NAS papers. Experiment 3 makes use of the first row of Table 3 as its search space on CIFRAR-10. “skip connect” symbolizes identity operation if stride size is 1 or ReLU-Conv-Conv-BN operation. “conv”, “sep”, and “dil conv” signifies convolution, depthwise-separable convolutions, and dilated depthwise-separable convolutions, respectively. “none” means the zero operation. Note that differentiable NAS baselines (DARTS, SNAS, PC-DARTS) also utilize the first row of Table 3 as their search space."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Neural architecture search (NAS) has revolutionized architecture designs of deep learning from manually to automatically in various applications, such as image classification (Zoph & Le, 2016) and semantic segmentation (Liu et al., 2019a). Reinforcement learning (Zoph & Le, 2016; Zoph et al., 2018; Pham et al., 2018), evolutionary algorithms (Real et al., 2017; 2019), and differentiable algorithms (Liu et al., 2019b; Cai et al., 2019) have been applied to discover the optimal architecture from a large search space of candidate network structures. Supernets (Zoph et al., 2018; Pham et al., 2018) comprising all possible networks reduce search spaces from complete network architectures to cell structures. Recent acceleration techniques of differentiable NAS (Xie et al., 2019; Yao et al., 2020; Chen et al., 2019; Xu et al., 2020) further diminish search costs to affordable computation overheads (e.g., half GPU day). Prior work (Xu et al., 2020) randomly samples partial channels of intermediate feature maps in the mixed operations.\\nHowever, supernets of differentiable NAS consume gigantic GPU memory, which constrains NAS from using large batch sizes and imposes restrictions on supernet architectures’ complexity. For example, NAS determines networks in shallow supernets (e.g., 8 layers) for deep compact networks (e.g., 20 layers). The cell structures are also required to remain identical for the same type of cells. Data parallelism can increase the search efficiency of NAS by using large batch sizes, such as SNAS (Xie et al., 2019), but it requires supernet complexity low enough to fit in a single GPU. In contrast, model parallelism can parallelize complex supernets, which distributes partial models to multiple devices. Nevertheless, model parallelism suffers from low hardware utilization. Only one device executes its model partition, while other devices stay idle. How to take advantage of multiple GPUs for large supernets efficiently is an open problem.\\n1Search and evaluation code are released at link\\nIn this paper, we propose a simple and efficient solution, binary neural architecture search (NASB) using consecutive model parallel (CMP), to tackle the above limitations. Specifically, supernets have two forward and two backward phases to learn architecture parameters and network weights. CMP distributes several sub-tasks split from the four phases in multiple GPUs and executes the sub-tasks of all forward/backward phases together. Figure 1 illustrates that sub-tasks of forward/backward phases will be overlapped to reduce waiting cycles. Nevertheless, CMP consumes large GPU memory due to two computation graphs existing at the same time. Thus, we introduce NASB to declines GPU memory occupation. NASB utilizes binary and sparse architecture parameters (1 or 0) for mixed operations. It excludes inactive operations in the computation graph and computes feature maps of inactive operations for architecture gradients during the back-propagation. In this way, NASB-CMP can increase hardware utilization of model parallelism with efficient GPU memory in differentiable NAS.\\nIn our experiments on CIFAR-10, NASB-CMP runs 1.2× faster than using model parallel and pipeline parallel, TorchGPipe (Kim et al., 2020) in a server with 4 GPUs 2. It can achieve the test error of 2.53 ± 0.06% by searching for only 1.48 hours. Our contribution can be summarized as follows:\\n• NASB-CMP is the first NAS algorithm that can parallelize large supernets with large batch sizes. We analyze the acceleration ratio between CMP and traditional model parallelism. Even though complex supernets (e.g., large layers and different cell structures) will not boost NAS performance, NASB-CMP paves the way to explore the supernet architecture design in the future.\\n• NASB utilizes binary architecture parameters and extra architecture gradients computation to reduce GPU usage. It can save memory consumption by accepting twice batch sizes larger than the other memory saving algorithm, PC-DARTS (Xu et al., 2020).\\n• We fairly compare NASB-CMP with state-of-the-art differentiable NAS in the same hardware and search space. Extensive experiments show that NASB-CMP can achieve competitive test error in short search time.\\n2NVIDIA GTX 1080 Ti.\"\n },\n {\n \"heading\": \"2 METHODOLOGY\",\n \"text\": \"We first describe the fundamental concepts of one-shot neural architecture search (NAS) in Section 2.1. We then portray the consecutive model parallel to enhance NAS search efficiency in multiple devices in Section 2.2. Finally, we explain how we binarize the architectural weights and compute their gradients to cut down the GPU memory consumption in Section 2.3.\"\n },\n {\n \"heading\": \"2.1 ONE-SHOT NEURAL ARCHITECTURE SEARCH\",\n \"text\": \"One-shot neural NAS (Zoph et al., 2018) is built on a supernet (a.k.a. meta graph) in which we stack normal cells and reduce cells sequentially in Figure 2 (a). Normal cells are analogous to convolutional layers to extract images features. Reduce cells are equivalent to pooling layers to reduce the spatial dimension of feature maps. All normal cells share the same structure, but each cell still has its network weights. So do all reduce cells. One-shot approaches are required to design two cell structures instead of complete neural networks. Figure 2 (b) illustrates one popular cell structure (Pham et al., 2018), an N -node directed-acyclic-graph (DAG) with total edges E, not counting the “concat” node. In the h-th cell, the first two nodes are the (h − 2)-th and (h − 1)-th cells having no inbound edges. The other nodes accept previous nodes whose index is lower than the current index. Total edges E (red lines of Figure 2 (b)) is (N +1)(N −2)/2. We denote the h-th cell’s output as yh = concat(nj), where 2 ≤ j ≤ N − 1 and nj is a DAG node signified in Eq. 1.\\nnj = yh−2, if j = 0, yh−1, j = 1,∑\\ni<j mO(ni), 2 ≤ j ≤ N − 1. (1)\\nA mixed operation mO is the edge between node i and j in the DAG. Let O be a set of candidate operations (e.g., convolution, pooling, identity, zero) and A ∈ RE×|O| be a matrix of architecture parameters. Eq. 2 formulates the mixed operation mO from node i to j as the weighted sum of all operations ok (Liu et al., 2019b).\\nmjO(ni) = |O|∑ k=1 Ae,kok(ni), j ≥ 2, i < j, (2)\\nwhere e = (j+1)(j− 2)/2+ i is the edge index. The mixed operations transform the cell structure search to the problem of learning two matrices, AN and AR, for the normal and reduce cell.\\nGiven that Lval and Ltrain is the loss function L beyond a training and validation dataset, respectively. Let A comprise AN and AR. Mathematically, one-shot NAS can be formulated in the following optimization problem,\\nminA Lval(w ∗,A)\\ns.t. w∗ = argmin w Ltrain(w,A). (3)\\nNAS leverages the validation performance to choose well-trained networks that outperform others. After training A, we derive the compact network by pruning unused operations in the supernet. Since the whole paper follows the image classification setting (Liu et al., 2019b; Cai et al., 2019), we assume each node is assigned two inputs and two operations. And we prune node inputs of cells of the supernet by the largest two values of A associated with that node. For simplicity, we use A in replace of A in the following discussion.\"\n },\n {\n \"heading\": \"2.2 CONSECUTIVE MODEL PARALLEL\",\n \"text\": \"Data parallelism can scale up supernets with large batch sizes, but it cannot handle large supernets (e.g., deep supernets with different cell structures). Model parallelism (MP) is able to amortize such large supernets across multiple GPUs, but its hardware utilization is low. MP would generate unwanted waiting cycles across devices. Figure 1 displays that the first device becomes idle until the second device finishes its forward and backward phases. The parallelization gets worse as we use large available GPUs.\\nMotivated by pipeline parallelism (Huang et al., 2019), we propose consecutive model parallel (CMP) to decrease GPU idle time. Let FA and BA signify the forward and backward phase to update A, and Fw and Bw be two phases to update w. CMP divides the four phases into several sub-tasks and performs sub-tasks of FA and Fw consecutively, followed by sub-tasks of Bw and BA. Figure 1 illustrates that the execution order change by CMP overlaps sub-tasks without waiting for others to finish. Given the number of available GPUs M , Eq. 4 reveals the ratio of execution time between CMP and MP in theory.\\nTime of CMP Time of MP = 1 M [4M − 2(M − 1)] 4 = 1− M − 1 2M . (4)\\nWe assume FA, BA, Fw, and Bw take the same time unit. MP will complete an iteration in 4 units. For CMP, the total sub-tasks is 4M , and 2(M − 1) sub-tasks can be overlapped. If a sub-task takes 1/M ideally, CMP will finish an iteration in 1/M(4M − 2(M − 1)) units. According to Eq. 4, CMP with two devices could reduce (2-1)/(2*2)=25% time from MP. In practice, Experiment 3.1 demonstrates that NASB-CMP runs 1.2× faster than model parallelism without sacrificing test error. The theoretical value for 4 GPU is 1.6 (or reduce 37.5% time). We believe communication overhead and uneven model balance cause the deviation. Communication overhead comes from the intermediate tensors transfer from one to another GPU when models are split into different GPUs. Moreover, the main thread is responsible for loading data and backward propagation. The GPU with the main thread always consumes the most GPU memory, which causes uneven model balance.\\nCMP is a general model parallel approach for any existing differentiable NAS algorithm. However, runningBA andBw consecutively asks for two computation graphs, which doubles GPU utilization and deteriorates CMP efficiency. To address the problem of great GPU consumption, we introduce a memory-efficient NAS to CMP, called binary neural architecture search (NASB).\"\n },\n {\n \"heading\": \"2.3 BINARY NEURAL ARCHITECTURE SEARCH\",\n \"text\": \"Binary neural architecture search (NASB) harnesses binary mixed operations mBO (Yao et al., 2020) that convert the real-valued A into sparse binary matrix G, as illustrated in Figure 2. Among rows Ae,: associate node j, mBO enforces the two largest elements to 1 (active) and the rest elements to 0 (inactive). The row indexes of active elements indicate selected edges to node j, while column indexes indicate chosen operations. Notice that NASB does not directly multiply G with candidate operations in Eq. 5. Instead, NASB constructs a set of active operations O(active) based on active elements in G. Only those active operations oa ∈ O(active) are included in the forward phase. This technique could stop inactive operations being stored in the computation graph and decrease roughly\\nAlgorithm 1: NASB - Consecutive Model Parallel 1: Initialize architecture weights A and network weights w 2: while not stopped do 3: Gt = binarize(At) 4: Create mBO using Gt and Eq. 5 5: Compute Lvalid(wt,Gt) and Ltrain(wt,Gt) consecutively // model parallel 6: Compute∇wLtrain(wt,Gt) and∇ALvalid(wt,Gt) consecutively // model parallel 7: Update wt+1 by descending∇wLtrain(wt,Gt) 8: Update At+1 by descending∇ALvalid(wt,Gt) 9: end while\\n|O| times GPU memory compared to using the multiplication by G.\\nmBO(ni) = |O|∑ k=1 Ge,kok(ni) = oa(ni). (5)\\nNASB computes gradients of network weights w using standard back-propagation in the supernet. For the gradients of A, NASB estimates ∂L/∂A approximately by ∂L/∂G:\\n∂L ∂Ae,k = ∂L ∂mO ∂mO ∂Ae,k ≈ ∂L ∂mBO ∂mBO ∂Ge,k = ∂L ∂mBO × ok(n) = ∂L ∂Ge,k . (6)\\nEq. 6 states that gradients of elements in A come from ∂L/∂mBO × ok(n). However, inactive operations are not in the computation graph. NASB saves inputs of inactive operations n in PyTorch Context that is used for backward computation. During the backward phase, NASB will compute inactive operations ok′(n) on the fly and multiply the results with the ∂L/∂mBO.\\nApart from saving unneeded GPU FLOPS and memory, mBO can avoid performance bias between supernets and compact networks. Supernets using mO assume that the performance of supernets can represent derived compact networks, but non-linear operations (e.g., ReLU-Conv-BN) break the representation that causes performance bias (Xie et al., 2019). Instead, the sparse matrix of mBO activates one operation. The performance of supernets during the search is only for one compact network. Thus, NASB can mitigate the bias caused by non-linear operations.\\nAlgorithm 1 describes how CMP works with NASB. Note that NASB-CMP does not update any parameter (including A and w) until FA, BA, Fw, and Bw complete. Ltrain will use the current binary architecture matrix Gt rather than updated Gt+1, which is the major difference from the alternate algorithm (See Appendix A). Experiment 3.2 demonstrates NASB could save substantial GPU memory than PC-DARTS (Xu et al., 2020), which reduces GPU memory by partial channels of feature maps in mixed operations.\\nComparison with other methods. NASP (Yao et al., 2020) binarizes A based on A itself, while ProxylessNAS (Cai et al., 2019) binarizes A based on the softmax results of A. The two binarization approaches are equivalent, but how they handle binary mixed operations (Eq. 5) is different. NASP multiplies G with all operations (i.e., saving active and inactive operations in the computation graph). ProxylessNAS selects two sampled operations (paths) in the computation graph according to multinomial distribution. NASB utilizes the same binarization as NASP but only keeps one active operation in the computation graph according to G.\"\n },\n {\n \"heading\": \"3 EXPERIMENTS\",\n \"text\": \"We compare NASB-CMP with other parallelisms on the CIFAR-10 in Section 3.1. We then inspect the quality of NASB and compare NASB-CMP with state-of-the-art NAS in Section 3.2. Finally, we investigate the design of supernet architectures using large layers and different cell structures in Section 3.3, which cannot be conducted without saving GPU consumption or model parallel.\\nDataset. CIFAR-10 Krizhevsky & Hinton (2009) is a color-image dataset for image classification, composed of 50,000 training images and 10,000 test images for 10 classes. The dataset preparation can be found in Append E.\\nSearch Space. The DAG (See Section 2.1) has N = 6 intermediate nodes and E = 14 total edges. The set of candidate operations follows NASP (Yao et al., 2020), where normal operations |ON |=8 and reduce operations |OR|=5. Notice that our baselines also use the same operation sets rather than their original one (|ON | = |OR| = 8). All operations are included in Appendix F.\"\n },\n {\n \"heading\": \"3.1 PARALLELISM COMPARISON ON CIFAR-10\",\n \"text\": \"The performance of NASB-CMP is compared with other parallel approaches on CIFAR-10, including data parallelism, model parallelism, and GPipe (Huang et al., 2019), the state-of-the-art model parallel that pipelines chunks of data into several model partitions. The implementation of Parallel NASB is written in Appendix B, and the search and evaluation settings are in Appendix C and D.\\nFigure 3 compares the performance of different parallelizations in NASB in varied GPUs. CMP runs 1.2× faster than model parallel (MP) and GPipe especially running in 3 and 4 GPUs. According to Eq. 4, four GPUs should run 1.6X faster (or reduce 37.5% search time) than MP. In practice, communication overhead and uneven model partitions reduce the ideal speedup ratio. Compared with all parallel approaches, CMP’s execution order change does not degrade the test error. Data parallel takes the lowest search cost, but it does not generate as low test error as other model parallel approaches. The reason might be that model replicas in data parallel utilize partial batches to compute architectural gradients, while model parallel can make use of the whole batches. Therefore, CMP is an efficient model parallel approach that helps NAS to utilize large batches.\\nDespite the competitive performance, the scalability of CMP is inferior. CMP disallows batch sizes from linearly scaling up as large GPUs are involved. For example, 2 GPUs should use 448 (we used 416 instead) if 1 GPU uses 224. Besides, 1-GPU NASB can utilize batch size 448, but NASB-CMP needs 4 GPUs to double batch sizes. The main reason is that CMP keeps two computation graphs (for BA and Bw) simultaneously for overlapping computations, resulting in twice GPU consumption. We believe that a mixed parallel combining CMP and data parallel can mitigate the drawback by merging two advantages, accuracy of CMP and scalability of data parallel.\"\n },\n {\n \"heading\": \"3.2 STATE-OF-THE-ART NAS COMPARISON ON CIFAR-10\",\n \"text\": \"Following experiment settings in Appendix C and D, we compare NASB and NASB-CMP with several NAS algorithms on CIFAR-10. DARTS (Liu et al., 2019b), SNAS (Xie et al., 2019), NASP (Yao et al., 2020), and PC-DARTS (Xu et al., 2020) are selected as our baselines. DARTS is the pioneer of differentiable NAS. SNAS points out the performance bias between a supernet and derived networks in DARTS. Both NASP and PC-DARTS reduce GPU memory, which overlaps the scope of this paper. We should select ProxylessNAS (Cai et al., 2019) as a baseline, but their search code on CIFAR-10 is not released. We prefer not to ruin their performance with improper implementation.\\nInstead of directly using their reported results, we re-run the baselines from scratch to ensure their hardware and search space are the same. So, we can fairly compare them in terms of test error and search cost.\\nThe test error and search cost on CIFAR-10 are stated in Table 1, where “c/o” signifies Cutout (DeVries & Taylor, 2017) used in the evaluation phase. The first row (human designed networks) and the second group of rows are extracted from their papers. The third group compares NASB with differentiable NAS baselines. The fourth group compares NAS algorithms using large batch sizes. Notably, ProxylessNAS attains the outstanding test error, but its supernet structure and search space are different from what we use, which might bias the comparison.\\nIn the third group, NASB significantly takes the cheapest search cost, roughly 4 hours, to reach a comparable test error of 2.64% with SNAS 2.58% and PC-DARTS 2.59%, not to mention the search cost is smaller than the second group. NASB and NASP use similar mixed binary operations, but NASB outperforms NASP in both search cost (3.92 versus 6.44) and test error (2.64 versus 2.76). The GPU memory utilization of NASB and NASP is 2,117 MB and 9,587 MB, respectively. These three comparisons indicate that the additional gradient computation for inactive operations is a useful technique. Notice that DARTS, SNAS, and PC-DARTS use different search space (See Appendix F), so the original test errors (second group) differ from what we report in the third group of Table 1. Especially, DARTS tends to overfit the validation set by selecting “skip connect” in our search space. Its results are not as good as their original search space. Even though NASP uses the same search space, the different batch sizes and random seeds for the search and retrain setting still lead to different results.\\nThe fourth group points out that NASB can considerably reduce GPU memory by using twice batch sizes larger than PC-DARTS within 1.64 hours to attain a test error of 2.49. PC-DARTS, however, becomes worse when using large batch sizes. We consider that the Hessian approximation in PC-DARTS fluctuates greatly with large batch sizes, which misleads PC-DARTS to easily select “skip connect”. NASB-CMP using four GPUs enables twice batch sizes for NASB to finish its search in 1.48 hours without severe test error degradation. Its test error 2.53% also performs better than other differentiable NAS. The empirical results in the third and fourth groups demonstrate the high efficient NASB with significant memory saving and the strong performance of NASB-CMP.\"\n },\n {\n \"heading\": \"3.3 LARGE SUPERNETS ON CIFAR-10\",\n \"text\": \"One-shot NAS embraces two limitations, (1) searching 8-layer supernets for 20-layer compact networks and (2) same cell structures (Liu et al., 2019b; Pham et al., 2018). We hypothesize that\\n20-layer supernets with different cell structures can build suitable compact networks. Thanks to NASB and NASB-CMP that reduce GPU utilization and exploit multiple GPUs, we can examine how supernet architectures affect NAS. The search and retrain setting follow Appendix C and D. Table 2 shows test errors on CIFAR-10 with various supernet architectures, where the 1st and 2nd rows indicate cell diversity and layers (cells) numbers. Since 8-layer supernets could have six varying normal cells, we magnify each normal cell three times to construct compact networks.\\nFirst, supernets with large layers do not benefit NAS to discover high-quality compact networks. Test errors in NASB (3rd row) and NASB-CMP (4th row) show that most 8-layer supernets can generate lower test errors than 20-layer supernets. The reason is 20-layer supernets have numerous architecture parameters and network weights, and they should ask for more search epochs than 8- layer supernets to train. Insufficient search epochs for deep supernets do not help NAS reach strong compact networks. Furthermore, supernets with different cell structures are not beneficial for NAS as well. When we compare results in 2nd and 4th columns (or 3rd and 5th columns), most supernets using the same cell structures can generate similar or lower test errors than using different cell structures. The reason is close to the previous one. Different cell structures demand extra search epochs to train high-dimensional architecture parameters compared to homogeneous cell structures. Not enough epochs for different cell structures do not produce low test error. Although the results contradict the hypothesis, NASB-CMP shows its potential to explore complicated supernet architectures, which paves the way for designing supernet architectures.\"\n },\n {\n \"heading\": \"4 RELATED WORK\",\n \"text\": \"Parallelism has been applied to NAS for acceleration (Zoph & Le, 2016; Xie et al., 2019; Cai et al., 2019; Mei et al., 2019). Parameter servers in NAS (Zoph & Le, 2016) train several child networks in parallel to speed up the learning process of the controller. ProxylessNAS (Cai et al., 2019) speed up its retrain phase by a distributed framework, Horovod (Sergeev & Del Balso, 2018). SNAS (Xie et al., 2019) and AtomNAS (Mei et al., 2019) have accelerated the search phase by data parallelism. Data parallelism runs data partitions simultaneously across multiple devices, but it cannot parallelize large models exceeding the memory of a single device, especially complicated supernets with large batch sizes. In contrast, model parallelism (Lee et al., 2014; Harlap et al., 2018; Huang et al., 2019; Kim et al., 2020) excels at parallelizing large models. GPipe (Huang et al., 2019) splits mini-batches to micro-batches and execute micro-batches in the pipeline of model partitions. The pipeline manner mitigates low hardware utilization in model parallelism. Consecutive model parallel is motivated by pipeline parallelism to overlap sub-tasks of forward/backward phases. We found that batch splitting and re-materialization (Chen et al., 2016) of GPipe increase NAS search time because frequently updating A and w intensifies extra computation. To the best of our knowledge, CMP is the most efficient model parallelism for NAS.\\nReducing GPU utilization to enlarge search batch sizes is another acceleration techniques (Xu et al., 2020; Chen et al., 2019; Xie et al., 2019; Yao et al., 2020; Cai et al., 2019). PC-DARTS (Xu et al., 2020) samples channels of feature maps in mixed operations. P-DARTS (Chen et al., 2019) reduce search space as it progressively increases layers of supernets in the search phase. ProxylessNAS (Cai et al., 2019) and NASP (Yao et al., 2020) binarize A to reduce all operations saved in GPU. NASB uses the same binarization as NASP but saves one active operation in the mixed operations. Thus, NASB can reduce GPU consumption substantially and give CMP more space to keep two computation graphs in GPUs.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"We proposed a simple and efficient model parallel approach, NASB-CMP, which overlaps sub-tasks of forward and backward phases to reduce idle time across GPUs and utilize binary architecture parameters to reduce GPU utilization for heavy supernets. Experiments on CIFAR-10 show NASBCMP runs 1.2× faster with a large batch size of 896 than other model parallel approaches in 4 GPUs and only took 1.48 hours to attain a test error of 2.53, surpassing state-of-the-art differentiable NAS. Moreover, NASB-CMP is able to accommodate high complicated supernets for search, which paves the way for supernet network architecture design. In the future, we will combine the data parallel with NASB-CMP to overcome its inferior scalability, investigate effective and complicated supernet architectures, and analyze the communication overhead of NASB-CMP in a multi-node GPU cluster.\"\n },\n {\n \"heading\": \"A ALTERNATE ALGORITHM OF BINARY NEURAL ARCHITECTURE SEARCH\",\n \"text\": \"Algorithm 2 displays the alternate fashion to update A and w in the NAS: updating A with fixed w and then updating w with fixed A. Note that Line 9 of Algorithm 2 computes the gradients ∇wLtrain with updated Gt+1, which is different from the consecutive algorithm (Algorithm 1).\\nAlgorithm 2: NASB 1: Initialize architecture weights A and network weights w 2: while not stopped do 3: Gt = binarize(At) 4: Create mBO using Gt and Eq. 5 5: Compute∇ALvalid(wt,Gt) using Eq. 6 // handle the gradients of inactive elements 6: Update At+1 by descending∇ALvalid(wt,Gt) 7: Gt+1 = binarize(At+1) 8: Create mBO using Gt+1 and Eq 5 9: Compute∇wLtrain(wt,Gt+1) // standard back-propagation 10: Update wt+1 by descending ∇wLtrain(wt,Gt+1) 11: end while\\nB IMPLEMENTATION OF PARALLEL NAS\\nThe data parallel leverages PyTorch (Paszke et al., 2017) distributed module providing communication interfaces to update parameter tensors between multiple processes. Model parallel and CMP are implemented in multi-threading. Each GPU has a specialized thread responsible for its model partition. Those threads enable different model partitions to run simultaneously. Without multi-threading, only assigning model partitions to specific devices do not automatically overlap sub-tasks. For GPipe, we adopt the corresponding PyTorch package, torchgpipe (Kim et al., 2020), in replace of GPipe, since GPipe is written in Tensorflow. The chunk setting to split mini-batch size to micro-batch size is disabled in the experiment, because enabling the setting increases the search cost.\"\n },\n {\n \"heading\": \"C SEARCH DETAILS ON CIFAR-10\",\n \"text\": \"Our platform is a server with 4 GPUs of NVIDIA GTX 1080 Ti, in which all search experiments are executed. Supernets consist of 8 cells in which the 3rd and 6th cells are reduce cells, and others are normal cells with initial channels 16. The optimizer for network weights w is momentum SGD with moment 0.9, L2 penalty 3e − 4, and cosine anneal learning rate initialized by 0.025 and minimal 0.001. The optimizer for architecture parameter A is Adam with learning rate 3e − 4, L2 penalty 1e− 3, and (β1, β2) = (0.5, 0.999). PC-DARTS with large batch sizes (Xu et al., 2020) has unique configurations: initial learning rate 0.1 and minimal 0.0 for SGD optimizer and learning rate 6e− 4 for Adam optimizer.\\nAll NAS algorithms will search networks for 50 epochs with varied batch sizes and random seeds. In Experiment 3.1, NABS-CMP is specified search batch size 224, 416, 512, 896 for 1, 2, 3, 4 GPUs, respectively It random seed is 2. In Experiment 3.2, The batch size is 60 determined by DARTS because DARTS consumes the largest GPU memory. We want all NAS algorithms to use the same batch size in order to compare each other fairly. Since PC-DARTS is proposed to reduce GPU memory consumption, we also compare the performance of PC-DARTS using a large batch size 224 with NASB and NASB-CMP. NASB is specified with its allowable maximal batch size 448 in a single GPU, and NASB-CMP uses a batch size of 896 in 4 GPUs. All NAS baselines and NASB use 2, 3, 4, 5, 6 as random seeds, and NASP-CMP uses 2, 3, 9, 11, 18 instead. In Experiment 3.3, NASB and NASB-CMP exploit batch size 160 and 256, respectively, for 50 epochs. We ran search experiments twice using random seed 2 and 3 and reported the average test error among the two searches in Table 2.\"\n },\n {\n \"heading\": \"D EVALUATION DETAILS ON CIFAR-10\",\n \"text\": \"The compact networks used in the retrain (evaluation) phase have 20 cells (layers), where the onethird and two-thirds of the depth are reduce cells and others are normal cells. We retrain the compact networks from scratch for 600 epochs with the batch size 96, dropout path of probability 0.2, and initial channels of 36. We also add the auxiliary layer in the network with a loss weight 0.4. During the evaluation phase, the cutout length 16 is additionally applied for image transformation. The optimizer setting for network weights w is the same as the searching setting. The retrain random seed is assigned to 0, which is different from the search seeds.\"\n },\n {\n \"heading\": \"E PRE-PROCESSING SUMMARY ON CIFAR-10\",\n \"text\": \"We preprocess the training images in the following techniques: padding 32×32 images with 4 pixels, and then randomly cropping them back to 32 × 32; randomly flipping images in the horizontal direction; normalizing image pixels by the channel mean and standard deviation. The processed training set is split evenly: the first half serves as the final training set, and the other serves as the validation set. SNAS merely relies on the training set to search, so its training set is not split.\"\n },\n {\n \"heading\": \"F CANDIDATE OPERATIONS ON CIFAR-10\",\n \"text\": \"Table 3 summarizes candidate operations for mixed operations used in NAS papers. Experiment 3 makes use of the first row of Table 3 as its search space on CIFRAR-10. “skip connect” symbolizes identity operation if stride size is 1 or ReLU-Conv-Conv-BN operation. “conv”, “sep”, and “dil conv” signifies convolution, depthwise-separable convolutions, and dilated depthwise-separable convolutions, respectively. “none” means the zero operation. Note that differentiable NAS baselines (DARTS, SNAS, PC-DARTS) also utilize the first row of Table 3 as their search space.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":96,"cells":{"paper_id":{"kind":"string","value":"SP:d10957cc11891e1aad6ecac21a73d589bfac341d"},"summaries":{"kind":"list like","value":["This paper proposes a method called Temporal Abstract Latent Dynamics (TALD). TALD is built up on RSSM (Hafner et al. 2019) but with hierarchical dynamics. The experiments are conducted on moving MNIST, GQN 3D Mazes, and KTH. Results are qualitatively better than other methods in term of maintaining long-term consistent prediction. Quantitative comparison is reported only on KTH dataset (Figure 5). Written presentation is clear and easy to understand."],"string":"[\n \"This paper proposes a method called Temporal Abstract Latent Dynamics (TALD). TALD is built up on RSSM (Hafner et al. 2019) but with hierarchical dynamics. The experiments are conducted on moving MNIST, GQN 3D Mazes, and KTH. Results are qualitatively better than other methods in term of maintaining long-term consistent prediction. Quantitative comparison is reported only on KTH dataset (Figure 5). Written presentation is clear and easy to understand.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Deep learning has shown promise for accurately predicting high-dimensional video sequences. Existing video prediction models succeeded in generating sharp but often short video sequences. Toward improving long-term video prediction, we study hierarchical latent variable models with levels that process at different time scales. To gain insights into the representations of such models, we study the information stored at each level of the hierarchy via the KL divergence, predictive entropy, datasets of varying speed, and generative distributions. Our analysis confirms that faster changing details are generally captured by lower levels, while slower changing facts are remembered by higher levels. On synthetic datasets where common methods fail after 25 frames, we show that temporally abstract latent variable models can make accurate predictions for up to 200 frames."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Mohammad Babaeizadeh","Chelsea Finn","Dumitru Erhan","Roy H. Campbell","Sergey Levine"],"title":"Stochastic variational video","venue":"prediction. CoRR,","year":2017},{"authors":["Lars Buesing","Theophane Weber","Sébastien Racanière","S.M. Ali Eslami","Danilo Jimenez Rezende","David P. 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Reichert\",\n \"Fabio Viola\",\n \"Frederic Besse\",\n \"Karol Gregor\",\n \"Demis Hassabis\",\n \"Daan Wierstra\"\n ],\n \"title\": \"Learning and querying fast generative models for reinforcement learning\",\n \"venue\": \"CoRR, abs/1802.03006,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Lluís Castrejón\",\n \"Nicolas Ballas\",\n \"Aaron C. 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Zhang\",\n \"P. Isola\",\n \"A.A. Efros\",\n \"E. Shechtman\",\n \"O. Wang\"\n ],\n \"title\": \"The unreasonable effectiveness of deep features as a perceptual metric\",\n \"venue\": \"IEEE/CVF Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Shengjia Zhao\",\n \"Jiaming Song\",\n \"Stefano Ermon\"\n ],\n \"title\": \"Learning hierarchical features from generative models\",\n \"venue\": \"CoRR, abs/1702.08396,\",\n \"year\": 2017\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":null,"text":"1 INTRODUCTION\nDeep learning has enabled predicting video sequences from large datasets (Chiappa et al., 2017; Oh et al., 2015; Vondrick et al., 2016). For high-dimensional inputs such as video, there likely exists a more compact representation of the scene that facilitates long term prediction. Instead of learning dynamics in pixel space, latent dynamics models predict ahead in a more compact feature space (Doerr et al., 2018; Buesing et al., 2018; Karl et al., 2016; Hafner et al., 2019). This has the added benefit of increased computational efficiency and a lower memory footprint, allowing to predict thousands of sequences in parallel using a large batch size.\nA lot of work in deep learning has focused on spatial abstraction, following the advent of convolutional networks (LeCun et al., 1989), such as the Variational Ladder Autoencoder (Zhao et al., 2017) that learns a hierarchy of features in images using networks of different capacities, along with playing an important role in the realm of video\nprediction models (Castrejón et al., 2019). Recent sequential models have incorporated temporal abstraction for learning dependencies in temporally distant observations (Koutník et al., 2014; Chung et al., 2016). Kim et al. (2019) proposed Variational Temporal Abstraction (VTA), in which they explored one level of temporal abstraction above the latent states, the transition of which was modeled using a Bernoulli random variable. In this paper, we intend to work in a more controlled setup than VTA for a qualitative and quantitative analysis of temporally abstract latent variable models.\nIn this paper, we study the benefits of temporal abstraction using a hierarchical latent dynamics model, trained using a variational objective. Each level in the hierarchy of this model temporally abstracts the level below by an adjustable factor. This model can perform long-horizon video prediction of 200 frames, while predicting accurate low-level information for a 6 times longer duration than the baseline model. We study the information stored at different levels of the hierarchy via KL divergence, predictive entropy, datasets of varying speeds, and generative distributions. In our experiments we show that this amounts to object location and identities for the Moving MNIST dataset, and the wall or floor patterns for the GQN mazes dataset (Eslami et al., 2018), stored at different levels.\nOur key contributions are summarized as follows:\n• Temporal Abstract Latent Dynamics (TALD) We introduce a simple model with different clock speeds at every level to study the properties of variational hierarchical dynamics.\n• Accurate long-term predictions Our form of temporal abstraction substantially improves for how long the model can accurately predict video frames into the future.\n• Adaptation to sequence speed We demonstrate that our model automatically adapts the amount of information processed at each level to the speed of the video sequence.\n• Separation of information We visualize the content represented at each level of the hierarchy to find location information in lower levels and object identity in higher levels."},{"heading":"2 RELATED WORK","text":"Generative video models A variety of methods have successfully approached video prediction using large datasets (Chiappa et al., 2017; Oh et al., 2015; Vondrick et al., 2016; Babaeizadeh et al., 2017; Gemici et al., 2017; Ha & Schmidhuber, 2018). Denton & Fergus (2018) proposed a stochastic video generation model with a learned prior that transitions in time, and is conditioned on past observations. Lee et al. (2018) proposed to use an adversarial loss with a variational latent variable model to produce naturalistic images, while Kumar et al. (2019) used flow-based generative modeling to directly optimize the likelihood of a video generation model. Recently, Weissenborn et al. (2020) scaled autoregressive models for video prediction using a three-dimensional self-attention mechanism and showed competitive results on real-world video datasets. On similar lines, Xu et al. (2018) proposed to use an entirely CNN-based architecture for modeling dependencies between sequential inputs. Latent dynamics models Latent dynamics models have evolved from latent space models that had access to low-dimensional features (Deisenroth & Rasmussen, 2011; Higuera et al., 2018), to models that can build a compact representation of visual scenes and facilitate video prediction purely in the latent space (Doerr et al., 2018; Buesing et al., 2018; Karl et al., 2016; Franceschi et al., 2020). The Variational RNN (Chung et al., 2015) uses an auto-regressive state transition that takes inputs from observations, making it computationally expensive to be used as an imagination module. Hafner et al. (2019) proposed a latent dynamics model, which is a combination of deterministic and stochastic states, that enables the model to deterministically remember all previous states and filter that information to obtain a distribution over the current state. Hierarchical latent variables Learning per-frame hierarchical structures has proven to be helpful in generating videos on long-term horizon (Wichers et al., 2018). Zhao et al. (2017) proposed the Variational Ladder Autoencoder (VLAE) that uses networks of different capacities at different levels of the hierarchy, encouraging the model to store high-level image features at the top level, and simple features at the bottom. Other recently proposed hierarchical models use a purely bottom-up inference approach with no interaction between the inference and generative models (Kingma & Welling,\n2014; Rezende & Mohamed, 2015; Rezende et al., 2014). In contrast, Sønderby et al. (2016, LVAE) and Vahdat & Kautz (2020, NVAE) proposed to use a combination of bottom-up and top-down inference procedures, sharing parameters between the inference and generative distributions during the top-down pass. We incorporate this conditional structure in our model design as well.\nTemporal abstraction Identifying complex dependencies between temporally distant observations is a challenging task and has inspired a variety of fundamental work in recurrent models (Koutník et al., 2014; Chung et al., 2016). However, relatively few works have demonstrated modeling longterm dependencies using temporally abstract latent dynamics models (Wichers et al., 2018; Jaderberg et al., 2018). Recently, Kim et al. (2019) introduced Variational Temporal Abstraction (VTA) to learn temporally abstract latent spaces. They explored one level of temporal abstraction above the latent states, the transition of which was modeled using a Bernoulli random variable, that chose between ‘copy’ or ‘update’ steps. Inspired by this work, we aim to gain a deeper understanding of such temporally-abstract latent dynamics models. We perform our analysis on a model that is simplified to using fixed time scales for every level. Moreover, the lower level is a continuing chain in our model, whereas VTA resets transitions at a lower level when transitioning at a higher level."},{"heading":"3 TEMPORALLY ABSTRACT LATENT DYNAMICS","text":"Long video sequences contain both information that is local to a few frames as well as global information that is shared among many frames. Traditional video prediction models that predict ahead at the frame rate of the video can struggle to retain information long enough to learn such longterm dependencies. We introduce Temporally Abstract Latent Dynamics (TALD) to learn long-term correlations of videos. Our model predicts ahead on multiple time scales to learn dependencies at different temporal levels, as visualized in Figure 3. We build our work upon the recurrent state-space model (RSSM; Hafner et al., 2019), the details of which can be found in Appendix A.\nTALD consists of a hierarchy of recurrent latent variables, where each level transitions at a different clock speed. We slow down the transitions exponentially as we go up in the hierarchy, i.e. every level being slower than the level below by a factor of k. We denote a set of active timesteps for every level l ∈ [1, L] as those steps in time where the state transition generates a new latent state,\nActive timesteps: Tl . = {t ∈ [1, T ] | tmod kl−1 = 1}. (1)\nAt each level, we condition every window of k latent states on a single latent variable in the level above. This can also be thought of as a hierarchy of latent variables where each level has the same clock speed, but performs a state transition every kl−1 timesteps and copies the same state variable otherwise, so that ∀t /∈ Tl:\nInactive states: slt . = slmax\nτ {τ∈Tl|τ≤t}. (2)\nJoint distribution We can factorize the joint distribution of a sequence of observations and (active) latents at every level into two terms: (1) a decoder term conditioned on the latent states in the lowest\nlevel, and (2) state transitions at all levels conditioned on the latent state of the last active timestep at the current level and the level above,\np(x1:T , s 1:L 1:T )\n. = (∏T t=1 p(xt | s1t ) )(∏L l=1 ∏ t∈Tl p(s l t | slt−1, sl+1t ) ) . (3)\nInference For inference, TALD embeds observed frames using a CNN. A hierarchical recurrent network then summarizes the input embeddings, for which each (active) latent state at a level l receives embeddings from kl−1 observation frames (dashed lines in Figure 3). The latent state at the previous timestep at the current level, and the state belief at the level above also condition the posterior belief (solid lines in Figure 3). The input embeddings combined with this top-down and temporal context together condition the posterior belief qlt over the latent state.\nGeneration The prior transition plt is computed by conditioning over the latent state at the previous timestep at the current level, and the state belief at the level above (solid lines in Figure 3).\nDecoding Finally, the state beliefs at the bottom-most level are decoded using a transposed CNN to provide a training signal. To summarize, we utilize the following components in our model, ∀ l ∈ [1, L], t ∈ Tl, Encoder: elt = e(xt:t+kl−1−1)\nPosterior transition qlt: q(s l t | slt−1, sl+1t , elt)\nPrior transition plt: p(s l t | slt−1, sl+1t )\nDecoder: p(xt | s1t ).\n(4)\nTraining objective Since we cannot compute the likelihood of the training data under the model in closed form, we use the ELBO as our training objective. This training objective optimizes a reconstruction loss at the lowest level, and a KL regularizer at every level in the hierarchy summed across active timesteps,\nmax e,h,q,p\n∑T t=1 Eq1t [ln p(xt | s 1 t )]− ∑L l=1 ∑ t∈Tl KL[q l t ‖ plt]. (5)\nThe KL regularizer at each level limits the amount of information that filters through the encoder and stays in the posterior at that level. This encourages the model to utilize the state transitions and context from the level above as much as possible. Since the number of active timesteps decreases as we go higher in the hierarchy, the number of KL terms per level decreases as well. Hence it is easier for the model to push global information high up in the hierarchy and pay lesser KL penalty, instead of transitioning those bits with an identity transformation at a lower level.\nStochastic and Deterministic Path As illustrated in Figure 3 (right), we split the state slt into stochastic (zlt) and deterministic (h l t) parts (Hafner et al., 2019). The deterministic state is computed using the top-down and temporal context, which then conditions the stochastic state at that level.\nThe stochastic states follow a diagonal Gaussian, with mean and variance predicted by a neural network. We use a GRU (Cho et al., 2014) per level to update the deterministic state at every active timestep. All components in Equation 4 are trained jointly by optimizing Equation 5 using stochastic backpropagation with reparameterized sampling. Please refer to Appendix B for architectures details.\n4 EXPERIMENTS\nWe aim to evaluate temporally-abstract latent dynamics models at modeling long-term dependencies in video. Moreover, we aim to understand how they separate information into different levels of the hierarchy. To investigate these questions, we train TALD described in Section 3, the temporally-abstract VTA model (Kim et al., 2019), the RSSM model without temporal abstraction (Hafner et al., 2019), and the image-space video prediction model SVGLP (Denton & Fergus, 2018) on four datasets of varying complexity. We consider the well-\nestablished Moving MINST dataset (Srivastava et al., 2015), the KTH Action dataset (Schuldt et al.,\n2004), the GQN mazes dataset (Eslami et al., 2018), and the MineRL Navigate dataset (Guss et al., 2019). We evaluate open-loop video predictions on these datasets using four quantitative metrics: Structural Similarity index (SSIM; higher is better), Peak Signal-to-Noise Ratio (PSNR; higher is better), Learned Perceptual Image Patch Similarity (LPIPS; lower is better) (Zhang et al., 2018), and Frechet Video Distance (FVD; lower is better) (Unterthiner et al., 2018). In Section 4.5, we investigate how the amount of information stored at different levels of a temporal hierarchy adapts to changes in sequence speed. In Section 4.6, we visualize the information stored at different levels by resetting individual levels of the hierarchy.\nWe trained all our models using sequences of length 100. We used convolutional frame encoders and decoders, with architectures very similar to the DCGAN (Radford et al., 2016) discriminator and generator, respectively. Our implementations made use of TensorFlow Probability (Dillon et al., 2017) and CuDNN, and used the Adam optimizer (Kingma & Ba, 2014) for training. The training time for a 3-level TALD model with temporal abstraction 6 amounted to around 24 hours for 100 epochs on a single NVIDIA TITAN Xp GPU. Refer to Appendix C for hyperparameters and experimental setup."},{"heading":"4.1 MOVING MNIST DATASET","text":"The Moving MNIST dataset consists of two digits moving in a square with velocities sampled in the range of 2 to 6 pixels per frame. We trained different versions of TALD with 3 levels in the hierarchy and temporal abstraction factors 1, 2, 4, 6 and 8, all of which use the same number of model parameters. We compare samples of long-horizon open-loop video predictions of 900 frames with RSSM and SVG-LP in Figure 4. All samples were conditioned using posterior beliefs inferred after observing 36 context frames. Please refer to Appendix C for more details and discussion.\nWe observe that SVG typically forgets object identity within 50 timesteps, while TALD with abstraction factor 6 maintains digit identity over 900 timesteps. RSSM clearly outperforms SVG, however starts to forget object identities after 250 time frames. TALD with abstraction factor 1 (i.e. no temporal abstraction) also starts to forget object identity after around 600 frames. With regards to the object positions, TALD with abstraction factor 6 predicts accurate digit positions until around 90 steps, and predicts a plausible sequence thereafter. RSSM and TALD without temporal abstraction predict the correct location of digits for at least as long as TALD with temporal abstraction. However, SVG starts to lose track of positions much sooner. We also note that our predictions are a bit blurry compared to those generated by SVG. Please refer to Appendix D for more experimental results.\nWe report the KL divergence value per level (summed over active time steps) for our TALD models in Table 1. Each value was obtained after training over sequences of length 100, for 200 epochs. The 2 and 3-level models were trained with a temporal abstraction factor of 6, and the 4-level model with a factor of 4 (to fit into memory). Figure 1 compares the Structural Similarity index (SSIM) for different versions of TALD with RSSM and SVG-LP. We note that SSIM decreases at a lower rate for models with higher temporal abstractions. As a baseline, we compute SSIM between ground truths and random sequences from the training set. It is interesting to note that quality of video predictions from TALD stay better than random for a 6 times longer duration than SVG."},{"heading":"4.2 KTH ACTION DATASET","text":"We trained a 3-level TALD model with temporal abstraction factor 6 for the KTH Action dataset. In Figure 6, we report the SSIM, PSNR, and LPIPS, of TALD compared to SVGLP, RSSM, and VTA. We also illustrate open-loop video predictions in Figure 5, conditioned using 36 context frames. While TALD predicts plausible frames for 50 timesteps, we observe jumpy transitions with VTA, probably because of breaks in the transition chain at the lower level. We also observe that SVG predicts accurately for 18 frames, while switching to a different task thereafter. We also note that SVG uses the DCGAN architecture for MNIST and the much larger VGG for KTH, whereas TALD works well even with the smaller DCGAN encoder/decoder. Refer to Appendix D for more results."},{"heading":"4.3 GQN 3D MAZES DATASET","text":"We trained a 2-level TALD model with temporal abstraction factor 6, and compared it with RSSM and VTA, on the GQN mazes dataset. Figure 7 shows open-loop video prediction samples, conditioned using 36 context frames. We observe that while our model can maintain global information of wall and floor colors for 200 frames, RSSM starts to forget the same after ∼50 frames. Even though the open-loop predictions from TALD differ from ground truth in terms of camera viewpoints, the model does not forget the wall and floor patterns. We\ncompare quantitative metrics on this dataset in Figure 12d. Please refer to Appendix D for detailed experimental results."},{"heading":"4.4 MINERL NAVIGATE DATASET","text":"We trained a 3-level TALD model with temporal abstraction factor 6, and compared it with RSSM, SVG-LP and VTA, on a Minecraft dataset (MineRL Navigate) (Guss et al., 2019). This dataset features videos in a variety of world environments with complex moving backgrounds. We show long-horizon open-loop video predictions upto 420 frames (conditioned on 36 context frames) on this dataset in Figure 2. We also compare quantitative metrics on this dataset in Figure 12c in Appendix D. Please refer to Figure 15 in Appendix D for more open-loop video prediction examples."},{"heading":"4.5 ADAPTING TO SEQUENCE SPEED","text":"In order to understand how our model adapts to changing temporal correlations in the dataset, we trained our model with slower and faster versions of moving MNIST, with speeds varied by factors of 3. For this experiment, our model consisted of 3 levels in the hierarchy, with each level temporally abstracting the level below by a factor of 6.\nFigure 8 shows the KL divergence summed across the active timesteps at every level in the hierarchy. We observe that there is a correlation between the KL divergence at every level and the speed at which the digits move. There is more information at level 1 when the digits move faster, and consequently lesser information at the levels above it. Also, even though the KL divergence at level 3 is small, it still follows the same trend as the other two levels. It is also important to note that the KL divergence between the prior and the posterior is only an upper bound on the information stored by the encoder in a posterior belief state."},{"heading":"4.6 RESETTING INDIVIDUAL LEVELS","text":"We visualize the information stored at a certain level by replacing the posterior belief at one level with the prior belief, i.e. all but one level receive observations (Zhao et al., 2017). Conditioned on those posterior beliefs, we sample open-loop video predictions using the trained prior model, which should show variations in the attributes learned at that level. We expect our model to store global information high up in the hierarchy, allowing the model to perform fewer transitions over that information, making it easier to pay less cost in the form of KL divergence during training.\nFigure 9 shows video predictions with different levels reset to the prior for the GQN mazes dataset. With a 3-level model and temporal abstraction factor 2, we observe that when level 3 was not fed with observations, the conditioned open-loop predictions started with the correct viewpoint, but differed w.r.t. wall and floor colors. This suggests that those characteristics were stored in the higher level of the hierarchy.\nFigure 11 shows video predictions with different levels reset to the prior for the Moving MNIST dataset. The 3-level model with temporal abstraction factor 6 obtained a separation of information in the bottom two levels, while the posterior at the third level nearly collapsed to the prior. When level 1 was not fed with observations, we observe that the conditioned open-loop predictions maintained the same digit identity, but showed variations w.r.t. digit positions. On the other hand, when level 2\nwas not fed with observations, the samples maintained the same digit positions but produced digits with different identities in every sample. This suggests that lower level stored digit positions, high frequency details which changed frequently in time, while the level above it stored the digit identities, i.e. long-term information. We also observed that, when resetting level 2 to the prior, the digits start to differ in position sooner (∼60 frames) than when all levels receive observations (∼80 frames). This suggests that this level does have some information about digit positions, and that there is still mixing of information between different levels of the hierarchy.\nPredictive entropy To corroborate our understanding of the latent representations, we observe the entropy of the prior distribution as it varies over time during open-loop video generation in Figure 10. With GQN mazes, the entropy at the top level remained relatively constant as the model remained certain about the high-level details. With Moving MNIST, the top two levels showed a relatively constant entropy, with level 2 storing the digit identities and level 3 suffering from posterior collapse. Please refer to Appendix D.6 for a more detailed analysis."},{"heading":"5 DISCUSSION","text":"In this work, we presented a hierarchical latent dynamics model with temporal abstraction (TALD), where each level in the hierarchy temporally abstracted the level below by an adjustable factor.\n• We evaluated long-horizon open-loop predictions using our model, and observed that TALD was able to predict far into the future while accurately maintaining global information.\n• We also observed that the amount of information at the higher levels decreased as the speed of the sequence was increased.\n• We analyzed the separation of information at different levels of the hierarchy, by generating open-loop video predictions with different levels reset to the prior. With Moving MNIST, the bottom level in the hierarchy stored high frequency details (digit positions) and the level above stored more global information (digit identities). With the GQN mazes dataset, TALD stored wall and floor patterns at the top level in the hierarchy.\nTemporally abstract models are an intuitive approach to obtaining high-level representations of complex datasets and environments. We hope that our work can refuel interest in temporally abstract latent dynamics models and motivate the development of effective deep learning systems for highdimensional data more generally."},{"heading":"A BACKGROUND","text":"We build our work upon the recurrent state-space model (RSSM; Hafner et al., 2019) that has been shown to successfully learn environment dynamics from raw pixels in reinforcement learning. This model acts as an important baseline for our evaluation. RSSM explains the video sequence x1:T using a latent sequence of compact Markovian states s1:T . Importantly, the model is autoregressive in latent space but not in image space, allowing us to predict into the future without generating images along the way,\np(x1:T , s1:T ) . = ∏T t=1 p(xt | st)p(st | st−1). (6)\nGiven a training sequence, RSSM first individually embeds the frames using a CNN. A recurrent network with deterministic and stochastic components then summarizes the image embeddings. Hafner et al. (2019) argued that the stochastic component helps with modeling multiple futures, while the deterministic state helps remember information over many timesteps so that it is not erased by noise. Finally, the recurrent states are decoded using a transposed CNN to provide a training signal.\nEncoder: et = enc(xt) Posterior transition qt: q(st | st−1, et) Prior transition pt: p(st | st−1) Decoder: p(xt | st).\n(7)\nThe posterior and prior transition models share the same recurrent model. The difference is that the posterior incorporates images, while the prior tries to predict ahead without knowing the corresponding images. This lets us to predict ahead purely in latent space at inference time. As typical with deep latent variable models, we cannot compute the likelihood of the training data under the model in closed form. Instead, we use the evidence lower bound (ELBO) as training objective. The ELBO encourages to accurately reconstruct each image from its corresponding latent state, while regularizing the latent state distributions to stay close to the prior dynamics,\nmax q,p\n∑T t=1 Eqt [ln p(xt | st)]− ∑T t=1 KL[qt ‖ pt]. (8)\nThe KL regularizer limits the amount of information that the posterior transition incorporates into the latent state at each time step, thus encouraging the model to mostly rely on information from past time steps and only extract information from each image that cannot be predicted from the preceding images already. All components jointly optimize Equation 8 using stochastic backpropagation with reparameterized sampling (Kingma & Welling, 2013; Rezende et al., 2014)."},{"heading":"B MODEL ARCHITECTURES","text":"We use convolutional frame encoders and decoders, with architectures very similar to the DCGAN (Radford et al., 2016) discriminator and generator, respectively. To obtain the input embeddings elt at a particular level, kl−1 input embeddings are pre-processed using a feed-forward network and then added up to obtain a single embedding. We also want to emphasize that we do not use any skip connections between the encoder and decoder bypassing the latent states as we believe this would motivate the model to make better use of the temporal hierarchy."},{"heading":"C HYPER PARAMETERS AND EXPERIMENTAL SETUP","text":"We kept the output dimensionality of the encoder at each level of TALD as |elt| = 1024, that of the stochastic states as |plt| = |qlt| = 20, and that of the deterministic states as |hlt| = 200. All hidden layers inside the cell, both for prior and posterior transition, were set to 200. We trained our models using the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 5× 10−4 and = 10−4 for the Moving MNIST dataset. With the KTH Action and GQN mazes datasets, we lowered the learning rate to 3× 10−4. For all three datasets, batch size was set to be 100 sequences, each sequence being of length 100.\nThe decision of using 36 context frames for conditioning all open-loop video predictions was taken considering the minimum number of observation frames required to transition at least once in the highest level of the hierarchy. With 3 levels in the hierarchy and a temporal abstraction factor of 6, each latent state at the highest level corresponds to 36 images in the sequence, and thus its encoder network expects 36 images as input."},{"heading":"D ADDITIONAL EXPERIMENTS","text":"D.1 QUANTITATIVE METRICS FOR LONG-HORIZON VIDEO PREDICTION\nD.2 LONG-HORIZON PREDICTION OF MOVING MNIST\nD.3 LONG-HORIZON PREDICTION OF 3D MAZES\nD.4 LONG-HORIZON PREDICTION OF MINERL NAVIGATE\nD.5 UNDERSTANDING THE LATENT REPRESENTATIONS - 3D MAZES\nD.6 ANALYSING THE ENTROPY CURVES Figure 10b shows the entropy of the prior distribution at all levels of the 3-level TALD model with temporal abstraction factor 2, as it varies over time, with the Moving MNIST dataset. We observe that the entropy at the lowest level varies between 9 to 13 nats, whereas for the higher levels in the hierarchy, the entropy remained relatively constant. In Figure 11 we presented some visualizations that supported the claim that the higher level in the hierarchy stores global information (digit identities) which, in this case, does not change over time, and hence the model should remain rather certain about as time progresses. Figure 10b corroborates that claim showing that the model’s uncertainty at higher levels does not dither significantly over time as compared to the lowest level in the hierarchy.\nFigure 10a shows the entropy of the prior distribution at all levels of the 3-level TALD model with temporal abstraction factor 2, as it varies over time, with the GQN mazes dataset. While the bottom two levels showed a similar variance in entropy with time, the entropy of the prior distribution at the top level did not change significantly. We showed in Figure 9 how the top level in the hierarchy stores wall and floor colors, which the model should remain certain about over time. In support of this claim, here we also see that the entropy of the prior distribution at the topmost level does not change significantly over time."}],"string":"[\n {\n \"heading\": null,\n \"text\": \"1 INTRODUCTION\\nDeep learning has enabled predicting video sequences from large datasets (Chiappa et al., 2017; Oh et al., 2015; Vondrick et al., 2016). For high-dimensional inputs such as video, there likely exists a more compact representation of the scene that facilitates long term prediction. Instead of learning dynamics in pixel space, latent dynamics models predict ahead in a more compact feature space (Doerr et al., 2018; Buesing et al., 2018; Karl et al., 2016; Hafner et al., 2019). This has the added benefit of increased computational efficiency and a lower memory footprint, allowing to predict thousands of sequences in parallel using a large batch size.\\nA lot of work in deep learning has focused on spatial abstraction, following the advent of convolutional networks (LeCun et al., 1989), such as the Variational Ladder Autoencoder (Zhao et al., 2017) that learns a hierarchy of features in images using networks of different capacities, along with playing an important role in the realm of video\\nprediction models (Castrejón et al., 2019). Recent sequential models have incorporated temporal abstraction for learning dependencies in temporally distant observations (Koutník et al., 2014; Chung et al., 2016). Kim et al. (2019) proposed Variational Temporal Abstraction (VTA), in which they explored one level of temporal abstraction above the latent states, the transition of which was modeled using a Bernoulli random variable. In this paper, we intend to work in a more controlled setup than VTA for a qualitative and quantitative analysis of temporally abstract latent variable models.\\nIn this paper, we study the benefits of temporal abstraction using a hierarchical latent dynamics model, trained using a variational objective. Each level in the hierarchy of this model temporally abstracts the level below by an adjustable factor. This model can perform long-horizon video prediction of 200 frames, while predicting accurate low-level information for a 6 times longer duration than the baseline model. We study the information stored at different levels of the hierarchy via KL divergence, predictive entropy, datasets of varying speeds, and generative distributions. In our experiments we show that this amounts to object location and identities for the Moving MNIST dataset, and the wall or floor patterns for the GQN mazes dataset (Eslami et al., 2018), stored at different levels.\\nOur key contributions are summarized as follows:\\n• Temporal Abstract Latent Dynamics (TALD) We introduce a simple model with different clock speeds at every level to study the properties of variational hierarchical dynamics.\\n• Accurate long-term predictions Our form of temporal abstraction substantially improves for how long the model can accurately predict video frames into the future.\\n• Adaptation to sequence speed We demonstrate that our model automatically adapts the amount of information processed at each level to the speed of the video sequence.\\n• Separation of information We visualize the content represented at each level of the hierarchy to find location information in lower levels and object identity in higher levels.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Generative video models A variety of methods have successfully approached video prediction using large datasets (Chiappa et al., 2017; Oh et al., 2015; Vondrick et al., 2016; Babaeizadeh et al., 2017; Gemici et al., 2017; Ha & Schmidhuber, 2018). Denton & Fergus (2018) proposed a stochastic video generation model with a learned prior that transitions in time, and is conditioned on past observations. Lee et al. (2018) proposed to use an adversarial loss with a variational latent variable model to produce naturalistic images, while Kumar et al. (2019) used flow-based generative modeling to directly optimize the likelihood of a video generation model. Recently, Weissenborn et al. (2020) scaled autoregressive models for video prediction using a three-dimensional self-attention mechanism and showed competitive results on real-world video datasets. On similar lines, Xu et al. (2018) proposed to use an entirely CNN-based architecture for modeling dependencies between sequential inputs. Latent dynamics models Latent dynamics models have evolved from latent space models that had access to low-dimensional features (Deisenroth & Rasmussen, 2011; Higuera et al., 2018), to models that can build a compact representation of visual scenes and facilitate video prediction purely in the latent space (Doerr et al., 2018; Buesing et al., 2018; Karl et al., 2016; Franceschi et al., 2020). The Variational RNN (Chung et al., 2015) uses an auto-regressive state transition that takes inputs from observations, making it computationally expensive to be used as an imagination module. Hafner et al. (2019) proposed a latent dynamics model, which is a combination of deterministic and stochastic states, that enables the model to deterministically remember all previous states and filter that information to obtain a distribution over the current state. Hierarchical latent variables Learning per-frame hierarchical structures has proven to be helpful in generating videos on long-term horizon (Wichers et al., 2018). Zhao et al. (2017) proposed the Variational Ladder Autoencoder (VLAE) that uses networks of different capacities at different levels of the hierarchy, encouraging the model to store high-level image features at the top level, and simple features at the bottom. Other recently proposed hierarchical models use a purely bottom-up inference approach with no interaction between the inference and generative models (Kingma & Welling,\\n2014; Rezende & Mohamed, 2015; Rezende et al., 2014). In contrast, Sønderby et al. (2016, LVAE) and Vahdat & Kautz (2020, NVAE) proposed to use a combination of bottom-up and top-down inference procedures, sharing parameters between the inference and generative distributions during the top-down pass. We incorporate this conditional structure in our model design as well.\\nTemporal abstraction Identifying complex dependencies between temporally distant observations is a challenging task and has inspired a variety of fundamental work in recurrent models (Koutník et al., 2014; Chung et al., 2016). However, relatively few works have demonstrated modeling longterm dependencies using temporally abstract latent dynamics models (Wichers et al., 2018; Jaderberg et al., 2018). Recently, Kim et al. (2019) introduced Variational Temporal Abstraction (VTA) to learn temporally abstract latent spaces. They explored one level of temporal abstraction above the latent states, the transition of which was modeled using a Bernoulli random variable, that chose between ‘copy’ or ‘update’ steps. Inspired by this work, we aim to gain a deeper understanding of such temporally-abstract latent dynamics models. We perform our analysis on a model that is simplified to using fixed time scales for every level. Moreover, the lower level is a continuing chain in our model, whereas VTA resets transitions at a lower level when transitioning at a higher level.\"\n },\n {\n \"heading\": \"3 TEMPORALLY ABSTRACT LATENT DYNAMICS\",\n \"text\": \"Long video sequences contain both information that is local to a few frames as well as global information that is shared among many frames. Traditional video prediction models that predict ahead at the frame rate of the video can struggle to retain information long enough to learn such longterm dependencies. We introduce Temporally Abstract Latent Dynamics (TALD) to learn long-term correlations of videos. Our model predicts ahead on multiple time scales to learn dependencies at different temporal levels, as visualized in Figure 3. We build our work upon the recurrent state-space model (RSSM; Hafner et al., 2019), the details of which can be found in Appendix A.\\nTALD consists of a hierarchy of recurrent latent variables, where each level transitions at a different clock speed. We slow down the transitions exponentially as we go up in the hierarchy, i.e. every level being slower than the level below by a factor of k. We denote a set of active timesteps for every level l ∈ [1, L] as those steps in time where the state transition generates a new latent state,\\nActive timesteps: Tl . = {t ∈ [1, T ] | tmod kl−1 = 1}. (1)\\nAt each level, we condition every window of k latent states on a single latent variable in the level above. This can also be thought of as a hierarchy of latent variables where each level has the same clock speed, but performs a state transition every kl−1 timesteps and copies the same state variable otherwise, so that ∀t /∈ Tl:\\nInactive states: slt . = slmax\\nτ {τ∈Tl|τ≤t}. (2)\\nJoint distribution We can factorize the joint distribution of a sequence of observations and (active) latents at every level into two terms: (1) a decoder term conditioned on the latent states in the lowest\\nlevel, and (2) state transitions at all levels conditioned on the latent state of the last active timestep at the current level and the level above,\\np(x1:T , s 1:L 1:T )\\n. = (∏T t=1 p(xt | s1t ) )(∏L l=1 ∏ t∈Tl p(s l t | slt−1, sl+1t ) ) . (3)\\nInference For inference, TALD embeds observed frames using a CNN. A hierarchical recurrent network then summarizes the input embeddings, for which each (active) latent state at a level l receives embeddings from kl−1 observation frames (dashed lines in Figure 3). The latent state at the previous timestep at the current level, and the state belief at the level above also condition the posterior belief (solid lines in Figure 3). The input embeddings combined with this top-down and temporal context together condition the posterior belief qlt over the latent state.\\nGeneration The prior transition plt is computed by conditioning over the latent state at the previous timestep at the current level, and the state belief at the level above (solid lines in Figure 3).\\nDecoding Finally, the state beliefs at the bottom-most level are decoded using a transposed CNN to provide a training signal. To summarize, we utilize the following components in our model, ∀ l ∈ [1, L], t ∈ Tl, Encoder: elt = e(xt:t+kl−1−1)\\nPosterior transition qlt: q(s l t | slt−1, sl+1t , elt)\\nPrior transition plt: p(s l t | slt−1, sl+1t )\\nDecoder: p(xt | s1t ).\\n(4)\\nTraining objective Since we cannot compute the likelihood of the training data under the model in closed form, we use the ELBO as our training objective. This training objective optimizes a reconstruction loss at the lowest level, and a KL regularizer at every level in the hierarchy summed across active timesteps,\\nmax e,h,q,p\\n∑T t=1 Eq1t [ln p(xt | s 1 t )]− ∑L l=1 ∑ t∈Tl KL[q l t ‖ plt]. (5)\\nThe KL regularizer at each level limits the amount of information that filters through the encoder and stays in the posterior at that level. This encourages the model to utilize the state transitions and context from the level above as much as possible. Since the number of active timesteps decreases as we go higher in the hierarchy, the number of KL terms per level decreases as well. Hence it is easier for the model to push global information high up in the hierarchy and pay lesser KL penalty, instead of transitioning those bits with an identity transformation at a lower level.\\nStochastic and Deterministic Path As illustrated in Figure 3 (right), we split the state slt into stochastic (zlt) and deterministic (h l t) parts (Hafner et al., 2019). The deterministic state is computed using the top-down and temporal context, which then conditions the stochastic state at that level.\\nThe stochastic states follow a diagonal Gaussian, with mean and variance predicted by a neural network. We use a GRU (Cho et al., 2014) per level to update the deterministic state at every active timestep. All components in Equation 4 are trained jointly by optimizing Equation 5 using stochastic backpropagation with reparameterized sampling. Please refer to Appendix B for architectures details.\\n4 EXPERIMENTS\\nWe aim to evaluate temporally-abstract latent dynamics models at modeling long-term dependencies in video. Moreover, we aim to understand how they separate information into different levels of the hierarchy. To investigate these questions, we train TALD described in Section 3, the temporally-abstract VTA model (Kim et al., 2019), the RSSM model without temporal abstraction (Hafner et al., 2019), and the image-space video prediction model SVGLP (Denton & Fergus, 2018) on four datasets of varying complexity. We consider the well-\\nestablished Moving MINST dataset (Srivastava et al., 2015), the KTH Action dataset (Schuldt et al.,\\n2004), the GQN mazes dataset (Eslami et al., 2018), and the MineRL Navigate dataset (Guss et al., 2019). We evaluate open-loop video predictions on these datasets using four quantitative metrics: Structural Similarity index (SSIM; higher is better), Peak Signal-to-Noise Ratio (PSNR; higher is better), Learned Perceptual Image Patch Similarity (LPIPS; lower is better) (Zhang et al., 2018), and Frechet Video Distance (FVD; lower is better) (Unterthiner et al., 2018). In Section 4.5, we investigate how the amount of information stored at different levels of a temporal hierarchy adapts to changes in sequence speed. In Section 4.6, we visualize the information stored at different levels by resetting individual levels of the hierarchy.\\nWe trained all our models using sequences of length 100. We used convolutional frame encoders and decoders, with architectures very similar to the DCGAN (Radford et al., 2016) discriminator and generator, respectively. Our implementations made use of TensorFlow Probability (Dillon et al., 2017) and CuDNN, and used the Adam optimizer (Kingma & Ba, 2014) for training. The training time for a 3-level TALD model with temporal abstraction 6 amounted to around 24 hours for 100 epochs on a single NVIDIA TITAN Xp GPU. Refer to Appendix C for hyperparameters and experimental setup.\"\n },\n {\n \"heading\": \"4.1 MOVING MNIST DATASET\",\n \"text\": \"The Moving MNIST dataset consists of two digits moving in a square with velocities sampled in the range of 2 to 6 pixels per frame. We trained different versions of TALD with 3 levels in the hierarchy and temporal abstraction factors 1, 2, 4, 6 and 8, all of which use the same number of model parameters. We compare samples of long-horizon open-loop video predictions of 900 frames with RSSM and SVG-LP in Figure 4. All samples were conditioned using posterior beliefs inferred after observing 36 context frames. Please refer to Appendix C for more details and discussion.\\nWe observe that SVG typically forgets object identity within 50 timesteps, while TALD with abstraction factor 6 maintains digit identity over 900 timesteps. RSSM clearly outperforms SVG, however starts to forget object identities after 250 time frames. TALD with abstraction factor 1 (i.e. no temporal abstraction) also starts to forget object identity after around 600 frames. With regards to the object positions, TALD with abstraction factor 6 predicts accurate digit positions until around 90 steps, and predicts a plausible sequence thereafter. RSSM and TALD without temporal abstraction predict the correct location of digits for at least as long as TALD with temporal abstraction. However, SVG starts to lose track of positions much sooner. We also note that our predictions are a bit blurry compared to those generated by SVG. Please refer to Appendix D for more experimental results.\\nWe report the KL divergence value per level (summed over active time steps) for our TALD models in Table 1. Each value was obtained after training over sequences of length 100, for 200 epochs. The 2 and 3-level models were trained with a temporal abstraction factor of 6, and the 4-level model with a factor of 4 (to fit into memory). Figure 1 compares the Structural Similarity index (SSIM) for different versions of TALD with RSSM and SVG-LP. We note that SSIM decreases at a lower rate for models with higher temporal abstractions. As a baseline, we compute SSIM between ground truths and random sequences from the training set. It is interesting to note that quality of video predictions from TALD stay better than random for a 6 times longer duration than SVG.\"\n },\n {\n \"heading\": \"4.2 KTH ACTION DATASET\",\n \"text\": \"We trained a 3-level TALD model with temporal abstraction factor 6 for the KTH Action dataset. In Figure 6, we report the SSIM, PSNR, and LPIPS, of TALD compared to SVGLP, RSSM, and VTA. We also illustrate open-loop video predictions in Figure 5, conditioned using 36 context frames. While TALD predicts plausible frames for 50 timesteps, we observe jumpy transitions with VTA, probably because of breaks in the transition chain at the lower level. We also observe that SVG predicts accurately for 18 frames, while switching to a different task thereafter. We also note that SVG uses the DCGAN architecture for MNIST and the much larger VGG for KTH, whereas TALD works well even with the smaller DCGAN encoder/decoder. Refer to Appendix D for more results.\"\n },\n {\n \"heading\": \"4.3 GQN 3D MAZES DATASET\",\n \"text\": \"We trained a 2-level TALD model with temporal abstraction factor 6, and compared it with RSSM and VTA, on the GQN mazes dataset. Figure 7 shows open-loop video prediction samples, conditioned using 36 context frames. We observe that while our model can maintain global information of wall and floor colors for 200 frames, RSSM starts to forget the same after ∼50 frames. Even though the open-loop predictions from TALD differ from ground truth in terms of camera viewpoints, the model does not forget the wall and floor patterns. We\\ncompare quantitative metrics on this dataset in Figure 12d. Please refer to Appendix D for detailed experimental results.\"\n },\n {\n \"heading\": \"4.4 MINERL NAVIGATE DATASET\",\n \"text\": \"We trained a 3-level TALD model with temporal abstraction factor 6, and compared it with RSSM, SVG-LP and VTA, on a Minecraft dataset (MineRL Navigate) (Guss et al., 2019). This dataset features videos in a variety of world environments with complex moving backgrounds. We show long-horizon open-loop video predictions upto 420 frames (conditioned on 36 context frames) on this dataset in Figure 2. We also compare quantitative metrics on this dataset in Figure 12c in Appendix D. Please refer to Figure 15 in Appendix D for more open-loop video prediction examples.\"\n },\n {\n \"heading\": \"4.5 ADAPTING TO SEQUENCE SPEED\",\n \"text\": \"In order to understand how our model adapts to changing temporal correlations in the dataset, we trained our model with slower and faster versions of moving MNIST, with speeds varied by factors of 3. For this experiment, our model consisted of 3 levels in the hierarchy, with each level temporally abstracting the level below by a factor of 6.\\nFigure 8 shows the KL divergence summed across the active timesteps at every level in the hierarchy. We observe that there is a correlation between the KL divergence at every level and the speed at which the digits move. There is more information at level 1 when the digits move faster, and consequently lesser information at the levels above it. Also, even though the KL divergence at level 3 is small, it still follows the same trend as the other two levels. It is also important to note that the KL divergence between the prior and the posterior is only an upper bound on the information stored by the encoder in a posterior belief state.\"\n },\n {\n \"heading\": \"4.6 RESETTING INDIVIDUAL LEVELS\",\n \"text\": \"We visualize the information stored at a certain level by replacing the posterior belief at one level with the prior belief, i.e. all but one level receive observations (Zhao et al., 2017). Conditioned on those posterior beliefs, we sample open-loop video predictions using the trained prior model, which should show variations in the attributes learned at that level. We expect our model to store global information high up in the hierarchy, allowing the model to perform fewer transitions over that information, making it easier to pay less cost in the form of KL divergence during training.\\nFigure 9 shows video predictions with different levels reset to the prior for the GQN mazes dataset. With a 3-level model and temporal abstraction factor 2, we observe that when level 3 was not fed with observations, the conditioned open-loop predictions started with the correct viewpoint, but differed w.r.t. wall and floor colors. This suggests that those characteristics were stored in the higher level of the hierarchy.\\nFigure 11 shows video predictions with different levels reset to the prior for the Moving MNIST dataset. The 3-level model with temporal abstraction factor 6 obtained a separation of information in the bottom two levels, while the posterior at the third level nearly collapsed to the prior. When level 1 was not fed with observations, we observe that the conditioned open-loop predictions maintained the same digit identity, but showed variations w.r.t. digit positions. On the other hand, when level 2\\nwas not fed with observations, the samples maintained the same digit positions but produced digits with different identities in every sample. This suggests that lower level stored digit positions, high frequency details which changed frequently in time, while the level above it stored the digit identities, i.e. long-term information. We also observed that, when resetting level 2 to the prior, the digits start to differ in position sooner (∼60 frames) than when all levels receive observations (∼80 frames). This suggests that this level does have some information about digit positions, and that there is still mixing of information between different levels of the hierarchy.\\nPredictive entropy To corroborate our understanding of the latent representations, we observe the entropy of the prior distribution as it varies over time during open-loop video generation in Figure 10. With GQN mazes, the entropy at the top level remained relatively constant as the model remained certain about the high-level details. With Moving MNIST, the top two levels showed a relatively constant entropy, with level 2 storing the digit identities and level 3 suffering from posterior collapse. Please refer to Appendix D.6 for a more detailed analysis.\"\n },\n {\n \"heading\": \"5 DISCUSSION\",\n \"text\": \"In this work, we presented a hierarchical latent dynamics model with temporal abstraction (TALD), where each level in the hierarchy temporally abstracted the level below by an adjustable factor.\\n• We evaluated long-horizon open-loop predictions using our model, and observed that TALD was able to predict far into the future while accurately maintaining global information.\\n• We also observed that the amount of information at the higher levels decreased as the speed of the sequence was increased.\\n• We analyzed the separation of information at different levels of the hierarchy, by generating open-loop video predictions with different levels reset to the prior. With Moving MNIST, the bottom level in the hierarchy stored high frequency details (digit positions) and the level above stored more global information (digit identities). With the GQN mazes dataset, TALD stored wall and floor patterns at the top level in the hierarchy.\\nTemporally abstract models are an intuitive approach to obtaining high-level representations of complex datasets and environments. We hope that our work can refuel interest in temporally abstract latent dynamics models and motivate the development of effective deep learning systems for highdimensional data more generally.\"\n },\n {\n \"heading\": \"A BACKGROUND\",\n \"text\": \"We build our work upon the recurrent state-space model (RSSM; Hafner et al., 2019) that has been shown to successfully learn environment dynamics from raw pixels in reinforcement learning. This model acts as an important baseline for our evaluation. RSSM explains the video sequence x1:T using a latent sequence of compact Markovian states s1:T . Importantly, the model is autoregressive in latent space but not in image space, allowing us to predict into the future without generating images along the way,\\np(x1:T , s1:T ) . = ∏T t=1 p(xt | st)p(st | st−1). (6)\\nGiven a training sequence, RSSM first individually embeds the frames using a CNN. A recurrent network with deterministic and stochastic components then summarizes the image embeddings. Hafner et al. (2019) argued that the stochastic component helps with modeling multiple futures, while the deterministic state helps remember information over many timesteps so that it is not erased by noise. Finally, the recurrent states are decoded using a transposed CNN to provide a training signal.\\nEncoder: et = enc(xt) Posterior transition qt: q(st | st−1, et) Prior transition pt: p(st | st−1) Decoder: p(xt | st).\\n(7)\\nThe posterior and prior transition models share the same recurrent model. The difference is that the posterior incorporates images, while the prior tries to predict ahead without knowing the corresponding images. This lets us to predict ahead purely in latent space at inference time. As typical with deep latent variable models, we cannot compute the likelihood of the training data under the model in closed form. Instead, we use the evidence lower bound (ELBO) as training objective. The ELBO encourages to accurately reconstruct each image from its corresponding latent state, while regularizing the latent state distributions to stay close to the prior dynamics,\\nmax q,p\\n∑T t=1 Eqt [ln p(xt | st)]− ∑T t=1 KL[qt ‖ pt]. (8)\\nThe KL regularizer limits the amount of information that the posterior transition incorporates into the latent state at each time step, thus encouraging the model to mostly rely on information from past time steps and only extract information from each image that cannot be predicted from the preceding images already. All components jointly optimize Equation 8 using stochastic backpropagation with reparameterized sampling (Kingma & Welling, 2013; Rezende et al., 2014).\"\n },\n {\n \"heading\": \"B MODEL ARCHITECTURES\",\n \"text\": \"We use convolutional frame encoders and decoders, with architectures very similar to the DCGAN (Radford et al., 2016) discriminator and generator, respectively. To obtain the input embeddings elt at a particular level, kl−1 input embeddings are pre-processed using a feed-forward network and then added up to obtain a single embedding. We also want to emphasize that we do not use any skip connections between the encoder and decoder bypassing the latent states as we believe this would motivate the model to make better use of the temporal hierarchy.\"\n },\n {\n \"heading\": \"C HYPER PARAMETERS AND EXPERIMENTAL SETUP\",\n \"text\": \"We kept the output dimensionality of the encoder at each level of TALD as |elt| = 1024, that of the stochastic states as |plt| = |qlt| = 20, and that of the deterministic states as |hlt| = 200. All hidden layers inside the cell, both for prior and posterior transition, were set to 200. We trained our models using the Adam optimizer (Kingma & Ba, 2014), with a learning rate of 5× 10−4 and = 10−4 for the Moving MNIST dataset. With the KTH Action and GQN mazes datasets, we lowered the learning rate to 3× 10−4. For all three datasets, batch size was set to be 100 sequences, each sequence being of length 100.\\nThe decision of using 36 context frames for conditioning all open-loop video predictions was taken considering the minimum number of observation frames required to transition at least once in the highest level of the hierarchy. With 3 levels in the hierarchy and a temporal abstraction factor of 6, each latent state at the highest level corresponds to 36 images in the sequence, and thus its encoder network expects 36 images as input.\"\n },\n {\n \"heading\": \"D ADDITIONAL EXPERIMENTS\",\n \"text\": \"D.1 QUANTITATIVE METRICS FOR LONG-HORIZON VIDEO PREDICTION\\nD.2 LONG-HORIZON PREDICTION OF MOVING MNIST\\nD.3 LONG-HORIZON PREDICTION OF 3D MAZES\\nD.4 LONG-HORIZON PREDICTION OF MINERL NAVIGATE\\nD.5 UNDERSTANDING THE LATENT REPRESENTATIONS - 3D MAZES\\nD.6 ANALYSING THE ENTROPY CURVES Figure 10b shows the entropy of the prior distribution at all levels of the 3-level TALD model with temporal abstraction factor 2, as it varies over time, with the Moving MNIST dataset. We observe that the entropy at the lowest level varies between 9 to 13 nats, whereas for the higher levels in the hierarchy, the entropy remained relatively constant. In Figure 11 we presented some visualizations that supported the claim that the higher level in the hierarchy stores global information (digit identities) which, in this case, does not change over time, and hence the model should remain rather certain about as time progresses. Figure 10b corroborates that claim showing that the model’s uncertainty at higher levels does not dither significantly over time as compared to the lowest level in the hierarchy.\\nFigure 10a shows the entropy of the prior distribution at all levels of the 3-level TALD model with temporal abstraction factor 2, as it varies over time, with the GQN mazes dataset. While the bottom two levels showed a similar variance in entropy with time, the entropy of the prior distribution at the top level did not change significantly. We showed in Figure 9 how the top level in the hierarchy stores wall and floor colors, which the model should remain certain about over time. In support of this claim, here we also see that the entropy of the prior distribution at the topmost level does not change significantly over time.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":97,"cells":{"paper_id":{"kind":"string","value":"SP:6082a5b51b24315dfdbfe147de1aef2c53cd113d"},"summaries":{"kind":"list like","value":["This paper extends the Wasserstein autoencoder for learning disentangled representations from sequential data. The latent variable model considered contains separate latent variables capturing global and local information respectively, each of which is regularized by a divergence measuring the marginal posterior $Q_z$ and the prior $P_z$. An optional auxiliary discrete latent is introduced to incorporate inductive bias for discrete local features (e.g., type of actions). To estimate the divergence terms, the authors propose to use MMD for the recurrent local latents since the prior distribution evolves over time; for the global latent, the authors presented two options: discriminator-based Jenson-Shannon Divergence estimate (the same as adversarial autoencoder proposed in Makhzani et al., 2016) and scaled MMD (Arbel et al., 2018). The connection between the proposed objective and mutual information maximization is outlined in Section 4. Experimental results show that the proposed R-WAE model outperforms baseline DS-VAE/FHVAE/MocoGAN"],"string":"[\n \"This paper extends the Wasserstein autoencoder for learning disentangled representations from sequential data. The latent variable model considered contains separate latent variables capturing global and local information respectively, each of which is regularized by a divergence measuring the marginal posterior $Q_z$ and the prior $P_z$. An optional auxiliary discrete latent is introduced to incorporate inductive bias for discrete local features (e.g., type of actions). To estimate the divergence terms, the authors propose to use MMD for the recurrent local latents since the prior distribution evolves over time; for the global latent, the authors presented two options: discriminator-based Jenson-Shannon Divergence estimate (the same as adversarial autoencoder proposed in Makhzani et al., 2016) and scaled MMD (Arbel et al., 2018). The connection between the proposed objective and mutual information maximization is outlined in Section 4. Experimental results show that the proposed R-WAE model outperforms baseline DS-VAE/FHVAE/MocoGAN\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Learning disentangled representations leads to interpretable models and facilitates data generation with style transfer, which has been extensively studied on static data such as images in an unsupervised learning framework. However, only a few works have explored unsupervised disentangled sequential representation learning due to challenges of generating sequential data. In this paper, we propose recurrent Wasserstein Autoencoder (R-WAE), a new framework for generative modeling of sequential data. R-WAE disentangles the representation of an input sequence into static and dynamic factors (i.e., time-invariant and time-varying parts). Our theoretical analysis shows that, R-WAE minimizes an upper bound of a penalized form of the Wasserstein distance between model distribution and sequential data distribution, and simultaneously maximizes the mutual information between input data and different disentangled latent factors, respectively. This is superior to (recurrent) VAE which does not explicitly enforce mutual information maximization between input data and disentangled latent representations. When the number of actions in sequential data is available as weak supervision information, R-WAE is extended to learn a categorical latent representation of actions to improve its disentanglement. Experiments on a variety of datasets show that our models outperform other baselines with the same settings in terms of disentanglement and unconditional video generation both quantitatively and qualitatively."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"DISENTANGLED RECURRENT"},{"affiliations":[],"name":"WASSERSTEIN AUTOEN"},{"affiliations":[],"name":"Jun Han"},{"affiliations":[],"name":"Martin Renqiang Min"},{"affiliations":[],"name":"Ligong Han"},{"affiliations":[],"name":"Li Erran Li"},{"affiliations":[],"name":"Xuan Zhang"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"DISENTANGLED RECURRENT\"\n },\n {\n \"affiliations\": [],\n \"name\": \"WASSERSTEIN AUTOEN\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Jun Han\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Martin Renqiang Min\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Ligong Han\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Li Erran Li\"\n },\n {\n \"affiliations\": [],\n \"name\": 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Lakshminarayanan\",\n \"Stephan Hoyer\",\n \"Rémi Munos\"\n ],\n \"title\": \"The cramer distance as a solution to biased wasserstein gradients\",\n \"venue\": \"arXiv preprint arXiv:1705.10743,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yoshua Bengio\",\n \"Aaron Courville\",\n \"Pascal Vincent\"\n ],\n \"title\": \"Representation learning: A review and new perspectives\",\n \"venue\": \"IEEE transactions on pattern analysis and machine intelligence,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Mikołaj Bińkowski\",\n \"Dougal J Sutherland\",\n \"Michael Arbel\",\n \"Arthur Gretton\"\n ],\n \"title\": \"Demystifying mmd gans\",\n \"venue\": \"arXiv preprint arXiv:1801.01401,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Olivier Bousquet\",\n \"Sylvain Gelly\",\n \"Ilya Tolstikhin\",\n \"Carl-Johann Simon-Gabriel\",\n \"Bernhard Schoelkopf\"\n ],\n \"title\": \"From optimal transport to generative modeling: the vegan cookbook\",\n \"venue\": \"arXiv preprint arXiv:1705.07642,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Andrew Brock\",\n \"Jeff Donahue\",\n \"Karen Simonyan\"\n ],\n \"title\": \"Large scale gan training for high fidelity natural image synthesis\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Tian Qi Chen\",\n \"Xuechen Li\",\n \"Roger B Grosse\",\n \"David K Duvenaud\"\n ],\n \"title\": \"Isolating sources of disentanglement in variational autoencoders\",\n \"venue\": \"NeurIPS,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Xi Chen\",\n \"Yan Duan\",\n \"Rein Houthooft\",\n \"John Schulman\",\n \"Ilya Sutskever\",\n \"Pieter Abbeel\"\n ],\n \"title\": \"Infogan: Interpretable representation learning by information maximizing generative adversarial nets\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Junyoung Chung\",\n \"Kyle Kastner\",\n \"Laurent Dinh\",\n \"Kratarth Goel\",\n \"Aaron C Courville\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"A recurrent latent variable model for sequential data\",\n \"venue\": \"Advances in Neural Information Processing Systems\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Najim Dehak\",\n \"Patrick J Kenny\",\n \"Réda Dehak\",\n \"Pierre Dumouchel\",\n \"Pierre Ouellet\"\n ],\n \"title\": \"Front-end factor analysis for speaker verification\",\n \"venue\": \"IEEE Transactions on Audio, Speech, and Language Processing,\",\n \"year\": 2010\n },\n {\n \"authors\": [\n \"Emily L Denton\"\n ],\n \"title\": \"Unsupervised learning of disentangled representations from video\",\n \"venue\": \"In NIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Emilien Dupont\"\n ],\n \"title\": \"Learning disentangled joint continuous and discrete representations\",\n \"venue\": \"In NIPS,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Peyman Mohajerin Esfahani\",\n \"Daniel Kuhn\"\n ],\n \"title\": \"Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations\",\n \"venue\": \"Mathematical Programming,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"John S Garofolo\"\n ],\n \"title\": \"Timit acoustic phonetic continuous speech corpus\",\n \"venue\": \"Linguistic Data Consortium,\",\n \"year\": 1993\n },\n {\n \"authors\": [\n \"Ian Goodfellow\",\n \"Jean Pouget-Abadie\",\n \"Mehdi Mirza\",\n \"Bing Xu\",\n \"David Warde-Farley\",\n \"Sherjil Ozair\",\n \"Aaron Courville\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Generative adversarial nets\",\n \"venue\": \"In NIPS,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Arthur Gretton\",\n \"Karsten Borgwardt\",\n \"Malte Rasch\",\n \"Bernhard Schölkopf\",\n \"Alex J Smola\"\n ],\n \"title\": \"A kernel method for the two-sample-problem\",\n \"venue\": \"In NIPS,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Ishaan Gulrajani\",\n \"Faruk Ahmed\",\n \"Martin Arjovsky\",\n \"Vincent Dumoulin\",\n \"Aaron C Courville\"\n ],\n \"title\": \"Improved training of wasserstein gans\",\n \"venue\": \"In NIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Jiawei He\",\n \"Andreas Lehrmann\",\n \"Joseph Marino\",\n \"Greg Mori\",\n \"Leonid Sigal\"\n ],\n \"title\": \"Probabilistic video generation using holistic attribute control\",\n \"venue\": \"In Proceedings of the European Conference on Computer Vision (ECCV),\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Junxian He\",\n \"Daniel Spokoyny\",\n \"Graham Neubig\",\n \"Taylor Berg-Kirkpatrick\"\n ],\n \"title\": \"Lagging inference networks and posterior collapse in variational autoencoders\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Kaiming He\",\n \"Xiangyu Zhang\",\n \"Shaoqing Ren\",\n \"Jian Sun\"\n ],\n \"title\": \"Deep residual learning for image recognition\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Martin Heusel\",\n \"Hubert Ramsauer\",\n \"Thomas Unterthiner\",\n \"Bernhard Nessler\",\n \"Sepp Hochreiter\"\n ],\n \"title\": \"Gans trained by a two time-scale update rule converge to a local nash equilibrium\",\n \"venue\": \"NeurIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Irina Higgins\",\n \"Loic Matthey\",\n \"Arka Pal\",\n \"Christopher Burgess\",\n \"Xavier Glorot\",\n \"Matthew Botvinick\",\n \"Shakir Mohamed\",\n \"Alexander Lerchner\"\n ],\n \"title\": \"beta-vae: Learning basic visual concepts with a constrained variational framework\",\n \"venue\": \"In ICLR,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"R Devon Hjelm\",\n \"Alex Fedorov\",\n \"Samuel Lavoie-Marchildon\",\n \"Karan Grewal\",\n \"Adam Trischler\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Learning deep representations by mutual information estimation and maximization\",\n \"venue\": \"arXiv preprint arXiv:1808.06670,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Jun-Ting Hsieh\",\n \"Bingbin Liu\",\n \"De-An Huang\",\n \"Li F Fei-Fei\",\n \"Juan Carlos Niebles\"\n ],\n \"title\": \"Learning to decompose and disentangle representations for video prediction\",\n \"venue\": \"NeurIPS,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Wei-Ning Hsu\",\n \"Yu Zhang\",\n \"James Glass\"\n ],\n \"title\": \"Unsupervised learning of disentangled and interpretable representations from sequential data\",\n \"venue\": \"In NIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Eric Jang\",\n \"Shixiang Gu\",\n \"Ben Poole\"\n ],\n \"title\": \"Categorical reparameterization with gumbel-softmax\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Hyunjik Kim\",\n \"Andriy Mnih\"\n ],\n \"title\": \"Disentangling by factorising\",\n \"venue\": \"In ICML,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Diederik P Kingma\",\n \"Jimmy Ba\"\n ],\n \"title\": \"Adam: A method for stochastic optimization\",\n \"venue\": \"ICLR,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Diederik P Kingma\",\n \"Max Welling\"\n ],\n \"title\": \"Auto-encoding variational bayes\",\n \"venue\": null,\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Chun-Liang Li\",\n \"Wei-Cheng Chang\",\n \"Yu Cheng\",\n \"Yiming Yang\",\n \"Barnabás Póczos\"\n ],\n \"title\": \"Mmd gan: Towards deeper understanding of moment matching\",\n \"venue\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Francesco Locatello\",\n \"Stefan Bauer\",\n \"Mario Lucic\",\n \"Gunnar Raetsch\",\n \"Sylvain Gelly\",\n \"Bernhard Schölkopf\",\n \"Olivier Bachem\"\n ],\n \"title\": \"Challenging common assumptions in the unsupervised learning of disentangled representations\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Laurens van der Maaten\",\n \"Geoffrey Hinton\"\n ],\n \"title\": \"Visualizing data using t-sne\",\n \"venue\": null,\n \"year\": 2008\n },\n {\n \"authors\": [\n \"Chris J Maddison\",\n \"Andriy Mnih\",\n \"Yee Whye Teh\"\n ],\n \"title\": \"The concrete distribution: A continuous relaxation of discrete random variables\",\n \"venue\": 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in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Henning Petzka\",\n \"Asja Fischer\",\n \"Denis Lukovnicov\"\n ],\n \"title\": \"On the regularization of wasserstein gans\",\n \"venue\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Ben Poole\",\n \"Sherjil Ozair\",\n \"Aaron Van Den Oord\",\n \"Alex Alemi\",\n \"George Tucker\"\n ],\n \"title\": \"On variational bounds of mutual information\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Guo-Jun Qi\"\n ],\n \"title\": \"Loss-sensitive generative adversarial networks on lipschitz densities\",\n \"venue\": \"arXiv preprint arXiv:1701.06264,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Kevin Roth\",\n \"Aurelien Lucchi\",\n \"Sebastian Nowozin\",\n \"Thomas Hofmann\"\n ],\n \"title\": \"Stabilizing training of generative adversarial networks through regularization\",\n \"venue\": \"In NIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Paul K Rubenstein\",\n \"Bernhard Schoelkopf\",\n \"Ilya Tolstikhin\"\n ],\n \"title\": \"On the latent space of wasserstein auto-encoders\",\n \"venue\": \"arXiv preprint arXiv:1802.03761,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Paul K Rubenstein\",\n \"Bernhard Schoelkopf\",\n \"Ilya Tolstikhin\"\n ],\n \"title\": \"Learning disentangled representations with wasserstein auto-encoders\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Tim Salimans\",\n \"Ian Goodfellow\",\n \"Wojciech Zaremba\",\n \"Vicki Cheung\",\n \"Alec Radford\",\n \"Xi Chen\"\n ],\n \"title\": \"Improved techniques for training gans\",\n \"venue\": \"In NIPS,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Casper Kaae Sønderby\",\n \"Tapani Raiko\",\n \"Lars Maaløe\",\n \"Søren Kaae Sønderby\",\n \"Ole Winther\"\n ],\n \"title\": \"Ladder variational autoencoders\",\n \"venue\": \"In NIPS,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Ximeng Sun\",\n \"Huijuan Xu\",\n \"Kate Saenko\"\n ],\n \"title\": \"A two-stream variational adversarial network for video generation\",\n \"venue\": \"arXiv preprint arXiv:1812.01037,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Hoang Thanh-Tung\",\n \"Truyen Tran\",\n \"Svetha Venkatesh\"\n ],\n \"title\": \"Improving generalization and stability of generative adversarial networks\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Michael Tschannen\",\n \"Josip Djolonga\",\n \"Paul K Rubenstein\",\n \"Sylvain Gelly\",\n \"Mario Lucic\"\n ],\n \"title\": \"On mutual information maximization for representation learning\",\n \"venue\": null,\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Sergey Tulyakov\",\n \"Ming-Yu Liu\",\n \"Xiaodong Yang\",\n \"Jan Kautz\"\n ],\n \"title\": \"Mocogan: Decomposing motion and content for video generation\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Li Yingzhen\",\n \"Stephan Mandt\"\n ],\n \"title\": \"Disentangled sequential autoencoder\",\n \"venue\": \"In ICML,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Shengjia Zhao\",\n \"Jiaming Song\",\n \"Stefano Ermon\"\n ],\n \"title\": \"Learning hierarchical features from deep generative models\",\n \"venue\": \"In ICML,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Brock\"\n ],\n \"title\": \"2019), named \\\"ResBlock down\\\". After each Resblock, we use a FC network to get latent feature h, for i = 0\",\n \"venue\": null,\n \"year\": 2019\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Unsupervised representation learning is an important research topic in machine learning. It embeds high-dimensional sensory data such as images and videos into a low-dimensional latent space in an unsupervised learning framework, aiming at extracting essential data variation factors to help downstream tasks such as classification and prediction (Bengio et al., 2013). In the last several years, disentangled representation learning, which further separates the latent embedding space into exclusive explainable factors such that each factor only interprets one of semantic attributes of sensory data, has received a lot of interest and achieved many empirical successes on static data such as images (Chen et al., 2016; Higgins et al., 2017; Dupont, 2018; Chen et al., 2018; Rubenstein et al., 2018b;a; Kim & Mnih, 2018). For example, the latent representation of handwritten digits can be disentangled into a content factor encoding digit identity and a style factor encoding handwriting style.\nIn spite of successes on static data, only a few works have explored unsupervised representation disentanglement of sequential data due to the challenges of developing generative models of sequential ∗Equal contribution. †Part of his work was done before joining Tencent. ‡His work was done before joining Amazon. §His work was done before joining Texas A&M University.\ndata. Learning disentangled representations of sequential data is important and has many applications. For example, the latent representation of a smiling-face video can be disentangled into a static part encoding the identity of the person (content factor) and a dynamic part encoding the smiling motion of the face (motion factor). The disentangled representation of the video can be potentially used for many downstream tasks such as classification, retrieval, and synthetic video generation with style transfer. Most of previous unsupervised representation disentanglement models for static data heavily rely on the KL-divergence regularization in a VAE framework (Higgins et al., 2017; Dupont, 2018; Chen et al., 2018; Kim & Mnih, 2018), which has been shown to be problematic due to matching individual instead of aggregated posterior distribution of the latent code to the same prior (Tolstikhin et al., 2018; Rubenstein et al., 2018b;a). Therefore, extending VAE or recurrent VAE (Chung et al., 2015) to disentangle sequential data in a generative model framework (Hsu et al., 2017; Yingzhen & Mandt, 2018) is not ideal. In addition, recent research (Locatello et al., 2019) has theoretically shown that it is impossible to perform unsupervised disentangled representation learning without inductive biases on both models and data, especially on static data. Fortunately, sequential data such as videos often have clear inductive biases for the disentanglement of content factor and motion factor as mentioned in (Locatello et al., 2019). Unlike static data, the learned static and dynamic factors of sequential data are not exchangeable.\nIn this paper, we propose a recurrent Wasserstein Autoencoder (R-WAE) to learn disentangled representations of sequential data. We employ a Wasserstein metric (Arjovsky et al., 2018; Gulrajani et al., 2017; Bellemare et al., 2017) induced from the optimal transport between model distribution and the underlying data distribution, which has some nicer properties (for e.g., sum invariance, scale sensitivity, applicable to distributions with non-overlapping supports, and better out-of-sample performance in the worst-case expectation (Esfahani & Kuhn, 2018)) than the KL divergence in VAE (Kingma & Welling, 2014) and β-VAE (Higgins et al., 2017). Leveraging explicit inductive biases in both sequential data and model, we encode an input sequence into two parts: a shared static latent code and a dynamic latent code, and sequentially decode each element of the sequence by combining both codes. We enforce a fixed prior distribution for the static code and learn a prior for the dynamic code to ensure the consistency of the sequence. The disentangled representations are learned by separately regularizing the posteriors of the latent codes with their corresponding priors.\nOur main contributions are summarized as follows: (1) We draw the first connection between minimizing a Wasserstein distance and maximizing mutual information for unsupervised representation disentanglement of sequential data from an information theory perspective; (2) We propose two sets of effective regularizers to learn the disentangled representation in a completely unsupervised manner with explicit inductive biases in both sequential data and models. (3) We incorporate a relaxed discrete latent variable to improve the disentangled learning of actions on real data. Experiments show that our models achieve state-of-the-art performance in both disentanglement of static and dynamic latent representations and unconditional video generation under the same settings as baselines (Yingzhen & Mandt, 2018; Tulyakov et al., 2018)."},{"heading":"2 BACKGROUND AND RELATED WORK","text":"Notation Let calligraphic letters (i.e. X ) be sets, capital letters (i.e. X) be random variables and lowercase letters be their values. Let D(PX , PG) be the divergence between the true (but unknown) data distribution PX (density p(x)) and the latent-variable generative model distribution PG specified by a prior distribution PZ (density p(z)) of latent variable Z. Let DKL be KL divergence, DJS be Jensen-Shannon divergence and MMD be Maximum Mean Discrepancy (MMD) (Gretton et al., 2007).\nOptimal Transport Between Distributions The optimal transport cost inducing a rich class of divergence between the distribution PX and the distribution PG is defined as follows,\nW (PX , PG):= inf Γ∼P(X∼PX ,Y∼PG) E(X,Y )∼Γ[c(X,Y )], (1)\nwhere c(X,Y ) is any measurable cost function and P(X ∼ PX , Y ∼ PG) is the set of joint distributions of (X, Y) with respective marginals PX and PG.\nComparison between WAE (Tolstikhin et al., 2018) and VAE (Kingma & Welling, 2014) Instead of optimizing over all couplings Γ between two random variables in X , Bousquet et al.\n(2017); Tolstikhin et al. (2018) show that it is sufficient to find Q(Z|X) such that the marginal Q(Z) := EX∼PX [Q(Z|X)] is identical to the prior P (Z), as given in the following definition,\nDefinition 1. For any deterministic PG(X|Z) and any function G : Z → X ,\nW (PX , PG) = inf Q:QZ=PZ EPXEQ(Z|X)[c(X,G(Z))]. (2)\nDefinition 1 leads to the following loss DWAE of WAE based on a Wasserstein distance,\ninf Q(Z|X) EPXEQ(Z|X)[c(X,G(Z))] + β D(QZ , PZ), (3)\nwhere the first term is data reconstruction loss, and the second one is a regularizer that forces the posterior QZ = ∫ Q(Z|X)dPX to match the prior PZ (Adversarial autoencoder (AAE) (Makhzani et al., 2015) shares a similar idea to WAE). In contrast, VAE has a different regularizer EX [DKL(Q(Z|X), PZ))] enforcing the latent posterior distribution of each input to match PZ . In (Rubenstein et al., 2018a;b), it is shown that WAE has better disentanglement than β-VAE (Higgins et al., 2017) on images, which inspires us to design a new representation disentanglement framework for sequential data with several innovations.\nUnsupervised disentangled representation learning Several generative models have been proposed to learn disentangled representations of sequential data (Denton et al., 2017; Hsu et al., 2017; Yingzhen & Mandt, 2018; Hsieh et al., 2018; Sun et al., 2018; Tulyakov et al., 2018). FHVAE in (Hsu et al., 2017) is a VAE-based hierarchical graphical model with factorized Gaussian priors and only focuses on speech or audio data. Our R-WAE employing a more powerful recurrent prior can be applied to both speech and video data. The models in (Sun et al., 2018; Denton et al., 2017; Hsieh et al., 2018) are based on the first several elements of a sequence to design disentanglement architectures for future sequence predictions.\nIn terms of representation learning by mutual information maximization, our work empirically demonstrates that explicit inductive biases in data and model architecture are necessary to the success of learning meaningful disentangled representations of sequential data, while the works in (Locatello et al., 2019; Poole et al., 2019; Tschannen et al., 2020; Ozair et al., 2019) are about general representation learning, especially on static data.\nThe most related works to ours are MoCoGAN (Tulyakov et al., 2018) and DS-VAE (Yingzhen & Mandt, 2018), which have the ability to disentangle variant and invariant parts of sequential data and perform unconditional sequence generation. Tulyakov et al. (2018) is a GAN-based model that can be only applied to the setting in which the number of motions is finite, and cannot encode the latent representation of sequences. Yingzhen & Mandt (2018) provides a disentangled sequential autoencoder based on VAE (Kingma & Welling, 2014). Training VAE is equivalent to minimizing a lower bound of the KL divergence between empirical data distribution and generated data distribution, which has been shown to produce inferior disentangled representations of static data than generative models employing the Wasserstein metric (Rubenstein et al., 2018a;b)."},{"heading":"3 PROPOSED APPROACH: DISENTANGLED RECURRENT WASSERSTEIN AUTOENCODER (R-WAE)","text":"Given a high-dimensional sequence x1:T , our goal is to learn a disentangled representation of timeinvariant latent code zc and time-variant latent code zmt , along the sequence. Let zt = (z\nc, zmt ) be the latent code of xt. Let Xt, Zt, Zc and Zmt be random variables with realizations xt, zt, z\nc and zmt respectively, and denote D = X1:T . To achieve this goal, we define the following probabilistic generative model by assuming Zmt and Z c are independent,\nP (X1:T , Z1:T ) = P (Z c) T∏ t=1 Pψ(Z m t |Zm<t)Pθ(Xt|Zt), (4)\nwhere P (Z1:T ) = P (Zc) ∏T t=1 Pψ(Z m t |Zm<t) is the prior in which Zt = (Zc, Zmt ), and the decoder model Pθ(Xt | Zt) is a Dirac delta distribution. In practice, P (Zc) is chosen as N (0, I)\nand Pψ(Zmt |Zm<t) = N (µψ(Zm<t),σ2ψ(Zm<t)), µψ and σψ are parameterized by Recurrent Neural Networks (RNNs). The inference model Q is defined as\nQφ(Z c, Zm1:T |X1:T ) = Qφ(Zc|X1:T ) T∏ t=1 Qφ(Z m t | Zm<t, Xt), (5)\nwhereQφ(Zc|X1:T ) andQφ(Zmt | Zm<t, Xt) are also Gaussian distributions parameterized by RNNs. The structures of the generative model (4) and the inference model (5) are provided in Fig. 1."},{"heading":"3.1 R-WAE MINIMIZES A PENALIZED FORM OF A WASSERSTEIN DISTANCE","text":"The optimal transport cost between two distributions PD and PG with respective sequential variables X1:T (X1:T∼PD) and Y1:T (Y1:T∼PG) is given as follows,\nW (PD, PG) := inf Γ∼P(X1:T∼PD,Y1:T∼PG) E(X1:T ,Y1:T )∼Γ[c(X1:T , Y1:T )], (6)\nP(X1:T∼PD,Y1:T∼PG) is a set of all joint distributions with marginals PD and PG respectively. When we choose c(x,y) = ‖x−y‖2 (2-Wasserstein distance) and c(X1:T , Y1:T ) = ∑ t ‖Xt−Yt‖2 by linearity, it is easy to derive the optimal transport cost for disentangled sequential variables.\nTheorem 1. With deterministic P (Xt|Zt) and any function Yt = G(Zt), we derive\nW (PD, PG) = inf Q:QZc=PZc ,QZm\n1:T =PZm 1:T ∑ t EPDEQ(Zt|Z<t,Xt)[c(Xt, G(Zt))], (7)\nwhere QZ1:T = QZcQZm1:T is the marginal distribution of Z1:T when X1:T ∼ PD and Z1:T ∼ Q(Z1:T |X1:T ) and PZ1:T is the prior. Based on the assumptions, we have an upper bound,\nW (PD, PG) ≤ inf Q∈S ∑ t EPDEQ(Zt|Z<t,Xt)[c(Xt, G(Zt))], (8)\nwhere the subset S is S = {Q : QZc = PZc , QZm1 = PZm1 , QZmt |Zm<t = PZmt |Zm<t} .\nIn practice, we have the following objective function of our proposed R-WAE based on Theorem 1,\nT∑ t=1 EQ(Zt|Z<t,Xt)[c(Xt, G(Zt))] + β1 D(QZc , PZc) + β2 T∑ t=1 D(QZmt |Zm<t , PZmt |Zm<t), (9)\nwhere D is an divergence between two distributions, and the second and third terms are, respectively, regularization terms for Zc and Zmt . In the following, we will present two different approaches to calculating the regularization terms in section 3.2 and 3.3. Because we cannot straightforwardly estimate the marginals Qφ(Zc) and Qφ(Zmt |Zm<t), we cannot directly use KL divergence in the two regularization terms, but we can optimize the RHS of (9) by likelihood-free optimizations (Gretton et al., 2007; Goodfellow et al., 2014; Nowozin et al., 2016; Arjovsky et al., 2018) when samples from all distributions are available.\n3.2 DJS PENALTY FOR Zc AND MMD PENALTY FOR Zm\nThe prior distribution of Zc is chosen as a multivariate unit-variance Gaussian, N (0, I). We can choose penalty DJS(QZc ,PZc) for Zc and apply min-max optimization by introducing a discriminator Dγ (Goodfellow et al., 2014). Instead of performing optimizations in high-dimensional input data space, we move the adversarial optimization to the latent representation space of the content with a much lower dimension. Because the prior distribution of {Zmt } is dynamically learned during training, it is challenging to use DJS to regularize {Zmt } with a discriminator, which will induce a third minimization within a min-max optimization. Therefore, we use MMD to regularize {Zmt } as samples from both distributions are easy to obtain (dimension of zmt is less than 20 in our experiments on videos). With a kernel k, MMDk(Q,P ) is approximated by samples from Q and P (Gretton et al., 2007). The regularization terms can be summarized as follows and we call the resulting model R-WAE(GAN) (see Algorithm 1 in Appendix for details):\nD(QZc ,PZc) = DJS(QZc ,PZc); D(QZmt |Zm<t ,PZmt |Zm<t) = MMDk(QZmt |Zm<t ,PZmt |Zm<t). (10)\n3.3 SCALED MMD PENALTY FOR Zc AND MMD PENALTY FOR Zm\nMMD with neural kernels for generative modeling of real-world data (Li et al., 2017; Bińkowski et al., 2018; Arbel et al., 2018) motivates us to use only MMD as regularization in Eq. (9),\nD(QZc ,PZc) = MMDkγ (QZc ,PZc); D(QZmt |Zm<t ,PZmt |Zm<t) = MMDk(QZmt |Zm<t ,PZmt |Zm<t), (11)\nwhere kγ is a parametrized family of kernels (Li et al., 2017; Bińkowski et al., 2018; Arbel et al., 2018) defined as kγ(x,y) = k(fγ(x), fγ(y)) and fγ(x) is a feature map, which is more expressive and used for Zc with equal or higher dimension than Zmt . The details of optimizing the first term MMDkγ (QZc ,PZc) in Eq. (11) is provided in Appendix D based on scaled MMD (Arbel et al., 2018), a principled and stable technique for training MMD-based critic. We call the resulting model R-WAE(MMD) (see Algorithm 2 in Appendix for details)."},{"heading":"3.4 WEAKLY SUPERVISED DISENTANGLEMENT WITH A KNOWN NUMBER OF ACTIONS","text":"When the number of actions (motions) in sequential data, denoted by A, is available, we incorporate a categorical latent variable a (a one-hot vector whose dimension is A) to enhance the disentanglement of the dynamic latent codes of the motions. The inference model for a is designed as qφ(a|x1:T , zm1:T ). Intuitively, the action is inferred from the motion sequence to recognize its label. Learning such a categorical distribution requires a continuous relaxation of the discrete random variable in order to backpropagate its gradient. Let α1, · · · , αA be the class probabilities, we can obtain a sample ã = (y1, · · · , yA) from its continuous relaxation by first sampling g = (g1, · · · , gA) with gj ∼ Gumbel(0, 1) and then applying transformation ãj = exp((logαj +gj)/τ) ∑ i exp((logαi+gi)/τ), where τ is a temperature parameter controlling the approximation. To learn the categorical distribution using the reparameterization trick, we use a regularizer DKL(qφ(ã|x1:T , zm1:T ), p(ã)), where p(ã) is a uniform Gumbel-Softmax prior distribution (Jang et al., 2016; Maddison et al., 2016). The motion variable is augmented as zRt = (z m t ,a), and learning joint continuous and discrete latent representation of image data has been extensively discussed in (Dupont, 2018) (see Fig. 1(c,d) for illustrations)."},{"heading":"4 ANALYZING R-WAE FROM AN INFORMATION THEORY PERSPECTIVE","text":"Theorem 2. If the mutual information (MI) between Z1:T and X1:T is defined in terms of the inference model Q, I(Z1:T ;X1:T ) = EQ(X1:T ,Z1:T )[logQ(Z1:T |X1:T ) − logQ(Z1:T )], where Q(X1:T , Z1:T ) = Q(Z1:T |X1:T )P (X1:T ) and Q(Z1:T ) = ∑ X1:T\nQ(X1:T , Z1:T ), we have a lower bound:\nI(Z1:T ;X1:T )≥ T∑ t=1 EPD [ EQφ [logPθ(Xt |Zt)−logP (D)]−EQφ(Zc|X1:T )[logQφ(Z c)−logP (Zc)] ]\n− T∑ t=1 EPD [ EQφ(Zmt |Zm<t,Xt)[logQφ(Z m t |Zm<t)−logP (Zmt |Zm<t) ] . (12)\nTheorem 2 shows that R-WAE maximizes a lower bound of the mutual information between X1:T and Z1:T , which theoretically guarantees that R-WAE learns semantically meaningful latent representations of input sequences. With constant removed, the RHS of (9) and (12) are the same if D is KL divergence. In spite of being theoretically important, Theorem 2 with KL divergence cannot be directly used for the regularization terms of R-WAE in practice, because we cannot straightforwardly estimate the marginals Qφ(Zc) and Qφ(Zmt |Zm<t) as discussed previously. From Eq. (9) and (12), we can obtain the following theorem.\nTheorem 3. When its distribution divergence is chosen as KL divergence, the regularization terms in Eq. (9) jointly minimize the KL divergence between the inference model Q(Z1:T |X1:T ) and the prior model P (Z1:T ) and maximize the mutual information between X1:T and Z1:T ,\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc), (13) KL(Q(Zmt |Zm<t)||P (Zmt |Zm<t))=EpD [KL(Q(Zmt |Zm<t, X1:T )||P (Zmt |Zm<t)]− I(X1:T ;Zmt |Zm<t),\nwhere the mutual information is defined in terms of the inference model as in Theorem 2.\nTheorem 3 shows that, even if adopting KL divergence, the regularization in the loss of R-WAE still improves over the one in vanilla VAE, which only has the first term in the RHS of Eq. (13). The two mutual information terms explicitly enforce mutual information maximization between input data and unexchangeable disentangled latent representations, Zc and Zmt . Therefore, R-WAE is superior to recurrent VAE (DS-VAE)."},{"heading":"5 EXPERIMENTS","text":"We conduct extensive experiments on four datasets to quantitatively and qualitatively validate our methods. The baseline methods for comparisons are DS-VAE (Yingzhen & Mandt, 2018) and MoCoGAN (Tulyakov et al., 2018). We train our models on Stochastic moving MNIST (SMMNIST), Sprites, and TIMIT datasets under a completely unsupervised setting. The number of actions (motions) is used as prior information for all methods on MUG facial dataset. The detailed descriptions of datasets, architectures, and hyperparameters are provided in Appendix C, D, and G, respectively."},{"heading":"5.1 QUALITATIVE RESULTS ON DISENTANGLEMENT","text":"We encode two original videos on the first and fourth row in Fig. 2 and generate videos on the second and third row by swapping corresponding {zc} and {zm1:T } between videos for style transfer. Fig. 2(left) shows that even testing on the long sequence (trained with T = 100), our R-WAE can disentangle content and motions exactly. In Fig. 2(right), we do the same swapping on Sprites. We can see that the generated swapped videos have exact same appearances and actions as the corresponding original ones. On the MUG\ndataset, it is interesting to see that we can swap different motions between different persons. PPPPPPPMethods Datasets EER zc = 16 ↓ zm = 16 ↑\nFHVAE 5.06% 22.77% DS-VAE 5.64% 19.20% R-WAE 4.73% 23.41%\nTable 1: EER on TIMIT speech dataset under the same dimension setting of segment-level zc and sequence-leve zm for FHVAE (Hsu et al., 2017), DS-VAE (full q) (Yingzhen & Mandt, 2018) and R-WAE(MMD), respectively. Small EER is better for zc and larger EER is better for zm.\nPPPPPPPMethods Datasets Sprites SM-MNIST\nactions content digits DS-VAE(S) 8.11% 3.98% 3.31% R-WAE(S) 5.83% 2.45% 1.78% DS-VAE(C) 10.37% 4.86% 4.26% R-WAE(C) 7.72% 3.31% 2.84%\nTable 2: Comparison of averaged classification errors. On Sprites dataset, fix one encoded attribute and randomly sample others. On SM-MNIST dataset, we fix the encoded zc and randomly sample the motion sequence from the learned prior pψ(zmt |zm<t). We cannot quantitatively verify the motion disentanglement on SM-MNIST."},{"heading":"5.2 QUANTITATIVE RESULTS","text":"SM-MNIST and Sprites Datasets We quantitatively evaluate the disentanglement of our RWAE(MMD). In Table 2, \"S\" denotes a simple encoder/decoder architecture, where the encoders in both our model and DS-VAE (Yingzhen & Mandt, 2018) only use 5 layers of convolutional and deconvolutional networks adopted from DS-VAE (Yingzhen & Mandt, 2018). \"C\" denotes a complex encoder/decoder architecture where we use Ladder network (Sønderby et al., 2016; Zhao et al., 2017) and ResBlock (He et al., 2016), provided in Appendix E. On SM-MNIST, we get the labeled latent codes {zc} of test videos {x1:T } with T = 10 and randomly sample motion variables {zm1:T } to get labeled new samples. We pretrain a classifier and predict the accuracy on these labeled new samples. The accuracy on SM-MNIST dataset is evaluated on 10000 test samples. On Sprites, the labels of each attribute(skin colors, pants, hair styles, tops and pants) are available. We get the latent codes by fixing one attribute and randomly sample other attributes. We train a classifier for each attribute and evaluate the disentanglement of each attribute. The accuracy is based on 296 × 9 test data. Both DS-VAE and R-WAE(MMD) have extremely high accuracy (99.94%) when fixing hair style attribute, which is not provided in Table 2 due to space limit. As R-WAE(GAN) and R-WAE(MMD) have similar performance on these datasets, we only provide the results and parameters of R-WAE(MMD) to save space. There are two interesting observations in Table 2. First, the simple architecture has better disentanglement than the complex architecture overall. The reason is that the simple architecture has sufficient ability to extract features and generate clear samples to be recognized by the pretrained classifiers. But the simple architecture cannot generate high-quality samples when applied to real data. Second, our proposed R-WAE(MMD) achieve better disentanglement than DS-VAE (Yingzhen & Mandt, 2018) on both corresponding architectures. The attributes within content latent variables are independent, our model can further disentangle the factors. Compared to DS-VAE, these results demonstrate the advantages of R-WAE with implicit mutual information maximization terms. Due to space limit, we also include similar comparisons on a new Moving-Shape dataset in Appendix I. As the number of possible motions in SM-MNIST is infinite and random, we cannot evaluate the disentanglement of motions by training a classifier. We fix the encoded motions {zm1:T } and randomly sample content variables {zc}. We also randomly sample a motion sequence {zm1:T } and randomly sample contents {zc}. We manually check the motions of these samples and almost all have the same corresponding motion even though the sequence is long (T = 100). TIMIT Speech Dataset We quantitatively compare our R-WAE with FHVAE and DS-VAE on the speaker verification task under the same setting as (Hsu et al., 2017; Yingzhen & Mandt, 2018). The evaluation metric is based on equal error rate (EER), which is explained in detail in Appendix C. The lower EER on zc encoding the timbre of speakers is better and the higher EER on zm encoding linguistic content is better. From Table 1, our model can disentangle zc and zm well. We can see that our R-WAE(MMD) has the best EER performance on both content attribute and motion attribute. In Appendix H we show by style transfer experiments that the learned dynamic factor encodes semantic content which is comparable to DS-VAE. MUG Facial Dataset We quantitatively evaluate the disentanglement and quality of generated samples. We train a 3D classifier on MUG facial dataset with accuracy 95.11% and Inception Score 5.20 on test data (Salimans et al., 2016). We calculate Inception score, intra-entropy H(y|v), where y is the predicted label and v is the generated video, and inter-entropy H(y) (He et al., 2018). For a comprehensive quantitative evaluation, Frame-level FID score, introduced by (Heusel et al., 2017), is also provided. From Table 2, our R-WAE(MMD) and R-WAE(GAN) have higher accuracy, which\nmeans the categorical variable best captures the actions, which indicates our models achieve the best disentanglement. In table 2, the Inception score of R-WAE(GAN) is very close to Inception Score of the exact test data, which means our models have the best sample quality. Our proposed R-WAE(GAN) and R-WAE(MMD) have the best frame-level FID scores, compared with DS-VAE and MoCoGAN. The orignal DS-VAE (DS-VAE(NA)) (Yingzhen & Mandt, 2018) without leveraging the number of actions performs worst, which shows that incorporating the number of actions as prior information does enhance the disentanglement of actions."},{"heading":"5.3 UNCONDITIONAL VIDEO GENERATION","text":"SM-MNIST dataset Fig. 4 in Appendix E provides generated samples on the SM-MNIST dataset by randomly sampling content {zc} from the prior p(zc) and motions {zm1:T } from the learned prior pψ(z m t |zm<t). The length of our generated videos is T = 100 and we only show randomly chosen videos of T = 20 to save file size. Our R-WAE(MMD) achieves the most consistent and visually best sequence even when T = 100. Samples from MoCoGAN (Tulyakov et al., 2018) usually change digit identity along the sequence. The reason is that MoCoGAN (Tulyakov et al., 2018) requires the number of actions be finite. Our generated Sprites videos also have the best results but are not provided due to page limit.\nMUG Facial Dataset Fig. 3 and Fig. 5 in Appendix E show generated samples on MUG dataset by randomly sampling content {zc} from the prior p(zc) and motions zRt = (a, zmt ) from the categorical prior p(a) (latent action variable a is a one-hot vector with dimension 6) and the learned prior pψ(zmt |zm<t). We show generated videos of length T = 10. DS-VAE (Yingzhen & Mandt, 2018) used the same structure as ours. Fig. 5 shows that DS-VAE (Yingzhen & Mandt, 2018) and\nMoCoGAN (Tulyakov et al., 2018) have blurry beginning frames {xt} and even more blurry frames as time t evolves. While our R-WAE(GAN) has much better frame quality and more consistent video sequences. To have a clear comparison among all three methods, we show the samples at time step T = 10 in Fig. 3, and we can see that DS-VAE has very blurry samples with large time steps."},{"heading":"6 CONCLUSION","text":"In this paper, we propose recurrent Wasserstein Autoencoder (R-WAE) to learn disentangled representations of sequential data based on the optimal transport between distributions with sequential variables. Our theoretical analysis shows that R-WAE simultaneously maximizes the mutual information between input sequential data and different disentangled latent factors. Experiments on a variety of datasets demonstrate that our models achieve state-of-the-art results on the disentanglement of static and dynamic latent representations and unconditional video generation. Future research includes exploring our framework in self-supervised learning and conditional settings for text-to-video and video-to-video synthesis.\nAcknowledgement Jun Han thanks Dr. Chen Fang at Tencent for insightful discussions and Prof. Qiang Liu at UT Austin for invaluable support."},{"heading":"APPENDIX FOR RECURRENT WASSERSTEIN AUTOENCODER","text":""},{"heading":"APPENDIX A: PROOF OF THEOREM 1","text":"In the following, we provide the proof of Theorem 1.\nTheorem 1 For PG defined with deterministic PG(X|Z) and any function Y = G(Z),\nW (PD, PG) = inf Q:QZc=PZc ,QZm\n1:T =PZm 1:T T∑ t=1 EPDEQ(Zt|Xt)[c(Xt, G(Zt))], (14)\nwhere QZ1:T is the marginal distribution of Z1:T when X1:T ∼ PD and Z1:T ∼ Q(Z1:T |X1:T ) and PZ1:T is the prior. Based on the assumptions, the constraint set is equivalent to\nW (PD, PG) ≤ inf Q∈S ∑ t EPDEQ(Zt|Xt)[c(Xt, G(Zt))], (15)\nwhere the set S = {Q : QZc = PZc , QZm1 = PZm1 , QZmt |Zm<t = PZmt |Zm<t}.\nProof: Consider the sequential random variables D = X1:T and Y1:T , the optimal transport between the distribution for D = X1:T and the distribution for Y1:T induces a rich class of divergence,\nW (PD, PG) := inf Γ∼P(X1:T∼PD,Y1:T∼PG) E(X1:T ,Y1:T )∼Γ[c(X1:T , Y1:T )] (16)\nwhere P(X1:T ∼ PD, Y1:T ∼ PG) is a set of all joint distributions of (X1:T , Y1:T ) with marginals PD and PG, respectively.\nWhen we choose c(x,y) = ‖x − y‖2, c(X1:T , Y1:T ) = ∑ t ‖Xt − Yt‖2 by linearity. It is easy to derive the optimal transport for distributions with sequential random variables,\nW (PD, PG) = inf Q:QZ1:T =PZ1:T ∑ t EPDEQ(Zt|Xt)[c(Xt, G(Zt))]. (17)\nBased on our assumption, the marginal distribution of the inference model satisfies Q(Z1, · · · , ZT ) = Q(Zc)Q(Zm1 , · · · , ZmT ) = Q(Zc) ∏ t Q(Zmt |Zm<t). (18) The prior distribution P (Z1, · · · , ZT ) satisfies P (Z1, · · · , ZT ) = P (Zc)P (Zm1 , · · · , ZmT ) = P (Zc)\n∏ t P (Zmt |Zm<t). (19)\nSince the set S is a subset of {Q : QZ1:T = PZ1:T }, we derive the inequality (15)."},{"heading":"APPENDIX B: PROOF OF THEOREM 2","text":"In the following, we provide the proof of Theorem 2. To make the notations easy to read, we use the density functions of corresponding distributions.\nThe joint generative distribution is p(x1:T , z1:T ) = pψ(z1:T )pθ(x1:T |z1:T ), where pψ(z1:T ) is the prior distribution and pθ(x1:T |z1:T ) is the decoder model. And the corresponding joint inference distribution is qφ(x1:T , z1:T ) = pD(x1:T )qφ(z1:T | x1:T ). If the MI between z1:T and x1:T is defined in terms of the inference model q, we have the following lower bound with step-by-step derivations:\nI(z1:T ;x1:T )= Eq(x1:T ,z1:T )[log qφ(z1:T |x1:T ) qφ(z1:T ) ] (20)\n= Eq(x1:T ,z1:T )[DKL(qφ(z1:T |x1:T ), p(z1:T |x1:T ))+log p(z1:T |x1:T )−log qφ(z1:T )] ≥ EpD [Eq(z1:T |x1:T )[log p(x1:T |z1:T ) + log p(z1:T )− log qφ(z1:T )−log p(D)]]\n≥ T∑ t=1 Ep(D) [ Eqφ(zt|xt)[log pθ(xt|zt)] ] −Ep(D)[Eqφ(zt|xt)[log qφ(z c)−log p(zc)] ]\n− T∑ t=1 Ep(D) [ Eqφ(zmt |xt)[log qφ(z m t |zm<t)− log p(zmt |zm<t) + log p(D) ] ,\nwhere we use Bayesian’s rule p(z1:T |x1:T ) = pθ(x1:T |z1:T )p(z1:T )p(D) . Maximizing the MI between z1:T and x1:T achieves state-of-the-art results in disentangled latent representation by using different regularizers for the static and dynamical latent variable with different priors (Hjelm et al., 2018). In practice, incorporating the mutual information I(zmt ,xt) between element xt and motion z m t might facilitate the disentanglement of the dynamical latent variable zmt .\nTheorem 3 When its distribution divergence is chosen as KL divergence, the regularization terms in Eq. (9) jointly minimize the KL divergence between the inference model Q(Z1:T |X1:T ) and the prior model P (Z1:T ) and maximize the mutual information between X1:T and Z1:T ,\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc). KL(Q(Zmt |Zm<t)||P (Zmt |Zm<t)) = EpD [KL(Q(Zmt |Zm<t, X1:T )||P (Zmt |Zm<t)]− I(X1:T ;Zmt |Zm<t).\nProof: Denote XD = X1:T . As in the proof of Theorem 2, the mutual information between Z1:T and X1:T is defined in terms of the inference model Q, and we use the density functions of corresponding distributions to make the notations easy to read. Thus,\nQ(Z1:T ) = EpDq(z1:T |x1:T ).\nAccording to the definition of mutual information, we have\nI(X1:T ;Z c) = EpD ∑ zc pD(x1:T )q(z c|x1:T ) log pD(x1:T )q(z c|x1:T ) pD(x1:T )q(zc)\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) q(zc)\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) p(zc) − EpD ∑ zc q(zc|x1:T ) log q(zc) p(zc)\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) p(zc) − ∑ zc q(zc) log q(zc) p(zc) = EpD [KL(Q(Zc|X1:T )||P (Zc))]−KL(Q(Zc)||P (Zc))\nTherefore,\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc).\nSimilarly, we can prove the second equality in the theorem."},{"heading":"APPENDIX C: DATASETS","text":"Stochastic Moving MNIST(SM-MNIST) Dataset Stochastic moving MNIST (SM-MNIST) consists of sequences of frames of size 64× 64× 1, containing one MNIST digit moving and bouncing off edges of the frame (walls). We use one digit instead of two digits because two moving digits may collide, which changes the content of the dynamics and is inconsistent with our assumption. The digits in SM-MNIST move with a constant velocity along a trajectory until they hit at wall at which point they bounce off with a random speed and direction.\nSprites Dataset We follow the same steps as in Yingzhen & Mandt (2018) to process Sprites dataset, which consists of animated cartoon characters whose clothing, hairstyle, skin color and action can be fully controlled. We use 6 variants in each of 4 attribute categories (skin colors, tops, pants and hair style) and there are 64 = 1296 unique characters in total, where 1000 of them are used for training and the rest of them are used for testing. We use 9 action categories (walking, casting spells and slashing, each with three different viewing angles.) The resulting dataset consists of video sequences with T = 8 frames of size 64× 64× 3. MUG Facial Dataset We use the MUG Facial Expression Database (Aifanti et al., 2010) for this experiment. The dataset consists of 86 subjects. Each video consists of 50 to 160 frames. To use the same network architecture for the whole video datasets in this paper, we cropped the face regions and scaled to the same size 64× 64× 3. We use six facial expressions (anger, fear, disgust, happiness,\nsadness, and surprise). To ensure there is sufficient change in the facial expression along a video sequence, we choose every other frame in the original video sequences to form training and test video sequences of length T = 10. 80% of the videos are used for training and 20% of the videos are used for testing.\nTIMIT Speech Dataset The TIMIT dataset (Garofolo, 1993) contains broadband 16k Hz of phonetically-balanced read speech. A total of 6300 utterances (5.4 hours) are presented with 10 sentences from each of 630 speakers. The data is preprocessed in the same way as in (Yingzhen & Mandt, 2018) and (Hsu et al., 2017). The raw speech waveforms are first split into sub-sequences of 200ms, and then preprocessed with sparse fast Fourier transform to obtain a 200 dimensional logmagnitude spectrum, computed every 10ms, i.e., we use T = 20 for sequence x1:T . The dimension of xt is 200.\nNow we explain the detail of the evaluation metric, equal error rate (EER), used on TIMIT dataset. Letwtest be the feature of test utterance xtest1:T andw\ntarget be the feature of test utterance xtarget1:T . The predicted identity is confirmed if the cosine similarity betweenwtest andwtarget, cos(wtest,wtarget) is greater than some threshold used in Dehak et al. (2010). The equal error rate (EER) means the false rejection rate equals the false acceptance rate (Dehak et al., 2010). In the following, we will discuss the two choices of feature wtest for evaluations of all methods,\nµc = 1\nN N∑ i=1 Eq(zc|xi1:T )[z c],\nwhich is used to evaluate the disentanglement of zc;\nµm = 1\nNT N∑ i=1 T∑ j=1 Eq(zmt |xi1:T )[z m t ],\nwhich is used to evaluate the disentanglement of zm. For more details, please refer to (Dehak et al., 2010; Yingzhen & Mandt, 2018; Hsu et al., 2017). We use the same network architecture as in Yingzhen & Mandt (2018) for a fair comparison on speech dataset. As the input dimension of speech is low, the encoder/decoder network is a 2-hidden-layer MLP with the hidden dimension 256."},{"heading":"APPENDIX D: CHOICES OF REGULARIZERS","text":"In the following, we will discuss the choice of regularizers in R-WAE. To make notations easy to read, we use density functions for corresponding distributions. In both R-WAE(GAN) and R-WAE(MMD), we use the same regularizer for D(q(zmt |zm<t), p(zmt |zm<t)). We also add a KLdivergence regularization term on zm to stabilize training. In the experiments, we assume inference model q(zc|x1:T ) is a Gaussian distribution with parameters mean µc and diagonal variance matrix σc. Inference model q(zmt |xt, zm<t) is a Gaussian distribution with parameters meanµm and diagonal variance matrix σm. For the prior distribution, we assume p(zmt |zm<t) is a Gaussian distribution with parameters mean µψm and diagonal covariance matrix σ ψ m. For regularizing the motion variables, we just use MMD without introducing any additional parameter, MMDk(q(zmt |zm<t), p(zmt |zm<t)), and we choose mixture of RBF kernel (Li et al., 2017), where RBF kernel is defined as k(x,y) = exp(−‖x−y‖ 2\n2σ2 ). With samples {z̃i} n i=1 from the posterior q(z̃ c) and samples {zi}ni=1 from the prior p(zc), MMDk(q(z̃ c), p(zc)) is defined as\nMMDk(q(z̃ c), p(zc))=\n1 n(n− 1) ∑ i6=j k(zi, zj)+ 1 n(n− 1) ∑ i 6=j k(z̃i, z̃j)− 1 n2 ∑ i,j k(z̃i, zj). (21)\nThe difference between R-WAE(MMD) and R-WAE(GAN) is how to choose metrics for the regularizer D(QZc ,PZc), where PZc is the prior distribution and QZc is the posterior distribution of the inference model.\nR-WAE(MMD) The regularizer D(QZc ,PZc) is chosen as,\nD(QZc ,PZc) = MMDkγ (Q(Zc),P (Zc)),\nwhere the scaled MMD MMDkγ (Q(Z c),P (Zc)) is chosen as\nMMDkγ (QZc ,PZc) = M̂MDkγ (QZc , PZc)\n1 + 10EP̂ [‖∇fγ(zc)‖2F ] ,\nwhere the function fγ(zc) is the kernel feature map and M̂MDkγ (QZc , PZc) is defined in the following. When we have samples {z̃ci}ni=1 from Q(Zc) and samples {zci}ni=1 from P (Zc),\nM̂MDkγ (Q(Z c), P (Zc)) =\n1 n(n− 1) ∑ i 6=j k(fγ(z c i ), fγ(z c j)) +\n1 n(n− 1) ∑ i 6=j k(fγ(z̃ c i ), fγ(z̃ c j))\n(22)\n− 1 n2 ∑ i,j k(fγ(z̃ c i ), fγ(z c j)),\nwhere the RBF kernel k is defined on scalar variables, k(x, y) = exp(−‖x−y‖ 2\n2 ). To avoid the situation where the generator gets stuck on a local optimum, we apply spectral parametrization for the weight matrix (Miyato et al., 2018). The feature map fγ is updated L steps at each iteration. To overcome posterior collapse and inference lagging, we will update the inference model per iteration of updating the decoder model for L steps during training (He et al., 2019). See Algorithm 1 for details.\nR-WAE(GAN) For the regularizer DJS(QZc ,PZc), we introduce a discriminator Dγ . The loss is as follows, L = Ezc∼p(zc)[logDγ(zc)] + Ez̃c∼q(z̃c)[log(1−Dγ(z̃c)))], (23) where p(zc) is the prior distribution and q(z̃c) is the posterior distribution of the inference model. To stabilize the training of the min-max problem in GAN-based optimization (23), a lot of stabilization techniques have been proposed (Thanh-Tung et al., 2019; Mescheder et al., 2018; Gulrajani et al., 2017; Petzka et al., 2017; Roth et al., 2017; Qi, 2017). Let samples {zc} are from the prior p(zc) and {z̃c} are from the inference posterior q(z̃c). In our R-WAE(GAN), we will adopt the regularization from Mescheder et al. (2018) and Thanh-Tung et al. (2019),\nL − λE[‖(∇Dγ)ẑc‖2], (24) where ẑc = αzc + (1− α)z̃c, α ∈ U(0, 1) and (∇Dγ)ẑc is evaluated its gradient at the point ẑc.\nAlgorithm 1 R-WAE(GAN) Input: regularization coefficient β and content prior p(zc) Goal: learn encoders qφ(zc|x1:T ) and qφ(z m t |xt, zm<t), prior pψ(zmt |zm<t), discrimi-\nnatorDγ , and decoder pθ(xt|zt), where zt = (zc, zmt ) while not converged do\nfor step 1 to L do Sample batch X = {xt} Sample {zc} from prior p(zc) and {zmt } from prior pψ Sample {z̃c, z̃mt } from encoders qφ Update discriminator Dγ and encoders qφ with loss given by (9), (10) end for Update pθ and prior pψ with loss given by (9) and (10).\nend while\nAlgorithm 2 R-WAE(MMD) Input: regularization coefficient β and content prior p(zc) Goal: learn encoders qφ(zc|x1:T ) and qφ(z m t |xt, zm<t), prior pψ(zmt |zm<t), feature\nmap fγ and decoder pθ(xt|zt), where zt = (zc, zmt ) while not converged do\nfor step 1 to L do Sample batch X = {xt} Sample {zc} from prior p(zc) and {zmt } from prior pψ Sample {z̃c, z̃mt } from encoders qφ Update feature map fγ and encoders qφ with loss given by (9), (11) end for Update pθ and prior pψ with loss given by (9) and (11).\nend while"},{"heading":"6.1 APPENDIX E: UNCONDITIONAL VIDEO GENERATION","text":"Fig. 4 provides generated samples on the SM-MNIST dataset by randomly sampling content {zc} from the prior p(zc) and motions {zm1:T } from the learned prior pψ(zmt |zm<t). The length of our\ngenerated videos is T = 100 and we only show randomly chosen videos of T = 20 to save file size. Our R-WAE(MMD) achieves the most consistent and visually best sequence even when T = 100. Samples from MoCoGAN (Tulyakov et al., 2018) usually change digit identity along the sequence. The reason is that MoCoGAN (Tulyakov et al., 2018) requires the number of actions be finite.\nFig. 5 shows unconditional video generation with T = 10 on MUG facial dataset. DS-VAE in (b) is improved by incorporating categorical latent variables. The figures should be viewed with Adobe Reader to see video.\n(a) R-WAE(GAN) (b) DS-VAE (c) MoCoGAN\nFigure 5: Unconditional video generation with T = 10 on MUG facial dataset. DS-VAE in (b) is improved by incorporating categorical latent variables. The figures should be viewed with Adobe Reader to see video."},{"heading":"APPENDIX F: LATENT MANIFOLD VISUALIZATION","text":"We encode the test data {x1:T } of SM-MNIST with T = 10 to get the content codes {zc} using our R-WAE(MMD). We visualize two-dimensional (2D) manifold of {zc} using t-SNE (Maaten & Hinton, 2008). In Fig. 6, different colors correspond to the digit identities of the latent codes {zc} of test videos on SM-MNIST. This indicates that {zc} encoded by our R-WAE(MMD) exactly captures the invariant information (digits) of the test data. The latent motion codes are sequential and cannot be visualized."},{"heading":"APPENDIX G: MODEL ARCHITECTURE AND HYPER-PARAMETERS","text":"In the inference model, we use an encoder network, defined in Fig. 7 (a) to extract latent feature ht defined in Fig.1. We use a decoder network to reconstruct x̂t from the hidden state ht, defined in Fig.1. For the discriminator Dγ in R-WAE(GAN), we use a 4-layer fully-connected neural\nnetwork (FC NN) with respective dimension (256, 256, 128, 1). For the feature map fγ with a scalar output for the RBF kernel of R-WAE(MMD), we use a 4-layer fully-connected neural network with respective dimension (256, 256, 128, 1). After encoding xt, we get extracted latent feature ht. We use Fig. 8(a) and Fig. 8(b) to infer the content variable zc and motion variables zmt . When the Gumbel latent variable is incorporated into our weakly-supervised inference model, we use Fig. 8(c) to infer the Gumbel latent variable a. The latent content variable zc and latent motion variable zmt are concatenated as input to an LSTM after an FC NN to output hidden state ht for reconstructing x̂t using the decoder. For our weakly-supervised model, the latent content variable zc, latent motion variable zmt and latent action variable a are concatenated as input to an LSTM after an FC NN to output hidden state ht for reconstructing x̂t using the decoder. We use Adam optimizer (Kingma & Ba, 2015) with β1 = 0.5 and β2 = 0.9.\nArchitecture on SM-MNIST, Sprites and TIMIT Datasets We use the same architecture on SM-MNIST and Sprites dataset, as shown in Fig. 9. The details of the parameters of the networks are provided in Fig. 9. As R-WAE(GAN) and R-WAE(MMD) have similar performance on SMMNIST and Sprites (see Sprites results in Table 4), we only provide the results and parameters of R-WAE(MMD) to save space. At each iteration of training the decoder pθ(xt|zt) and the prior pψ(z m t |zm<t), we train the encoder parameters qφ and the feature map fγ for R-WAE(MMD) with L steps. The results on SM-MNIST and Sprites datasets are evaluated after 500 epochs. On SM-MNIST dataset, we use a Bernoulli cross-entropy loss and choose L = 5. The penalty coefficients β1 and β2, are, respectively, 5 and 20. The learning rate for the decoder model is 5× 10−4 and the learning rate for the encoder is 1× 10−4. The learning rate for fγ is 1× 10−4. On Sprites dataset, we use an L2 reconstruction loss and choose L = 5 steps. The penalty coefficients β1 and β2 are, respectively, 10 and 60. The learning rate for the decoder model is 3× 10−4 and the learning rate for the encoder is 1× 10−4. The learning rate for Dγ in R-WAE(GAN) or fγ in R-WAE(MMD) is 1× 10−4. We use a decayed learning rate schedule on both datasets. After 50 epochs, we decrease all learning rates by a factor of 2 and after 80 epochs decrease further by a factor of 5. On TIMIT speech dataset, we use the same encoder and decoder architecture as that of DS-VAE. The dimension of hidden states is 256 and the dimensions of zc and zmt are both 16.\nResBlock1 down 64*3*3 self-attention\nResBlock2 down 128*3*3 ResBlock3 down 256*3*3 ResBlock4 down 512*3*3\nResBlock5 down 1024*3*3 Reshape output to (N, 1024× 2× 2)\nFC NN\nTable 5: Encoder Network Architecture.\nFC NN and Reshape input to (N, 2048, 2, 2) ResBlock1 up 1024*3*3 ResBlock2 up 512*3*3 ResBlock3 up 256*3*3 ResBlock4 up 128*3*3\nself-attention ResBlock5 up 64*3*3\nConv 3*3*3, activation=sigmoid\nTable 6: Decoder Network Architecture.\nTable 7: Encoder Network Architecture.\nFigure 10: Network parameters on encoder network and decoder network on MUG facial dataset. We adopt ResBlock down and up from Brock et al. (2019). The dimensions of zc, zmt , ht, a are 150, 16, 180 and 6 respectively. The batch size on MUG facial dataset are 30 and the length of video sequence for training is T = 8.\nArchitecture on MUG Facial Dataset The details of the architecture parameters of the networks for MUG facial dataset are provided in Fig. 9. The results on MUG facial dataset are evaluated after 800 epochs. For the regularizer DKL(qφ(a|x1:T , zm1:T ), p(a)), we choose the coefficient of this categorical regularizer to be 50. We use an L2 reconstruction loss and choose L = 5 steps. For R-WAE(MMD), the penalty coefficients β1 and β2 are, respectively, 10 and 50. For R-WAE(GAN), the coefficients β1 and β2 of the penalties are, respectively, 5 and 60. The learning rate for the decoder model is 5 × 10−4 and the learning rate for the encoder is 2 × 10−4. The learning rate for Dγ in R-WAE(GAN) or fγ in R-WAE(MMD) is 2× 10−4. We use the same decayed learning rate schedule as described on SM-MNIST and Sprites datasets. This architecture can be applied to improve the compression rate (?)."},{"heading":"APPENDIX H: ADDITIONAL RESULTS ON AUDIO DATA","text":"Swapping Static and Dynamic Factors on Audio Data Here we present results of swapping static and dynamic factors of given audio sequences. Results are given in Figure 11. Each heatmap subplot is of dimension 80 × 20 and visualizes the spectrum of 200ms of an audio clip, in which the mel-scale filter bank features are plotted in the frequency domain (x-axis represents temporal domain with 20 timesteps and y-axis is the value of frequencies). We collect these heatmaps in a matrix where the static factors in a row are kept the same and each column shares the same dynamic factor. It can be observed that in each column, the linguistic phonetic contents as reflected by the formants along x-axis are kept almost the same after swapping. Likewise, the timbres are reflected as the harmonics in the spectrum plot. This can be concluded by observing that the horizontal light stripes which represents the harmonics are kept consistent in a row. Moreover, we perform\nidentity verification experiment as conducted in DS-VAE (Yingzhen & Mandt, 2018). Similar to cross reconstruction, zcfemale and z c male (or f\nfemale and fmale in DS-VAE) are swapped for two sequences {xfemale} and {xmale}. By an informal listening test of the original-swapped speech sequence pairs, we confirm that the speech content is preserved and identity is transferred (i.e. female voice usually has higher frequency).\nAPPENDIX I: ADDITIONAL RESULTS ON A MOVING-SHAPE VIDEO DATA\nGeneration Results on Moving Shapes We report results on a Moving-Shape dataset in Table 9 and Fig. 12. The Moving-Shape synthetic dataset was introduced in Balaji et al. (2018) which has 5 control parameters: shape type (e.g. triangle and square), size (small and large), color (e.g. white and red), motion type (e.g. zig-zag, straight line and diagonal) and motion direction. In Table 9, TFGAN (Balaji et al., 2018) encoder and decoder architectures are considered less expressive compared with BigGAN (Brock et al., 2019) architectures. Similar to results in Table 2, with more complex and expressive architecture, learning disentangled representation is harder. The results in Table 9 and Fig. 12 demonstrate that R-WAE produces better disentanglement and generation performance than DS-VAE both quantitatively and qualitatively. Qualitative difference of fixing zm and sampling zc for DS-VAE and R-WAE is not that obvious and thus not shown."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Unsupervised representation learning is an important research topic in machine learning. It embeds high-dimensional sensory data such as images and videos into a low-dimensional latent space in an unsupervised learning framework, aiming at extracting essential data variation factors to help downstream tasks such as classification and prediction (Bengio et al., 2013). In the last several years, disentangled representation learning, which further separates the latent embedding space into exclusive explainable factors such that each factor only interprets one of semantic attributes of sensory data, has received a lot of interest and achieved many empirical successes on static data such as images (Chen et al., 2016; Higgins et al., 2017; Dupont, 2018; Chen et al., 2018; Rubenstein et al., 2018b;a; Kim & Mnih, 2018). For example, the latent representation of handwritten digits can be disentangled into a content factor encoding digit identity and a style factor encoding handwriting style.\\nIn spite of successes on static data, only a few works have explored unsupervised representation disentanglement of sequential data due to the challenges of developing generative models of sequential ∗Equal contribution. †Part of his work was done before joining Tencent. ‡His work was done before joining Amazon. §His work was done before joining Texas A&M University.\\ndata. Learning disentangled representations of sequential data is important and has many applications. For example, the latent representation of a smiling-face video can be disentangled into a static part encoding the identity of the person (content factor) and a dynamic part encoding the smiling motion of the face (motion factor). The disentangled representation of the video can be potentially used for many downstream tasks such as classification, retrieval, and synthetic video generation with style transfer. Most of previous unsupervised representation disentanglement models for static data heavily rely on the KL-divergence regularization in a VAE framework (Higgins et al., 2017; Dupont, 2018; Chen et al., 2018; Kim & Mnih, 2018), which has been shown to be problematic due to matching individual instead of aggregated posterior distribution of the latent code to the same prior (Tolstikhin et al., 2018; Rubenstein et al., 2018b;a). Therefore, extending VAE or recurrent VAE (Chung et al., 2015) to disentangle sequential data in a generative model framework (Hsu et al., 2017; Yingzhen & Mandt, 2018) is not ideal. In addition, recent research (Locatello et al., 2019) has theoretically shown that it is impossible to perform unsupervised disentangled representation learning without inductive biases on both models and data, especially on static data. Fortunately, sequential data such as videos often have clear inductive biases for the disentanglement of content factor and motion factor as mentioned in (Locatello et al., 2019). Unlike static data, the learned static and dynamic factors of sequential data are not exchangeable.\\nIn this paper, we propose a recurrent Wasserstein Autoencoder (R-WAE) to learn disentangled representations of sequential data. We employ a Wasserstein metric (Arjovsky et al., 2018; Gulrajani et al., 2017; Bellemare et al., 2017) induced from the optimal transport between model distribution and the underlying data distribution, which has some nicer properties (for e.g., sum invariance, scale sensitivity, applicable to distributions with non-overlapping supports, and better out-of-sample performance in the worst-case expectation (Esfahani & Kuhn, 2018)) than the KL divergence in VAE (Kingma & Welling, 2014) and β-VAE (Higgins et al., 2017). Leveraging explicit inductive biases in both sequential data and model, we encode an input sequence into two parts: a shared static latent code and a dynamic latent code, and sequentially decode each element of the sequence by combining both codes. We enforce a fixed prior distribution for the static code and learn a prior for the dynamic code to ensure the consistency of the sequence. The disentangled representations are learned by separately regularizing the posteriors of the latent codes with their corresponding priors.\\nOur main contributions are summarized as follows: (1) We draw the first connection between minimizing a Wasserstein distance and maximizing mutual information for unsupervised representation disentanglement of sequential data from an information theory perspective; (2) We propose two sets of effective regularizers to learn the disentangled representation in a completely unsupervised manner with explicit inductive biases in both sequential data and models. (3) We incorporate a relaxed discrete latent variable to improve the disentangled learning of actions on real data. Experiments show that our models achieve state-of-the-art performance in both disentanglement of static and dynamic latent representations and unconditional video generation under the same settings as baselines (Yingzhen & Mandt, 2018; Tulyakov et al., 2018).\"\n },\n {\n \"heading\": \"2 BACKGROUND AND RELATED WORK\",\n \"text\": \"Notation Let calligraphic letters (i.e. X ) be sets, capital letters (i.e. X) be random variables and lowercase letters be their values. Let D(PX , PG) be the divergence between the true (but unknown) data distribution PX (density p(x)) and the latent-variable generative model distribution PG specified by a prior distribution PZ (density p(z)) of latent variable Z. Let DKL be KL divergence, DJS be Jensen-Shannon divergence and MMD be Maximum Mean Discrepancy (MMD) (Gretton et al., 2007).\\nOptimal Transport Between Distributions The optimal transport cost inducing a rich class of divergence between the distribution PX and the distribution PG is defined as follows,\\nW (PX , PG):= inf Γ∼P(X∼PX ,Y∼PG) E(X,Y )∼Γ[c(X,Y )], (1)\\nwhere c(X,Y ) is any measurable cost function and P(X ∼ PX , Y ∼ PG) is the set of joint distributions of (X, Y) with respective marginals PX and PG.\\nComparison between WAE (Tolstikhin et al., 2018) and VAE (Kingma & Welling, 2014) Instead of optimizing over all couplings Γ between two random variables in X , Bousquet et al.\\n(2017); Tolstikhin et al. (2018) show that it is sufficient to find Q(Z|X) such that the marginal Q(Z) := EX∼PX [Q(Z|X)] is identical to the prior P (Z), as given in the following definition,\\nDefinition 1. For any deterministic PG(X|Z) and any function G : Z → X ,\\nW (PX , PG) = inf Q:QZ=PZ EPXEQ(Z|X)[c(X,G(Z))]. (2)\\nDefinition 1 leads to the following loss DWAE of WAE based on a Wasserstein distance,\\ninf Q(Z|X) EPXEQ(Z|X)[c(X,G(Z))] + β D(QZ , PZ), (3)\\nwhere the first term is data reconstruction loss, and the second one is a regularizer that forces the posterior QZ = ∫ Q(Z|X)dPX to match the prior PZ (Adversarial autoencoder (AAE) (Makhzani et al., 2015) shares a similar idea to WAE). In contrast, VAE has a different regularizer EX [DKL(Q(Z|X), PZ))] enforcing the latent posterior distribution of each input to match PZ . In (Rubenstein et al., 2018a;b), it is shown that WAE has better disentanglement than β-VAE (Higgins et al., 2017) on images, which inspires us to design a new representation disentanglement framework for sequential data with several innovations.\\nUnsupervised disentangled representation learning Several generative models have been proposed to learn disentangled representations of sequential data (Denton et al., 2017; Hsu et al., 2017; Yingzhen & Mandt, 2018; Hsieh et al., 2018; Sun et al., 2018; Tulyakov et al., 2018). FHVAE in (Hsu et al., 2017) is a VAE-based hierarchical graphical model with factorized Gaussian priors and only focuses on speech or audio data. Our R-WAE employing a more powerful recurrent prior can be applied to both speech and video data. The models in (Sun et al., 2018; Denton et al., 2017; Hsieh et al., 2018) are based on the first several elements of a sequence to design disentanglement architectures for future sequence predictions.\\nIn terms of representation learning by mutual information maximization, our work empirically demonstrates that explicit inductive biases in data and model architecture are necessary to the success of learning meaningful disentangled representations of sequential data, while the works in (Locatello et al., 2019; Poole et al., 2019; Tschannen et al., 2020; Ozair et al., 2019) are about general representation learning, especially on static data.\\nThe most related works to ours are MoCoGAN (Tulyakov et al., 2018) and DS-VAE (Yingzhen & Mandt, 2018), which have the ability to disentangle variant and invariant parts of sequential data and perform unconditional sequence generation. Tulyakov et al. (2018) is a GAN-based model that can be only applied to the setting in which the number of motions is finite, and cannot encode the latent representation of sequences. Yingzhen & Mandt (2018) provides a disentangled sequential autoencoder based on VAE (Kingma & Welling, 2014). Training VAE is equivalent to minimizing a lower bound of the KL divergence between empirical data distribution and generated data distribution, which has been shown to produce inferior disentangled representations of static data than generative models employing the Wasserstein metric (Rubenstein et al., 2018a;b).\"\n },\n {\n \"heading\": \"3 PROPOSED APPROACH: DISENTANGLED RECURRENT WASSERSTEIN AUTOENCODER (R-WAE)\",\n \"text\": \"Given a high-dimensional sequence x1:T , our goal is to learn a disentangled representation of timeinvariant latent code zc and time-variant latent code zmt , along the sequence. Let zt = (z\\nc, zmt ) be the latent code of xt. Let Xt, Zt, Zc and Zmt be random variables with realizations xt, zt, z\\nc and zmt respectively, and denote D = X1:T . To achieve this goal, we define the following probabilistic generative model by assuming Zmt and Z c are independent,\\nP (X1:T , Z1:T ) = P (Z c) T∏ t=1 Pψ(Z m t |Zm<t)Pθ(Xt|Zt), (4)\\nwhere P (Z1:T ) = P (Zc) ∏T t=1 Pψ(Z m t |Zm<t) is the prior in which Zt = (Zc, Zmt ), and the decoder model Pθ(Xt | Zt) is a Dirac delta distribution. In practice, P (Zc) is chosen as N (0, I)\\nand Pψ(Zmt |Zm<t) = N (µψ(Zm<t),σ2ψ(Zm<t)), µψ and σψ are parameterized by Recurrent Neural Networks (RNNs). The inference model Q is defined as\\nQφ(Z c, Zm1:T |X1:T ) = Qφ(Zc|X1:T ) T∏ t=1 Qφ(Z m t | Zm<t, Xt), (5)\\nwhereQφ(Zc|X1:T ) andQφ(Zmt | Zm<t, Xt) are also Gaussian distributions parameterized by RNNs. The structures of the generative model (4) and the inference model (5) are provided in Fig. 1.\"\n },\n {\n \"heading\": \"3.1 R-WAE MINIMIZES A PENALIZED FORM OF A WASSERSTEIN DISTANCE\",\n \"text\": \"The optimal transport cost between two distributions PD and PG with respective sequential variables X1:T (X1:T∼PD) and Y1:T (Y1:T∼PG) is given as follows,\\nW (PD, PG) := inf Γ∼P(X1:T∼PD,Y1:T∼PG) E(X1:T ,Y1:T )∼Γ[c(X1:T , Y1:T )], (6)\\nP(X1:T∼PD,Y1:T∼PG) is a set of all joint distributions with marginals PD and PG respectively. When we choose c(x,y) = ‖x−y‖2 (2-Wasserstein distance) and c(X1:T , Y1:T ) = ∑ t ‖Xt−Yt‖2 by linearity, it is easy to derive the optimal transport cost for disentangled sequential variables.\\nTheorem 1. With deterministic P (Xt|Zt) and any function Yt = G(Zt), we derive\\nW (PD, PG) = inf Q:QZc=PZc ,QZm\\n1:T =PZm 1:T ∑ t EPDEQ(Zt|Z<t,Xt)[c(Xt, G(Zt))], (7)\\nwhere QZ1:T = QZcQZm1:T is the marginal distribution of Z1:T when X1:T ∼ PD and Z1:T ∼ Q(Z1:T |X1:T ) and PZ1:T is the prior. Based on the assumptions, we have an upper bound,\\nW (PD, PG) ≤ inf Q∈S ∑ t EPDEQ(Zt|Z<t,Xt)[c(Xt, G(Zt))], (8)\\nwhere the subset S is S = {Q : QZc = PZc , QZm1 = PZm1 , QZmt |Zm<t = PZmt |Zm<t} .\\nIn practice, we have the following objective function of our proposed R-WAE based on Theorem 1,\\nT∑ t=1 EQ(Zt|Z<t,Xt)[c(Xt, G(Zt))] + β1 D(QZc , PZc) + β2 T∑ t=1 D(QZmt |Zm<t , PZmt |Zm<t), (9)\\nwhere D is an divergence between two distributions, and the second and third terms are, respectively, regularization terms for Zc and Zmt . In the following, we will present two different approaches to calculating the regularization terms in section 3.2 and 3.3. Because we cannot straightforwardly estimate the marginals Qφ(Zc) and Qφ(Zmt |Zm<t), we cannot directly use KL divergence in the two regularization terms, but we can optimize the RHS of (9) by likelihood-free optimizations (Gretton et al., 2007; Goodfellow et al., 2014; Nowozin et al., 2016; Arjovsky et al., 2018) when samples from all distributions are available.\\n3.2 DJS PENALTY FOR Zc AND MMD PENALTY FOR Zm\\nThe prior distribution of Zc is chosen as a multivariate unit-variance Gaussian, N (0, I). We can choose penalty DJS(QZc ,PZc) for Zc and apply min-max optimization by introducing a discriminator Dγ (Goodfellow et al., 2014). Instead of performing optimizations in high-dimensional input data space, we move the adversarial optimization to the latent representation space of the content with a much lower dimension. Because the prior distribution of {Zmt } is dynamically learned during training, it is challenging to use DJS to regularize {Zmt } with a discriminator, which will induce a third minimization within a min-max optimization. Therefore, we use MMD to regularize {Zmt } as samples from both distributions are easy to obtain (dimension of zmt is less than 20 in our experiments on videos). With a kernel k, MMDk(Q,P ) is approximated by samples from Q and P (Gretton et al., 2007). The regularization terms can be summarized as follows and we call the resulting model R-WAE(GAN) (see Algorithm 1 in Appendix for details):\\nD(QZc ,PZc) = DJS(QZc ,PZc); D(QZmt |Zm<t ,PZmt |Zm<t) = MMDk(QZmt |Zm<t ,PZmt |Zm<t). (10)\\n3.3 SCALED MMD PENALTY FOR Zc AND MMD PENALTY FOR Zm\\nMMD with neural kernels for generative modeling of real-world data (Li et al., 2017; Bińkowski et al., 2018; Arbel et al., 2018) motivates us to use only MMD as regularization in Eq. (9),\\nD(QZc ,PZc) = MMDkγ (QZc ,PZc); D(QZmt |Zm<t ,PZmt |Zm<t) = MMDk(QZmt |Zm<t ,PZmt |Zm<t), (11)\\nwhere kγ is a parametrized family of kernels (Li et al., 2017; Bińkowski et al., 2018; Arbel et al., 2018) defined as kγ(x,y) = k(fγ(x), fγ(y)) and fγ(x) is a feature map, which is more expressive and used for Zc with equal or higher dimension than Zmt . The details of optimizing the first term MMDkγ (QZc ,PZc) in Eq. (11) is provided in Appendix D based on scaled MMD (Arbel et al., 2018), a principled and stable technique for training MMD-based critic. We call the resulting model R-WAE(MMD) (see Algorithm 2 in Appendix for details).\"\n },\n {\n \"heading\": \"3.4 WEAKLY SUPERVISED DISENTANGLEMENT WITH A KNOWN NUMBER OF ACTIONS\",\n \"text\": \"When the number of actions (motions) in sequential data, denoted by A, is available, we incorporate a categorical latent variable a (a one-hot vector whose dimension is A) to enhance the disentanglement of the dynamic latent codes of the motions. The inference model for a is designed as qφ(a|x1:T , zm1:T ). Intuitively, the action is inferred from the motion sequence to recognize its label. Learning such a categorical distribution requires a continuous relaxation of the discrete random variable in order to backpropagate its gradient. Let α1, · · · , αA be the class probabilities, we can obtain a sample ã = (y1, · · · , yA) from its continuous relaxation by first sampling g = (g1, · · · , gA) with gj ∼ Gumbel(0, 1) and then applying transformation ãj = exp((logαj +gj)/τ) ∑ i exp((logαi+gi)/τ), where τ is a temperature parameter controlling the approximation. To learn the categorical distribution using the reparameterization trick, we use a regularizer DKL(qφ(ã|x1:T , zm1:T ), p(ã)), where p(ã) is a uniform Gumbel-Softmax prior distribution (Jang et al., 2016; Maddison et al., 2016). The motion variable is augmented as zRt = (z m t ,a), and learning joint continuous and discrete latent representation of image data has been extensively discussed in (Dupont, 2018) (see Fig. 1(c,d) for illustrations).\"\n },\n {\n \"heading\": \"4 ANALYZING R-WAE FROM AN INFORMATION THEORY PERSPECTIVE\",\n \"text\": \"Theorem 2. If the mutual information (MI) between Z1:T and X1:T is defined in terms of the inference model Q, I(Z1:T ;X1:T ) = EQ(X1:T ,Z1:T )[logQ(Z1:T |X1:T ) − logQ(Z1:T )], where Q(X1:T , Z1:T ) = Q(Z1:T |X1:T )P (X1:T ) and Q(Z1:T ) = ∑ X1:T\\nQ(X1:T , Z1:T ), we have a lower bound:\\nI(Z1:T ;X1:T )≥ T∑ t=1 EPD [ EQφ [logPθ(Xt |Zt)−logP (D)]−EQφ(Zc|X1:T )[logQφ(Z c)−logP (Zc)] ]\\n− T∑ t=1 EPD [ EQφ(Zmt |Zm<t,Xt)[logQφ(Z m t |Zm<t)−logP (Zmt |Zm<t) ] . (12)\\nTheorem 2 shows that R-WAE maximizes a lower bound of the mutual information between X1:T and Z1:T , which theoretically guarantees that R-WAE learns semantically meaningful latent representations of input sequences. With constant removed, the RHS of (9) and (12) are the same if D is KL divergence. In spite of being theoretically important, Theorem 2 with KL divergence cannot be directly used for the regularization terms of R-WAE in practice, because we cannot straightforwardly estimate the marginals Qφ(Zc) and Qφ(Zmt |Zm<t) as discussed previously. From Eq. (9) and (12), we can obtain the following theorem.\\nTheorem 3. When its distribution divergence is chosen as KL divergence, the regularization terms in Eq. (9) jointly minimize the KL divergence between the inference model Q(Z1:T |X1:T ) and the prior model P (Z1:T ) and maximize the mutual information between X1:T and Z1:T ,\\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc), (13) KL(Q(Zmt |Zm<t)||P (Zmt |Zm<t))=EpD [KL(Q(Zmt |Zm<t, X1:T )||P (Zmt |Zm<t)]− I(X1:T ;Zmt |Zm<t),\\nwhere the mutual information is defined in terms of the inference model as in Theorem 2.\\nTheorem 3 shows that, even if adopting KL divergence, the regularization in the loss of R-WAE still improves over the one in vanilla VAE, which only has the first term in the RHS of Eq. (13). The two mutual information terms explicitly enforce mutual information maximization between input data and unexchangeable disentangled latent representations, Zc and Zmt . Therefore, R-WAE is superior to recurrent VAE (DS-VAE).\"\n },\n {\n \"heading\": \"5 EXPERIMENTS\",\n \"text\": \"We conduct extensive experiments on four datasets to quantitatively and qualitatively validate our methods. The baseline methods for comparisons are DS-VAE (Yingzhen & Mandt, 2018) and MoCoGAN (Tulyakov et al., 2018). We train our models on Stochastic moving MNIST (SMMNIST), Sprites, and TIMIT datasets under a completely unsupervised setting. The number of actions (motions) is used as prior information for all methods on MUG facial dataset. The detailed descriptions of datasets, architectures, and hyperparameters are provided in Appendix C, D, and G, respectively.\"\n },\n {\n \"heading\": \"5.1 QUALITATIVE RESULTS ON DISENTANGLEMENT\",\n \"text\": \"We encode two original videos on the first and fourth row in Fig. 2 and generate videos on the second and third row by swapping corresponding {zc} and {zm1:T } between videos for style transfer. Fig. 2(left) shows that even testing on the long sequence (trained with T = 100), our R-WAE can disentangle content and motions exactly. In Fig. 2(right), we do the same swapping on Sprites. We can see that the generated swapped videos have exact same appearances and actions as the corresponding original ones. On the MUG\\ndataset, it is interesting to see that we can swap different motions between different persons. PPPPPPPMethods Datasets EER zc = 16 ↓ zm = 16 ↑\\nFHVAE 5.06% 22.77% DS-VAE 5.64% 19.20% R-WAE 4.73% 23.41%\\nTable 1: EER on TIMIT speech dataset under the same dimension setting of segment-level zc and sequence-leve zm for FHVAE (Hsu et al., 2017), DS-VAE (full q) (Yingzhen & Mandt, 2018) and R-WAE(MMD), respectively. Small EER is better for zc and larger EER is better for zm.\\nPPPPPPPMethods Datasets Sprites SM-MNIST\\nactions content digits DS-VAE(S) 8.11% 3.98% 3.31% R-WAE(S) 5.83% 2.45% 1.78% DS-VAE(C) 10.37% 4.86% 4.26% R-WAE(C) 7.72% 3.31% 2.84%\\nTable 2: Comparison of averaged classification errors. On Sprites dataset, fix one encoded attribute and randomly sample others. On SM-MNIST dataset, we fix the encoded zc and randomly sample the motion sequence from the learned prior pψ(zmt |zm<t). We cannot quantitatively verify the motion disentanglement on SM-MNIST.\"\n },\n {\n \"heading\": \"5.2 QUANTITATIVE RESULTS\",\n \"text\": \"SM-MNIST and Sprites Datasets We quantitatively evaluate the disentanglement of our RWAE(MMD). In Table 2, \\\"S\\\" denotes a simple encoder/decoder architecture, where the encoders in both our model and DS-VAE (Yingzhen & Mandt, 2018) only use 5 layers of convolutional and deconvolutional networks adopted from DS-VAE (Yingzhen & Mandt, 2018). \\\"C\\\" denotes a complex encoder/decoder architecture where we use Ladder network (Sønderby et al., 2016; Zhao et al., 2017) and ResBlock (He et al., 2016), provided in Appendix E. On SM-MNIST, we get the labeled latent codes {zc} of test videos {x1:T } with T = 10 and randomly sample motion variables {zm1:T } to get labeled new samples. We pretrain a classifier and predict the accuracy on these labeled new samples. The accuracy on SM-MNIST dataset is evaluated on 10000 test samples. On Sprites, the labels of each attribute(skin colors, pants, hair styles, tops and pants) are available. We get the latent codes by fixing one attribute and randomly sample other attributes. We train a classifier for each attribute and evaluate the disentanglement of each attribute. The accuracy is based on 296 × 9 test data. Both DS-VAE and R-WAE(MMD) have extremely high accuracy (99.94%) when fixing hair style attribute, which is not provided in Table 2 due to space limit. As R-WAE(GAN) and R-WAE(MMD) have similar performance on these datasets, we only provide the results and parameters of R-WAE(MMD) to save space. There are two interesting observations in Table 2. First, the simple architecture has better disentanglement than the complex architecture overall. The reason is that the simple architecture has sufficient ability to extract features and generate clear samples to be recognized by the pretrained classifiers. But the simple architecture cannot generate high-quality samples when applied to real data. Second, our proposed R-WAE(MMD) achieve better disentanglement than DS-VAE (Yingzhen & Mandt, 2018) on both corresponding architectures. The attributes within content latent variables are independent, our model can further disentangle the factors. Compared to DS-VAE, these results demonstrate the advantages of R-WAE with implicit mutual information maximization terms. Due to space limit, we also include similar comparisons on a new Moving-Shape dataset in Appendix I. As the number of possible motions in SM-MNIST is infinite and random, we cannot evaluate the disentanglement of motions by training a classifier. We fix the encoded motions {zm1:T } and randomly sample content variables {zc}. We also randomly sample a motion sequence {zm1:T } and randomly sample contents {zc}. We manually check the motions of these samples and almost all have the same corresponding motion even though the sequence is long (T = 100). TIMIT Speech Dataset We quantitatively compare our R-WAE with FHVAE and DS-VAE on the speaker verification task under the same setting as (Hsu et al., 2017; Yingzhen & Mandt, 2018). The evaluation metric is based on equal error rate (EER), which is explained in detail in Appendix C. The lower EER on zc encoding the timbre of speakers is better and the higher EER on zm encoding linguistic content is better. From Table 1, our model can disentangle zc and zm well. We can see that our R-WAE(MMD) has the best EER performance on both content attribute and motion attribute. In Appendix H we show by style transfer experiments that the learned dynamic factor encodes semantic content which is comparable to DS-VAE. MUG Facial Dataset We quantitatively evaluate the disentanglement and quality of generated samples. We train a 3D classifier on MUG facial dataset with accuracy 95.11% and Inception Score 5.20 on test data (Salimans et al., 2016). We calculate Inception score, intra-entropy H(y|v), where y is the predicted label and v is the generated video, and inter-entropy H(y) (He et al., 2018). For a comprehensive quantitative evaluation, Frame-level FID score, introduced by (Heusel et al., 2017), is also provided. From Table 2, our R-WAE(MMD) and R-WAE(GAN) have higher accuracy, which\\nmeans the categorical variable best captures the actions, which indicates our models achieve the best disentanglement. In table 2, the Inception score of R-WAE(GAN) is very close to Inception Score of the exact test data, which means our models have the best sample quality. Our proposed R-WAE(GAN) and R-WAE(MMD) have the best frame-level FID scores, compared with DS-VAE and MoCoGAN. The orignal DS-VAE (DS-VAE(NA)) (Yingzhen & Mandt, 2018) without leveraging the number of actions performs worst, which shows that incorporating the number of actions as prior information does enhance the disentanglement of actions.\"\n },\n {\n \"heading\": \"5.3 UNCONDITIONAL VIDEO GENERATION\",\n \"text\": \"SM-MNIST dataset Fig. 4 in Appendix E provides generated samples on the SM-MNIST dataset by randomly sampling content {zc} from the prior p(zc) and motions {zm1:T } from the learned prior pψ(z m t |zm<t). The length of our generated videos is T = 100 and we only show randomly chosen videos of T = 20 to save file size. Our R-WAE(MMD) achieves the most consistent and visually best sequence even when T = 100. Samples from MoCoGAN (Tulyakov et al., 2018) usually change digit identity along the sequence. The reason is that MoCoGAN (Tulyakov et al., 2018) requires the number of actions be finite. Our generated Sprites videos also have the best results but are not provided due to page limit.\\nMUG Facial Dataset Fig. 3 and Fig. 5 in Appendix E show generated samples on MUG dataset by randomly sampling content {zc} from the prior p(zc) and motions zRt = (a, zmt ) from the categorical prior p(a) (latent action variable a is a one-hot vector with dimension 6) and the learned prior pψ(zmt |zm<t). We show generated videos of length T = 10. DS-VAE (Yingzhen & Mandt, 2018) used the same structure as ours. Fig. 5 shows that DS-VAE (Yingzhen & Mandt, 2018) and\\nMoCoGAN (Tulyakov et al., 2018) have blurry beginning frames {xt} and even more blurry frames as time t evolves. While our R-WAE(GAN) has much better frame quality and more consistent video sequences. To have a clear comparison among all three methods, we show the samples at time step T = 10 in Fig. 3, and we can see that DS-VAE has very blurry samples with large time steps.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"In this paper, we propose recurrent Wasserstein Autoencoder (R-WAE) to learn disentangled representations of sequential data based on the optimal transport between distributions with sequential variables. Our theoretical analysis shows that R-WAE simultaneously maximizes the mutual information between input sequential data and different disentangled latent factors. Experiments on a variety of datasets demonstrate that our models achieve state-of-the-art results on the disentanglement of static and dynamic latent representations and unconditional video generation. Future research includes exploring our framework in self-supervised learning and conditional settings for text-to-video and video-to-video synthesis.\\nAcknowledgement Jun Han thanks Dr. Chen Fang at Tencent for insightful discussions and Prof. Qiang Liu at UT Austin for invaluable support.\"\n },\n {\n \"heading\": \"APPENDIX FOR RECURRENT WASSERSTEIN AUTOENCODER\",\n \"text\": \"\"\n },\n {\n \"heading\": \"APPENDIX A: PROOF OF THEOREM 1\",\n \"text\": \"In the following, we provide the proof of Theorem 1.\\nTheorem 1 For PG defined with deterministic PG(X|Z) and any function Y = G(Z),\\nW (PD, PG) = inf Q:QZc=PZc ,QZm\\n1:T =PZm 1:T T∑ t=1 EPDEQ(Zt|Xt)[c(Xt, G(Zt))], (14)\\nwhere QZ1:T is the marginal distribution of Z1:T when X1:T ∼ PD and Z1:T ∼ Q(Z1:T |X1:T ) and PZ1:T is the prior. Based on the assumptions, the constraint set is equivalent to\\nW (PD, PG) ≤ inf Q∈S ∑ t EPDEQ(Zt|Xt)[c(Xt, G(Zt))], (15)\\nwhere the set S = {Q : QZc = PZc , QZm1 = PZm1 , QZmt |Zm<t = PZmt |Zm<t}.\\nProof: Consider the sequential random variables D = X1:T and Y1:T , the optimal transport between the distribution for D = X1:T and the distribution for Y1:T induces a rich class of divergence,\\nW (PD, PG) := inf Γ∼P(X1:T∼PD,Y1:T∼PG) E(X1:T ,Y1:T )∼Γ[c(X1:T , Y1:T )] (16)\\nwhere P(X1:T ∼ PD, Y1:T ∼ PG) is a set of all joint distributions of (X1:T , Y1:T ) with marginals PD and PG, respectively.\\nWhen we choose c(x,y) = ‖x − y‖2, c(X1:T , Y1:T ) = ∑ t ‖Xt − Yt‖2 by linearity. It is easy to derive the optimal transport for distributions with sequential random variables,\\nW (PD, PG) = inf Q:QZ1:T =PZ1:T ∑ t EPDEQ(Zt|Xt)[c(Xt, G(Zt))]. (17)\\nBased on our assumption, the marginal distribution of the inference model satisfies Q(Z1, · · · , ZT ) = Q(Zc)Q(Zm1 , · · · , ZmT ) = Q(Zc) ∏ t Q(Zmt |Zm<t). (18) The prior distribution P (Z1, · · · , ZT ) satisfies P (Z1, · · · , ZT ) = P (Zc)P (Zm1 , · · · , ZmT ) = P (Zc)\\n∏ t P (Zmt |Zm<t). (19)\\nSince the set S is a subset of {Q : QZ1:T = PZ1:T }, we derive the inequality (15).\"\n },\n {\n \"heading\": \"APPENDIX B: PROOF OF THEOREM 2\",\n \"text\": \"In the following, we provide the proof of Theorem 2. To make the notations easy to read, we use the density functions of corresponding distributions.\\nThe joint generative distribution is p(x1:T , z1:T ) = pψ(z1:T )pθ(x1:T |z1:T ), where pψ(z1:T ) is the prior distribution and pθ(x1:T |z1:T ) is the decoder model. And the corresponding joint inference distribution is qφ(x1:T , z1:T ) = pD(x1:T )qφ(z1:T | x1:T ). If the MI between z1:T and x1:T is defined in terms of the inference model q, we have the following lower bound with step-by-step derivations:\\nI(z1:T ;x1:T )= Eq(x1:T ,z1:T )[log qφ(z1:T |x1:T ) qφ(z1:T ) ] (20)\\n= Eq(x1:T ,z1:T )[DKL(qφ(z1:T |x1:T ), p(z1:T |x1:T ))+log p(z1:T |x1:T )−log qφ(z1:T )] ≥ EpD [Eq(z1:T |x1:T )[log p(x1:T |z1:T ) + log p(z1:T )− log qφ(z1:T )−log p(D)]]\\n≥ T∑ t=1 Ep(D) [ Eqφ(zt|xt)[log pθ(xt|zt)] ] −Ep(D)[Eqφ(zt|xt)[log qφ(z c)−log p(zc)] ]\\n− T∑ t=1 Ep(D) [ Eqφ(zmt |xt)[log qφ(z m t |zm<t)− log p(zmt |zm<t) + log p(D) ] ,\\nwhere we use Bayesian’s rule p(z1:T |x1:T ) = pθ(x1:T |z1:T )p(z1:T )p(D) . Maximizing the MI between z1:T and x1:T achieves state-of-the-art results in disentangled latent representation by using different regularizers for the static and dynamical latent variable with different priors (Hjelm et al., 2018). In practice, incorporating the mutual information I(zmt ,xt) between element xt and motion z m t might facilitate the disentanglement of the dynamical latent variable zmt .\\nTheorem 3 When its distribution divergence is chosen as KL divergence, the regularization terms in Eq. (9) jointly minimize the KL divergence between the inference model Q(Z1:T |X1:T ) and the prior model P (Z1:T ) and maximize the mutual information between X1:T and Z1:T ,\\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc). KL(Q(Zmt |Zm<t)||P (Zmt |Zm<t)) = EpD [KL(Q(Zmt |Zm<t, X1:T )||P (Zmt |Zm<t)]− I(X1:T ;Zmt |Zm<t).\\nProof: Denote XD = X1:T . As in the proof of Theorem 2, the mutual information between Z1:T and X1:T is defined in terms of the inference model Q, and we use the density functions of corresponding distributions to make the notations easy to read. Thus,\\nQ(Z1:T ) = EpDq(z1:T |x1:T ).\\nAccording to the definition of mutual information, we have\\nI(X1:T ;Z c) = EpD ∑ zc pD(x1:T )q(z c|x1:T ) log pD(x1:T )q(z c|x1:T ) pD(x1:T )q(zc)\\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) q(zc)\\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) p(zc) − EpD ∑ zc q(zc|x1:T ) log q(zc) p(zc)\\n= EpD ∑ zc q(zc|x1:T ) log q(zc|x1:T ) p(zc) − ∑ zc q(zc) log q(zc) p(zc) = EpD [KL(Q(Zc|X1:T )||P (Zc))]−KL(Q(Zc)||P (Zc))\\nTherefore,\\nKL(Q(Zc)||P (Zc)) = EpD [KL(Q(Zc|X1:T )||P (Zc))]− I(X1:T ;Zc).\\nSimilarly, we can prove the second equality in the theorem.\"\n },\n {\n \"heading\": \"APPENDIX C: DATASETS\",\n \"text\": \"Stochastic Moving MNIST(SM-MNIST) Dataset Stochastic moving MNIST (SM-MNIST) consists of sequences of frames of size 64× 64× 1, containing one MNIST digit moving and bouncing off edges of the frame (walls). We use one digit instead of two digits because two moving digits may collide, which changes the content of the dynamics and is inconsistent with our assumption. The digits in SM-MNIST move with a constant velocity along a trajectory until they hit at wall at which point they bounce off with a random speed and direction.\\nSprites Dataset We follow the same steps as in Yingzhen & Mandt (2018) to process Sprites dataset, which consists of animated cartoon characters whose clothing, hairstyle, skin color and action can be fully controlled. We use 6 variants in each of 4 attribute categories (skin colors, tops, pants and hair style) and there are 64 = 1296 unique characters in total, where 1000 of them are used for training and the rest of them are used for testing. We use 9 action categories (walking, casting spells and slashing, each with three different viewing angles.) The resulting dataset consists of video sequences with T = 8 frames of size 64× 64× 3. MUG Facial Dataset We use the MUG Facial Expression Database (Aifanti et al., 2010) for this experiment. The dataset consists of 86 subjects. Each video consists of 50 to 160 frames. To use the same network architecture for the whole video datasets in this paper, we cropped the face regions and scaled to the same size 64× 64× 3. We use six facial expressions (anger, fear, disgust, happiness,\\nsadness, and surprise). To ensure there is sufficient change in the facial expression along a video sequence, we choose every other frame in the original video sequences to form training and test video sequences of length T = 10. 80% of the videos are used for training and 20% of the videos are used for testing.\\nTIMIT Speech Dataset The TIMIT dataset (Garofolo, 1993) contains broadband 16k Hz of phonetically-balanced read speech. A total of 6300 utterances (5.4 hours) are presented with 10 sentences from each of 630 speakers. The data is preprocessed in the same way as in (Yingzhen & Mandt, 2018) and (Hsu et al., 2017). The raw speech waveforms are first split into sub-sequences of 200ms, and then preprocessed with sparse fast Fourier transform to obtain a 200 dimensional logmagnitude spectrum, computed every 10ms, i.e., we use T = 20 for sequence x1:T . The dimension of xt is 200.\\nNow we explain the detail of the evaluation metric, equal error rate (EER), used on TIMIT dataset. Letwtest be the feature of test utterance xtest1:T andw\\ntarget be the feature of test utterance xtarget1:T . The predicted identity is confirmed if the cosine similarity betweenwtest andwtarget, cos(wtest,wtarget) is greater than some threshold used in Dehak et al. (2010). The equal error rate (EER) means the false rejection rate equals the false acceptance rate (Dehak et al., 2010). In the following, we will discuss the two choices of feature wtest for evaluations of all methods,\\nµc = 1\\nN N∑ i=1 Eq(zc|xi1:T )[z c],\\nwhich is used to evaluate the disentanglement of zc;\\nµm = 1\\nNT N∑ i=1 T∑ j=1 Eq(zmt |xi1:T )[z m t ],\\nwhich is used to evaluate the disentanglement of zm. For more details, please refer to (Dehak et al., 2010; Yingzhen & Mandt, 2018; Hsu et al., 2017). We use the same network architecture as in Yingzhen & Mandt (2018) for a fair comparison on speech dataset. As the input dimension of speech is low, the encoder/decoder network is a 2-hidden-layer MLP with the hidden dimension 256.\"\n },\n {\n \"heading\": \"APPENDIX D: CHOICES OF REGULARIZERS\",\n \"text\": \"In the following, we will discuss the choice of regularizers in R-WAE. To make notations easy to read, we use density functions for corresponding distributions. In both R-WAE(GAN) and R-WAE(MMD), we use the same regularizer for D(q(zmt |zm<t), p(zmt |zm<t)). We also add a KLdivergence regularization term on zm to stabilize training. In the experiments, we assume inference model q(zc|x1:T ) is a Gaussian distribution with parameters mean µc and diagonal variance matrix σc. Inference model q(zmt |xt, zm<t) is a Gaussian distribution with parameters meanµm and diagonal variance matrix σm. For the prior distribution, we assume p(zmt |zm<t) is a Gaussian distribution with parameters mean µψm and diagonal covariance matrix σ ψ m. For regularizing the motion variables, we just use MMD without introducing any additional parameter, MMDk(q(zmt |zm<t), p(zmt |zm<t)), and we choose mixture of RBF kernel (Li et al., 2017), where RBF kernel is defined as k(x,y) = exp(−‖x−y‖ 2\\n2σ2 ). With samples {z̃i} n i=1 from the posterior q(z̃ c) and samples {zi}ni=1 from the prior p(zc), MMDk(q(z̃ c), p(zc)) is defined as\\nMMDk(q(z̃ c), p(zc))=\\n1 n(n− 1) ∑ i6=j k(zi, zj)+ 1 n(n− 1) ∑ i 6=j k(z̃i, z̃j)− 1 n2 ∑ i,j k(z̃i, zj). (21)\\nThe difference between R-WAE(MMD) and R-WAE(GAN) is how to choose metrics for the regularizer D(QZc ,PZc), where PZc is the prior distribution and QZc is the posterior distribution of the inference model.\\nR-WAE(MMD) The regularizer D(QZc ,PZc) is chosen as,\\nD(QZc ,PZc) = MMDkγ (Q(Zc),P (Zc)),\\nwhere the scaled MMD MMDkγ (Q(Z c),P (Zc)) is chosen as\\nMMDkγ (QZc ,PZc) = M̂MDkγ (QZc , PZc)\\n1 + 10EP̂ [‖∇fγ(zc)‖2F ] ,\\nwhere the function fγ(zc) is the kernel feature map and M̂MDkγ (QZc , PZc) is defined in the following. When we have samples {z̃ci}ni=1 from Q(Zc) and samples {zci}ni=1 from P (Zc),\\nM̂MDkγ (Q(Z c), P (Zc)) =\\n1 n(n− 1) ∑ i 6=j k(fγ(z c i ), fγ(z c j)) +\\n1 n(n− 1) ∑ i 6=j k(fγ(z̃ c i ), fγ(z̃ c j))\\n(22)\\n− 1 n2 ∑ i,j k(fγ(z̃ c i ), fγ(z c j)),\\nwhere the RBF kernel k is defined on scalar variables, k(x, y) = exp(−‖x−y‖ 2\\n2 ). To avoid the situation where the generator gets stuck on a local optimum, we apply spectral parametrization for the weight matrix (Miyato et al., 2018). The feature map fγ is updated L steps at each iteration. To overcome posterior collapse and inference lagging, we will update the inference model per iteration of updating the decoder model for L steps during training (He et al., 2019). See Algorithm 1 for details.\\nR-WAE(GAN) For the regularizer DJS(QZc ,PZc), we introduce a discriminator Dγ . The loss is as follows, L = Ezc∼p(zc)[logDγ(zc)] + Ez̃c∼q(z̃c)[log(1−Dγ(z̃c)))], (23) where p(zc) is the prior distribution and q(z̃c) is the posterior distribution of the inference model. To stabilize the training of the min-max problem in GAN-based optimization (23), a lot of stabilization techniques have been proposed (Thanh-Tung et al., 2019; Mescheder et al., 2018; Gulrajani et al., 2017; Petzka et al., 2017; Roth et al., 2017; Qi, 2017). Let samples {zc} are from the prior p(zc) and {z̃c} are from the inference posterior q(z̃c). In our R-WAE(GAN), we will adopt the regularization from Mescheder et al. (2018) and Thanh-Tung et al. (2019),\\nL − λE[‖(∇Dγ)ẑc‖2], (24) where ẑc = αzc + (1− α)z̃c, α ∈ U(0, 1) and (∇Dγ)ẑc is evaluated its gradient at the point ẑc.\\nAlgorithm 1 R-WAE(GAN) Input: regularization coefficient β and content prior p(zc) Goal: learn encoders qφ(zc|x1:T ) and qφ(z m t |xt, zm<t), prior pψ(zmt |zm<t), discrimi-\\nnatorDγ , and decoder pθ(xt|zt), where zt = (zc, zmt ) while not converged do\\nfor step 1 to L do Sample batch X = {xt} Sample {zc} from prior p(zc) and {zmt } from prior pψ Sample {z̃c, z̃mt } from encoders qφ Update discriminator Dγ and encoders qφ with loss given by (9), (10) end for Update pθ and prior pψ with loss given by (9) and (10).\\nend while\\nAlgorithm 2 R-WAE(MMD) Input: regularization coefficient β and content prior p(zc) Goal: learn encoders qφ(zc|x1:T ) and qφ(z m t |xt, zm<t), prior pψ(zmt |zm<t), feature\\nmap fγ and decoder pθ(xt|zt), where zt = (zc, zmt ) while not converged do\\nfor step 1 to L do Sample batch X = {xt} Sample {zc} from prior p(zc) and {zmt } from prior pψ Sample {z̃c, z̃mt } from encoders qφ Update feature map fγ and encoders qφ with loss given by (9), (11) end for Update pθ and prior pψ with loss given by (9) and (11).\\nend while\"\n },\n {\n \"heading\": \"6.1 APPENDIX E: UNCONDITIONAL VIDEO GENERATION\",\n \"text\": \"Fig. 4 provides generated samples on the SM-MNIST dataset by randomly sampling content {zc} from the prior p(zc) and motions {zm1:T } from the learned prior pψ(zmt |zm<t). The length of our\\ngenerated videos is T = 100 and we only show randomly chosen videos of T = 20 to save file size. Our R-WAE(MMD) achieves the most consistent and visually best sequence even when T = 100. Samples from MoCoGAN (Tulyakov et al., 2018) usually change digit identity along the sequence. The reason is that MoCoGAN (Tulyakov et al., 2018) requires the number of actions be finite.\\nFig. 5 shows unconditional video generation with T = 10 on MUG facial dataset. DS-VAE in (b) is improved by incorporating categorical latent variables. The figures should be viewed with Adobe Reader to see video.\\n(a) R-WAE(GAN) (b) DS-VAE (c) MoCoGAN\\nFigure 5: Unconditional video generation with T = 10 on MUG facial dataset. DS-VAE in (b) is improved by incorporating categorical latent variables. The figures should be viewed with Adobe Reader to see video.\"\n },\n {\n \"heading\": \"APPENDIX F: LATENT MANIFOLD VISUALIZATION\",\n \"text\": \"We encode the test data {x1:T } of SM-MNIST with T = 10 to get the content codes {zc} using our R-WAE(MMD). We visualize two-dimensional (2D) manifold of {zc} using t-SNE (Maaten & Hinton, 2008). In Fig. 6, different colors correspond to the digit identities of the latent codes {zc} of test videos on SM-MNIST. This indicates that {zc} encoded by our R-WAE(MMD) exactly captures the invariant information (digits) of the test data. The latent motion codes are sequential and cannot be visualized.\"\n },\n {\n \"heading\": \"APPENDIX G: MODEL ARCHITECTURE AND HYPER-PARAMETERS\",\n \"text\": \"In the inference model, we use an encoder network, defined in Fig. 7 (a) to extract latent feature ht defined in Fig.1. We use a decoder network to reconstruct x̂t from the hidden state ht, defined in Fig.1. For the discriminator Dγ in R-WAE(GAN), we use a 4-layer fully-connected neural\\nnetwork (FC NN) with respective dimension (256, 256, 128, 1). For the feature map fγ with a scalar output for the RBF kernel of R-WAE(MMD), we use a 4-layer fully-connected neural network with respective dimension (256, 256, 128, 1). After encoding xt, we get extracted latent feature ht. We use Fig. 8(a) and Fig. 8(b) to infer the content variable zc and motion variables zmt . When the Gumbel latent variable is incorporated into our weakly-supervised inference model, we use Fig. 8(c) to infer the Gumbel latent variable a. The latent content variable zc and latent motion variable zmt are concatenated as input to an LSTM after an FC NN to output hidden state ht for reconstructing x̂t using the decoder. For our weakly-supervised model, the latent content variable zc, latent motion variable zmt and latent action variable a are concatenated as input to an LSTM after an FC NN to output hidden state ht for reconstructing x̂t using the decoder. We use Adam optimizer (Kingma & Ba, 2015) with β1 = 0.5 and β2 = 0.9.\\nArchitecture on SM-MNIST, Sprites and TIMIT Datasets We use the same architecture on SM-MNIST and Sprites dataset, as shown in Fig. 9. The details of the parameters of the networks are provided in Fig. 9. As R-WAE(GAN) and R-WAE(MMD) have similar performance on SMMNIST and Sprites (see Sprites results in Table 4), we only provide the results and parameters of R-WAE(MMD) to save space. At each iteration of training the decoder pθ(xt|zt) and the prior pψ(z m t |zm<t), we train the encoder parameters qφ and the feature map fγ for R-WAE(MMD) with L steps. The results on SM-MNIST and Sprites datasets are evaluated after 500 epochs. On SM-MNIST dataset, we use a Bernoulli cross-entropy loss and choose L = 5. The penalty coefficients β1 and β2, are, respectively, 5 and 20. The learning rate for the decoder model is 5× 10−4 and the learning rate for the encoder is 1× 10−4. The learning rate for fγ is 1× 10−4. On Sprites dataset, we use an L2 reconstruction loss and choose L = 5 steps. The penalty coefficients β1 and β2 are, respectively, 10 and 60. The learning rate for the decoder model is 3× 10−4 and the learning rate for the encoder is 1× 10−4. The learning rate for Dγ in R-WAE(GAN) or fγ in R-WAE(MMD) is 1× 10−4. We use a decayed learning rate schedule on both datasets. After 50 epochs, we decrease all learning rates by a factor of 2 and after 80 epochs decrease further by a factor of 5. On TIMIT speech dataset, we use the same encoder and decoder architecture as that of DS-VAE. The dimension of hidden states is 256 and the dimensions of zc and zmt are both 16.\\nResBlock1 down 64*3*3 self-attention\\nResBlock2 down 128*3*3 ResBlock3 down 256*3*3 ResBlock4 down 512*3*3\\nResBlock5 down 1024*3*3 Reshape output to (N, 1024× 2× 2)\\nFC NN\\nTable 5: Encoder Network Architecture.\\nFC NN and Reshape input to (N, 2048, 2, 2) ResBlock1 up 1024*3*3 ResBlock2 up 512*3*3 ResBlock3 up 256*3*3 ResBlock4 up 128*3*3\\nself-attention ResBlock5 up 64*3*3\\nConv 3*3*3, activation=sigmoid\\nTable 6: Decoder Network Architecture.\\nTable 7: Encoder Network Architecture.\\nFigure 10: Network parameters on encoder network and decoder network on MUG facial dataset. We adopt ResBlock down and up from Brock et al. (2019). The dimensions of zc, zmt , ht, a are 150, 16, 180 and 6 respectively. The batch size on MUG facial dataset are 30 and the length of video sequence for training is T = 8.\\nArchitecture on MUG Facial Dataset The details of the architecture parameters of the networks for MUG facial dataset are provided in Fig. 9. The results on MUG facial dataset are evaluated after 800 epochs. For the regularizer DKL(qφ(a|x1:T , zm1:T ), p(a)), we choose the coefficient of this categorical regularizer to be 50. We use an L2 reconstruction loss and choose L = 5 steps. For R-WAE(MMD), the penalty coefficients β1 and β2 are, respectively, 10 and 50. For R-WAE(GAN), the coefficients β1 and β2 of the penalties are, respectively, 5 and 60. The learning rate for the decoder model is 5 × 10−4 and the learning rate for the encoder is 2 × 10−4. The learning rate for Dγ in R-WAE(GAN) or fγ in R-WAE(MMD) is 2× 10−4. We use the same decayed learning rate schedule as described on SM-MNIST and Sprites datasets. This architecture can be applied to improve the compression rate (?).\"\n },\n {\n \"heading\": \"APPENDIX H: ADDITIONAL RESULTS ON AUDIO DATA\",\n \"text\": \"Swapping Static and Dynamic Factors on Audio Data Here we present results of swapping static and dynamic factors of given audio sequences. Results are given in Figure 11. Each heatmap subplot is of dimension 80 × 20 and visualizes the spectrum of 200ms of an audio clip, in which the mel-scale filter bank features are plotted in the frequency domain (x-axis represents temporal domain with 20 timesteps and y-axis is the value of frequencies). We collect these heatmaps in a matrix where the static factors in a row are kept the same and each column shares the same dynamic factor. It can be observed that in each column, the linguistic phonetic contents as reflected by the formants along x-axis are kept almost the same after swapping. Likewise, the timbres are reflected as the harmonics in the spectrum plot. This can be concluded by observing that the horizontal light stripes which represents the harmonics are kept consistent in a row. Moreover, we perform\\nidentity verification experiment as conducted in DS-VAE (Yingzhen & Mandt, 2018). Similar to cross reconstruction, zcfemale and z c male (or f\\nfemale and fmale in DS-VAE) are swapped for two sequences {xfemale} and {xmale}. By an informal listening test of the original-swapped speech sequence pairs, we confirm that the speech content is preserved and identity is transferred (i.e. female voice usually has higher frequency).\\nAPPENDIX I: ADDITIONAL RESULTS ON A MOVING-SHAPE VIDEO DATA\\nGeneration Results on Moving Shapes We report results on a Moving-Shape dataset in Table 9 and Fig. 12. The Moving-Shape synthetic dataset was introduced in Balaji et al. (2018) which has 5 control parameters: shape type (e.g. triangle and square), size (small and large), color (e.g. white and red), motion type (e.g. zig-zag, straight line and diagonal) and motion direction. In Table 9, TFGAN (Balaji et al., 2018) encoder and decoder architectures are considered less expressive compared with BigGAN (Brock et al., 2019) architectures. Similar to results in Table 2, with more complex and expressive architecture, learning disentangled representation is harder. The results in Table 9 and Fig. 12 demonstrate that R-WAE produces better disentanglement and generation performance than DS-VAE both quantitatively and qualitatively. Qualitative difference of fixing zm and sampling zc for DS-VAE and R-WAE is not that obvious and thus not shown.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2021,"string":"2,021"},"title":{"kind":"null"}}},{"rowIdx":98,"cells":{"paper_id":{"kind":"string","value":"SP:60894f74f40addd7a2a35a003dcdce6cf70ffef4"},"summaries":{"kind":"list like","value":["The paper extends prior work on equivalence between predictive coding and backprop in layered neural networks to arbitrary computation graphs. This is empirically tested first on a simple nonlinear scalar function, and then on a few commonly used architectures (CNNs, RNNs, LSTMs), confirming the theoretical results. The importance of this advance is highlighted by noting that the demonstrated equivalence shows how in principle modern architectures could be implemented in biological neural systems, and that the highly parallel nature of predictive coding could lead to efficient implementations in neuromorphic hardware."],"string":"[\n \"The paper extends prior work on equivalence between predictive coding and backprop in layered neural networks to arbitrary computation graphs. This is empirically tested first on a simple nonlinear scalar function, and then on a few commonly used architectures (CNNs, RNNs, LSTMs), confirming the theoretical results. The importance of this advance is highlighted by noting that the demonstrated equivalence shows how in principle modern architectures could be implemented in biological neural systems, and that the highly parallel nature of predictive coding could lead to efficient implementations in neuromorphic hardware.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. Recently it has been shown that backprop in multilayer-perceptrons (MLPs) can be approximated using predictive coding, a biologically-plausible process theory of cortical computation which relies solely on local and Hebbian updates. The power of backprop, however, lies not in its instantiation in MLPs, but rather in the concept of automatic differentiation which allows for the optimisation of any differentiable program expressed as a computation graph. Here, we demonstrate that predictive coding converges asymptotically (and in practice rapidly) to exact backprop gradients on arbitrary computation graphs using only local learning rules. We apply this result to develop a straightforward strategy to translate core machine learning architectures into their predictive coding equivalents. We construct predictive coding CNNs, RNNs, and the more complex LSTMs, which include a non-layer-like branching internal graph structure and multiplicative interactions. Our models perform equivalently to backprop on challenging machine learning benchmarks, while utilising only local and (mostly) Hebbian plasticity. Our method raises the potential that standard machine learning algorithms could in principle be directly implemented in neural circuitry, and may also contribute to the development of completely distributed neuromorphic architectures."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"APPROXIMATES BACKPROP"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"APPROXIMATES BACKPROP\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Mohamed Akrout","Collin Wilson","Peter Humphreys","Timothy Lillicrap","Douglas B Tweed"],"title":"Deep learning without weight transport","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Shun-Ichi Amari"],"title":"Information geometry of the em and em algorithms for neural networks","venue":"Neural networks,","year":1995},{"authors":["Yali Amit"],"title":"Deep learning with asymmetric connections and hebbian updates","venue":"Frontiers in computational 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No. 04CH37541),\",\n \"year\": 2004\n },\n {\n \"authors\": [\n \"Christian Szegedy\",\n \"Wei Liu\",\n \"Yangqing Jia\",\n \"Pierre Sermanet\",\n \"Scott Reed\",\n \"Dragomir Anguelov\",\n \"Dumitru Erhan\",\n \"Vincent Vanhoucke\",\n \"Andrew Rabinovich\"\n ],\n \"title\": \"Going deeper with convolutions\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Andrea Tacchetti\",\n \"Leyla Isik\",\n \"Tomaso Poggio\"\n ],\n \"title\": \"Invariant recognition drives neural representations of action sequences\",\n \"venue\": \"PLoS computational biology,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Corentin Tallec\",\n \"Yann Ollivier\"\n ],\n \"title\": \"Unbiased online recurrent optimization\",\n \"venue\": \"arXiv preprint arXiv:1702.05043,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Belinda Tzen\",\n \"Maxim Raginsky\"\n ],\n \"title\": \"Neural stochastic differential equations: Deep latent gaussian models in the diffusion limit\",\n \"venue\": \"arXiv preprint arXiv:1905.09883,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Ashish Vaswani\",\n \"Noam Shazeer\",\n \"Niki Parmar\",\n \"Jakob Uszkoreit\",\n \"Llion Jones\",\n \"Aidan N Gomez\",\n \"Łukasz Kaiser\",\n \"Illia Polosukhin\"\n ],\n \"title\": \"Attention is all you need\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Oriol Vinyals\",\n \"Igor Babuschkin\",\n \"Wojciech M Czarnecki\",\n \"Michaël Mathieu\",\n \"Andrew Dudzik\",\n \"Junyoung Chung\",\n \"David H Choi\",\n \"Richard Powell\",\n \"Timo Ewalds\",\n \"Petko Georgiev\"\n ],\n \"title\": \"Grandmaster level in starcraft ii using multi-agent reinforcement learning\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Eiji Watanabe\",\n \"Akiyoshi Kitaoka\",\n \"Kiwako Sakamoto\",\n \"Masaki Yasugi\",\n \"Kenta Tanaka\"\n ],\n \"title\": \"Illusory motion reproduced by deep neural networks trained for prediction\",\n \"venue\": \"Frontiers in psychology,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Veith Weilnhammer\",\n \"Heiner Stuke\",\n \"Guido Hesselmann\",\n \"Philipp Sterzer\",\n \"Katharina Schmack\"\n ],\n \"title\": \"A predictive coding account of bistable perception-a model-based fmri study\",\n \"venue\": \"PLoS computational biology,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Paul J Werbos\"\n ],\n \"title\": \"Applications of advances in nonlinear sensitivity analysis\",\n \"venue\": \"In System modeling and optimization,\",\n \"year\": 1982\n },\n {\n \"authors\": [\n \"James CR Whittington\",\n \"Rafal Bogacz\"\n ],\n \"title\": \"An approximation of the error backpropagation algorithm in a predictive coding network with local hebbian synaptic plasticity\",\n \"venue\": \"Neural computation,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"James CR Whittington\",\n \"Rafal Bogacz\"\n ],\n \"title\": \"Theories of error back-propagation in the brain\",\n \"venue\": \"Trends in cognitive sciences,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Ronald J Williams\",\n \"David Zipser\"\n ],\n \"title\": \"A learning algorithm for continually running fully recurrent neural networks\",\n \"venue\": \"Neural computation,\",\n \"year\": 1989\n },\n {\n \"authors\": [\n \"Daniel LK Yamins\",\n \"Ha Hong\",\n \"Charles F Cadieu\",\n \"Ethan A Solomon\",\n \"Darren Seibert\",\n \"James J DiCarlo\"\n ],\n \"title\": \"Performance-optimized hierarchical models predict neural responses in higher visual cortex\",\n \"venue\": \"Proceedings of the National Academy of Sciences,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Williams\",\n \"Zipser\",\n \"Lillicrap\",\n \"Santoro\"\n ],\n \"title\": \"While this is a fascinating area, we do not address it in this paper. We are solely concerned with the fact that predictive coding approximates backpropagation on feedforward computation graphs for which the unrolled RNN graph is a sufficient substrate\",\n \"venue\": \"Ollivier et al.,\",\n \"year\": 2004\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Deep learning has seen stunning successes in the last decade in computer vision (Krizhevsky et al., 2012; Szegedy et al., 2015), natural language processing and translation (Vaswani et al., 2017; Radford et al., 2019; Kaplan et al., 2020), and computer game playing (Mnih et al., 2015; Silver et al., 2017; Schrittwieser et al., 2019; Vinyals et al., 2019). While there is a great variety of architectures and models, they are all trained by gradient descent using gradients computed by automatic differentiation (AD). The key insight of AD is that it suffices to define a forward model which maps inputs to predictions according to some parameters. Then, using the chain rule of calculus, it is possible, as long as every operation of the forward model is differentiable, to differentiate back through the computation graph of the model so as to compute the sensitivity of every parameter in the model to the error at the output, and thus adjust every single parameter to best minimize the total loss. Early models were typically simple artificial neural networks where the computation graph is simply a composition of matrix multiplications and elementwise nonlinearities, and for which the implementation of automatic differentation has become known as ‘backpropagation’ (or ’backprop’). However, automatic differentiation allows for substantially more complicated graphs to be differentiated through, up to, and including, arbitrary programs (Griewank et al., 1989; Baydin et al., 2017; Paszke et al., 2017; Revels et al., 2016; Innes et al., 2019; Werbos, 1982; Rumelhart and Zipser, 1985; Linnainmaa, 1970). In recent years this has enabled the differentiation through differential equation solvers (Chen et al., 2018; Tzen and Raginsky, 2019; Rackauckas et al., 2019), physics engines (Degrave et al., 2019; Heiden et al., 2019), raytracers (Pal, 2019), and planning algorithms (Amos and Yarats, 2019; Okada et al., 2017). These advances allow the straightforward training of models which intrinsically embody complex processes and which can encode significantly more prior knowledge and structure about a given problem domain than previously possible.\nModern deep learning has also been closely intertwined with neuroscience (Hassabis et al., 2017; Hawkins and Blakeslee, 2007; Richards et al., 2019). The backpropagation algorithm itself arose\nas a technique for training multi-layer perceptrons – simple hierarchical models of neurons inspired by the brain (Werbos, 1982). Despite this origin, and its empirical successes, a consensus has emerged that the brain cannot directly implement backprop, since to do so would require biologically implausible connection rules (Crick, 1989). There are two principal problems. Firstly, backprop in the brain appears to require non-local information (since the activity of any specific neuron affects all subsequent neurons down to the final output neuron). It is difficult to see how this information could be transmitted ’backwards’ throughout the brain with the required fidelity without precise connectivity constraints. The second problem – the ‘weight transport problem’ is that backprop through MLP style networks requires identical forward and backwards weights. In recent years, however, a succession of models have been introduced which claim to implement backprop in MLP-style models using only biologically plausible connectivity schemes, and Hebbian learning rules (Liao et al., 2016; Guerguiev et al., 2017; Sacramento et al., 2018; Bengio and Fischer, 2015; Bengio et al., 2017; Ororbia et al., 2020; Whittington and Bogacz, 2019). Of particular significance is Whittington and Bogacz (2017) who show that predictive coding networks – a type of biologically plausible network which learn through a hierarchical process of prediction error minimization – are mathematically equivalent to backprop in MLP models. In this paper we extend this work, showing that predictive coding can not only approximate backprop in MLPs, but can approximate automatic differentiation along arbitrary computation graphs. This means that in theory there exist potentially biologically plausible algorithms for differentiating through arbitrary programs, utilizing only local connectivity. Moreover, in a class of models which we call parameter-linear, which includes many current machine learning models, the required update rules are Hebbian, raising the possibility that a wide range of current machine learning architectures may be faithfully implemented in the brain, or in neuromorphic hardware.\nIn this paper we provide two main contributions. (i) We show that predictive coding converges to automatic differentiation across arbitrary computation graphs. (ii) We showcase this result by implementing three core machine learning architectures (CNNs, RNNs, and LSTMs) in a predictive coding framework which utilises only local learning rules and mostly Hebbian plasticity."},{"heading":"2 PREDICTIVE CODING ON ARBITRARY COMPUTATION GRAPHS","text":"Predictive coding is an influential theory of cortical function in theoretical and computational neuroscience. Central to the theory is the idea that the core function of the brain is to minimize prediction errors between what is expected to happen and what actually happens. Predictive coding\nviews the brain as composed of multiple hierarchical layers which predict the activities of the layers below. Unpredicted activity is registered as prediction error which is then transmitted upwards for a higher layer to process. Over time, synaptic connections are adjusted so that the system improves at minimizing prediction error. Predictive coding possesses a wealth of empirical support (Friston, 2003; 2005; Bogacz, 2017; Whittington and Bogacz, 2019) and offers a single mechanism that accounts for diverse perceptual phenomena such as repetition-suppression (Auksztulewicz and Friston, 2016), endstopping (Rao and Ballard, 1999), bistable perception (Hohwy et al., 2008; Weilnhammer et al., 2017) and illusory motions (Lotter et al., 2016; Watanabe et al., 2018), and even attentional modulation of neural activity (Feldman and Friston, 2010; Kanai et al., 2015). Moreover, the central role of top-down predictions is consistent with the ubiquity, and importance of, top-down diffuse connections between cortical areas. Predictive coding is consistent with many known aspects of neurophysiology, and has been translated into biologically plausible process theories which define candidate cortical microcircuits which can implement the algorithm. (Spratling, 2008; Bastos et al., 2012; Kanai et al., 2015; Shipp, 2016).\nIn previous work, predictive coding has always been conceptualised as operating on hierarchies of layers (Bogacz, 2017; Whittington and Bogacz, 2017). Here we present a generalized form of predictive coding applied to arbitrary computation graphs. A computation graph G = {E,V} is a directed acyclic graph (DAG) which can represent the computational flow of essentially any program or computable function as a composition of elementary functions. Each edge ei ∈ E of the graph corresponds to an intermediate step – the application of an elementary function – while each vertex vi ∈ V is an intermediate variable computed by applying the functions of the edges to the values of their originating vertices. In this paper, vi denotes the vector of activations within a layer and we denote the set of all vertices as {vi}. Effectively, computation flows ’forward’ from parent nodes to all their children through the edge functions until the leaf nodes give the final output of the program as a whole (see Figure 1 and 2 for an example). Given a target T and a loss function L = g(T, vout), the graph’s output can be evaluated and, and if every edge function is differentiable, automatic differentiation can be performed on the computation graph.\nPredictive coding can be derived elegantly as a variational inference algorithm under a hierarchical Gaussian generative model (Friston, 2005; Buckley et al., 2017). We extend this approach to arbitrary computation graphs in a supervised setting by defining the inference problem to be solved as that of inferring the vertex value vi of each node in the graph given fixed start nodes v0 (the data), and end nodes vN (the targets). We define a generative model which parametrises the value of each vertex given the feedforward prediction of its parents, p({vi}) = p(v0 . . . vN ) = ∏N i p(vi|P(vi)) 1, and a factorised,\nvariational posterior Q({vi}|v0, vN ) = Q(v1 . . . vN−1|v0, vN ) = ∏N i Q(vi|P(vi), C(vi)), where P(vi) denotes the set of parents and C(vi) denotes the set of children of a given node vi. From this, we can define a suitable objective functional, the variational free-energy F (VFE), which acts as an upper bound on the divergence between the true and variational posteriors.\nF = KL[(Q(v1 . . . vN−1|v0, vN )‖p(v0 . . . vN )] ≥ KL[(Q(v1 . . . vN−1)|v0, vN )‖p(v1 . . . vN−1|v0, vN )]\n≈ N∑ i=0 Ti i\n(1) Under Gaussian assumptions for the generative model p({vi}) = ∏N i N (vi; v̂i,Σi), and the vari-\national posterior Q({vi}) = ∏N i N (vi), where the ‘predictions’ v̂i = f(P(vi); θi) are defined as the feedforward value of the vertex produced by running the graph forward, and all the precisions, or inverse variances, Σ−1i are fixed at the identity, we can write F as simply a sum of prediction errors (see Appendix D or (Friston, 2003; Bogacz, 2017; Buckley et al., 2017) for full derivations), with the prediction errors defined as i = vi − v̂i. These prediction errors play a core role in the framework and, in the biological process theories (Friston, 2005; Bastos et al., 2012), are generally considered to be represented by a distinct population of ‘error units’. Since F is an upper bound on the divergence between true and approximate posteriors, by minimizing F , we reduce this divergence, thus improving the quality of the variational posterior and approximating exact Bayesian inference. Predictive coding minimizes F by employing the Cauchy method of steepest descent to set the\n1This includes the prior p(v0), which simply has no parents.\ndynamics of the vertex variables vi as a gradient descent directly on F (Bogacz, 2017). dvi dt = ∂F ∂vi = i − ∑\nj∈C(vi)\nj ∂v̂j ∂vi\n(2)\nThe dynamics of the parameters of the edge functions θ such that v̂i = f(P(vi); θ), can also be derived as a gradient descent on F . Importantly these dynamics require only information (the current vertex value, prediction error, and prediction errors of child vertices) locally available at the vertex.\ndθi dt = ∂F ∂θi = i ∂v̂i ∂θi\n(3)\nTo run generalized predictive coding in practice on a given computation graph G = {E,V}, we augment the graph with error units ∈ E to obtain an augumented computation graph G̃ = {E,V, E}. The predictive coding algorithm then operates in two phases – a feedforward sweep and a backwards iteration phase. In the feedforward sweep, the augmented computation graph is run forward to obtain the set of predictions {v̂i}, and prediction errors { i} = {vi − v̂i} for every vertex. Following Whittington and Bogacz (2017), to achieve exact equivalence with the backprop gradients computed on the original computation graph, we initialize vi = v̂i in the initial feedforward sweep so that the output error computed by the predictive coding network and the original graph are identical.\nIn the backwards iteration phase, the vertex activities {vi} and prediction errors { i} are updated with Equation 2 for all vertices in parallel until the vertex values converge to a minimum of F . After convergence the parameters are updated according to Equation 3. Note we also assume, following Whittington and Bogacz (2017), that the predictions at each layer are fixed at the values assigned during the feedforward pass throughout the optimisation of the vs. We call this the fixed-prediction assumption. In effect, by removing the coupling between the vertex activities of the parents and the prediction at the child, this assumption separates the global optimisation problem into a local one for each vertex. We implement these dynamics with a simple forward Euler integration scheme so that the update rule for the vertices became vt+1i ← vti−η dFdvti where η is the step-size parameter. Importantly, if the edge function linearly combines the activities and the parameters followed by an elementwise nonlinearity – a condition which we call ‘parameter-linear’ – then both the update rule for the vertices (Equation 2) and the parameters (Equation 3) become Hebbian. Specifically, the update rules for the vertices and weights become dvidt = i − ∑ j jf ′(θj v̂j)θ T j and dθi dt = if ′(θiv̂i)v̂i T , respectively."},{"heading":"2.1 APPROXIMATION TO BACKPROP","text":"Here we show that at the equilibrium of the dynamics, the prediction errors ∗i converge to the correct backpropagated gradients ∂L∂vi , and consequently the parameter updates (Equation 3) become precisely those of a backprop trained network. Standard backprop works by computing the gradient of a vertex as the sum of the gradients of the child vertices. Beginning with the gradient of the output vertex ∂L ∂vL\n, it recursively computes the gradients of vertices deeper in the graph by the chain rule: ∂L\n∂vi = ∑ j=C(vi) ∂L ∂vj ∂vj ∂vi\n(4)\nIn comparison, in our predictive coding framework, at the equilibrium point (dvidt = 0) the prediction errors ∗i become,\n∗i = ∑\nj∈C(vi)\n∗j ∂v̂i ∂vj\n(5)\nImportantly, this means that the equilibrium value of the prediction error at a given vertex (Equation 5) satisfies the same recursive structure as the chain rule of backprop (Equation 4). Since this relationship is recursive, all that is needed for the prediction errors throughout the graph to converge to the backpropagated derivatives is for the prediction errors at the final layer to be equal to the output gradient: ∗L = ∂L ∂v̂L . To see this explicitly, consider a mean-squared-error loss function 2. at the\n2While the mean-squared-error loss function fits most nicely with the Gaussian generative model, other loss functions can be used in practice. If the loss function can be represented as a log probability distribution, then the generative model can be amended to simply set the output distribution to that distribution. If not, then there is no fully consistent generative model (although all nodes except the output remain Gaussian), but the algorithm will still work in practice. See (Figure 6) in Appendix A for results for CNNs trained with a crossentropy loss.\nAlgorithm 1: Generalized Predictive Coding Data: Dataset D = {X,L}, Augmented Computation Graph G̃ = {E,V, E}, inference learning rate ηv , weight learning rate ηθ begin\n/* For each minibatch in the dataset */ for (x, L) ∈ D do\n/* Fix start of graph to inputs */ v̂0 ← x /* Forward pass to compute predictions */ for v̂i ∈ V do\nv̂i ← f({P(v̂i); θ) /* Compute output error */ L ← L− v̂L /* Begin backwards iteration phase of the descent on the\nfree energy */ while not converged do\nfor (vi, i) ∈ G̃ do /* Compute prediction errors */ i ← vi − v̂i /* Update the vertex values */ vt+1i ← vti + ηv dFdvti\n/* Update weights at equilibrium */ for θi ∈ E do\nθt+1i ← θti + ηθ dFdθti\noutput layer L = 12 (T − v̂L) 2 with T as a vector of targets, and defining L = T − v̂L. We then consider the equilibrium value of the prediction error unit at a penultimate vertex L−1. By Equation 5, we can see that at equilibrium,\n∗L−1 = ∗ L ∂v̂L ∂vL−1 = (T − v̂∗L) ∂v̂L ∂vL−1\nsince, (T − v̂L) = ∂L∂v̂L , we can then write,\n∗L−1 = ∂L\n∂v̂L ∂v̂L ∂vL−1 = ∂L ∂vL−1 (6)\nThus the prediction errors of the penultimate nodes converge to the correct backpropagated gradient. Furthermore, recursing through the graph from children to parents allows the correct gradients to be computed3. Thus, by induction, we have shown that the fixed points of the prediction errors of the global optimization correspond exactly to the backpropagated gradients. Intuitively, if we imagine the computation-graph as a chain and the error as ’tension’ in the chain, backprop loads all the tension at the end (the output) and then systematically propagates it backwards. Predictive coding, however, spreads the tension throughout the entire chain until it reaches an equilibrium where the amount of tension at each link is precisely the backpropagated gradient. The full algorithm for training the predictive coding network is explicitly set out in Algorithm 1. Inference is just a forward pass through the network, and is identical to the corresponding ANN.\n3Some subtlety is needed here since vL−1 may have many children which each contribute to the loss. However, these different paths sum together at the node vL−1, thus propagating the correct gradient backwards.\nBy a similar argument, it is apparent that the dynamics of the parameters θi as a gradient descent on F also exactly match the backpropagated parameter gradients.\ndθi dt = dF dθi = ∗i d ∗i dθi\n= dL\ndv̂i dv̂i dθi = dL dθi (7)\nWhich follows from the fact that ∗i = dL dv̂i\nand that d ∗ i\ndθ = dv̂i dθi ."},{"heading":"3 RELATED WORK","text":"A number of recent works have tried to provide biologically plausible approximations to backprop. The requirement of symmetry between the forwards and backwards weights has been questioned by Lillicrap et al. (2016) who show that random fixed feedback weights suffice for effective learning. Recent additional work has shown that learning the backwards weights also helps (Amit, 2019; Akrout et al., 2019). Several schemes have also been proposed to approximate backprop using only local learning rules and/or Hebbian connectivity. These include target-prop (Lee et al., 2015) which approximate the backward gradients with trained inverse functions, but which fails to asymptotically compute the exact backprop gradients, and contrastive Hebbian (Seung, 2003; Scellier and Bengio, 2017; Scellier et al., 2018) approaches which do exactly approximate backprop, but which require two separate learning phases and the storing of information across successive phases. There are also dendritic error theories (Guerguiev et al., 2017; Sacramento et al., 2018) which are computationally similar to predictive coding (Whittington and Bogacz, 2019; Lillicrap et al., 2020). Whittington and Bogacz (2017) showed that predictive coding can approximate backprop in MLP models, and demonstrated comparable performance on MNIST. We advance upon this work by extending the proof to arbitrary computation graphs, enabling the design of predictive coding variants of a range of standard machine learning architectures, which we show perform comparably to backprop on considerably more difficult tasks than MNIST. Our algorithm evinces asymptotic (and in practice rapid) convergence to the exact backprop gradients, does not require separate learning phases, and utilises only local information and largely Hebbian plasticity."},{"heading":"4 RESULTS","text":""},{"heading":"4.1 NUMERICAL RESULTS","text":"To demonstrate the correctness of our derivation and empirical convergence to the true gradients, we present a numerical test in the simple scalar case, where we use predictive coding to derive the gradients of an arbitrary, highly nonlinear test function vL = tan( √ θv0) + sin(v 2 0) where θ is an arbitrary parameter. For our tests, we set v0 to 5 and θ to 2. The computation graph for this function is presented in Figure 2. Although simple, this is a good test of predictive coding because the function is highly nonlinear, and its computation graph does not follow a simple layer structure but includes some branching. An arbitrary target of T = 3 was set at the output and the gradient of the loss L = (vL − T )2 with respect to the input v0 was computed by predictive coding. We show (Figure 2) that the predictive coding optimisation rapidly converges to the exact numerical gradients computed by automatic differentiation, and that moreover this optimization is very robust and can handle even exceptionally high learning rates (up to 0.5) without divergence.\nIn summary, we have shown and numerically verified that at the equilibrium point of the global free-energy F on an arbitrary computation graph, the error units exactly equal the backpropagated gradients, and that this descent requires only local connectivity, does not require a separate phases or a sequential backwards sweep, and in the case of parameter-linear functions, requires only Hebbian plasticity. Our results provide a straightforward recipe for the direct implementation of predictive coding algorithms to approximate certain computation graphs, such as those found in common machine learning algorithms, in a potentially biologically plausible manner. Next, we showcase this capability by developing predictive coding variants of core machine learning architectures - convolutional neural networks (CNNs) recurrent neural networks (RNNs) and LSTMs (Hochreiter and Schmidhuber, 1997), and show performance comparable with backprop on tasks substantially more challenging than MNIST.\nθ\n* ( ⋅ , ⋅ ) ⋅ tan( ⋅ )\n+( ⋅ , ⋅ )\n( ⋅ )2 sin( ⋅ )\nμ1 = θ * v0 μ2 = ( ̂v1) μ3 = tan( ̂v2)\nμ4 = v20 μ5 = sin( ̂v4) μL = ̂v3 + ̂v5\n̂v1 ̂v2 ̂v3\n̂v4 ̂v5\nϵ1 ϵ2 ϵ3\nϵ5ϵ4\nT ϵL\nϵL dμL d ̂v3\nϵL dμL d ̂v5\nϵ5 dμ5 d ̂v4\nϵ3 dμ3 d ̂v2ϵ2 dμ2 d ̂v1ϵ1\ndμ1 dθ\nϵ1 dμ1 dv0\nϵ4 dμ4 dv0\nv0"},{"heading":"4.2 PREDICTIVE CODING CNN, RNN, AND LSTM","text":"First, we constructed predictive coding CNN models (see Appendix B for full implementation details). In the predictive coding CNN, each filter kernel was augmented with ‘error maps’ which measured the difference between the forward convolutional predictions and the backwards messages. Our CNN was composed of a convolutional layer, followed by a max-pooling layer, then two further convolutional layers followed by 3 fully-connected layers. We compared our predictive coding CNN to a backprop-trained CNN with the exact same architecture and hyperparameters. We tested our models on three image classification datasets significantly more challenging than MNIST – SVHN, CIFAR10, and CIFAR100. SVHN is a digit recognition task like MNIST, but has more naturalistic backgrounds, is in colour with continuously varying inputs and contains distractor digits. CIFAR10 and CIFAR100 are large image datasets composed of RGB 32x32 images. CIFAR10 has\n10 classes of image, while CIFAR100 is substantially more challenging with 100 possible classes. In general (Figure 3), performance was identical between the predictive coding and backprop CNNs and comparable to the standard performance of basic CNN models on these datasets, Moreover, the predictive coding gradient remained close to the true numerical gradient throughout training.\nWe also constructed predictive coding RNN and LSTM models, thus demonstrating the ability of predictive coding to scale to non-parameter-linear, branching, computation graphs. The RNN was trained on a character-level name classification task, while the LSTM was trained on a next-character prediction task on the full works of Shakespeare. Full implementation details can be found in Appendices B and C. LSTMs and RNNs are recurrent networks which are trained through backpropagation through time (BPTT). BPTT simply unrolls the network through time and backpropagates through the unrolled graph. Analogously we trained the predictive coding RNN and LSTM by applying predictive coding to the unrolled computation graph. The depth of the unrolled graph depends heavily on the sequence length, and in our tasks using a sequence length of 100 we still found that predictive coding evinced rapid convergence to the correct numerical gradient, and that the performance was approximately identical to the equivalent backprop-trained networks (Figure 3), thus showing that the algorithm is scalable even to very deep computation graphs."},{"heading":"5 DISCUSSION","text":"We have shown that predictive coding provides a local and potentially biologically plausible approximation to backprop on arbitrary, deep, and branching computation graphs. Moreover, convergence to the exact backprop gradients is rapid and robust, even in extremely deep graphs such as the unrolled LSTM. Our algorithm is fully parallelizable, does not require separate phases, and can produce equivalent performance to backprop in core machine-learning architectures. These results broaden the horizon of local approximations to backprop by demonstrating that they can be implemented on arbitrary computation graphs, not only simple MLP architectures. Our work prescribes a straightforward recipe for backpropagating through any computation graph with predictive coding using only local learning rules. In the future, this process could potentially be made fully automatic and translated onto neuromorphic hardware. Our results also raise the possibility that the brain may implement machine-learning type architectures much more directly than often considered. Many lines of work suggest a close correspondence between the representations and activations of CNNs and activity in higher visual areas (Yamins et al., 2014; Tacchetti et al., 2017; Eickenberg et al., 2017; Khaligh-Razavi and Kriegeskorte, 2014; Lindsay, 2020), for instance, and this similarity may be found to extend to other machine learning architectures.\nIt is important to note that predictive coding, as advanced here, still retains some biologically implausible features. Although using only local and Hebbian updates, the predictive coding algorithm still requires identical forward and backwards weights, as well as mandating a very precise oneto-one connectivity structure between value neurons vi and error neurons i. However, recent work (Millidge et al., 2020) has begun to show that these implausibilities can be relaxed using learnable backwards weights instead of requiring weight symmetry, and allowing for learnable dense connectivity between value and error neurons, without harm to performance in simple MLP settings. An additional limitation to the biological plausibility of our method is the fixed-prediction assumption, which requires that the feedforward pass values be somehow stored during the backwards\niteration phase. In biological neurons this could potentially be implemented by utilizing synaptic mechanisms for maintaining information over short periods, such as eligibility traces, or alternatively through synchronised phase locking (Buzsaki, 2006). Alternatively, it is important to note that this fixed-prediction assumption is only required for exact convergence to backprop, and predictive coding networks have been shown to be able to attain strong discriminative classification performance without it (Whittington and Bogacz, 2017).\nAlthough we have implemented three core machine learning architectures as predictive coding networks, we have nevertheless focused on relatively small and straightforward networks and thus both our backprop and predictive coding networks perform below the state of the art on the presented tasks. This is primarily because our focus was on demonstrating the theoretical convergence between the two algorithms. Nevertheless, we believe that due to the generality of our theoretical results, ’scaling up’ the existing architectures to implement performance-matched predictive coding versions of more advanced machine learning architectures such as resnets (He et al., 2016), GANs (Goodfellow et al., 2014), and transformers (Vaswani et al., 2017) should be relatively straightforward.\nIn terms of computational cost, one inference iteration in the predictive coding network is about as costly as a backprop backwards pass. Thus, due to using 100-200 iterations for full convergence, our algorithm is substantially more expensive than backprop which limits the scalability of our method. However, this serial cost is misleading when talking about highly parallel neural architectures. In the brain, neurons cannot wait for a sequential forward and backward sweep. By phrasing our algorithm as a global descent, our algorithm is fully parallel across layers. There is no waiting and no phases to be coordinated. Each neuron need only respond to its local driving inputs and downwards error signals. We believe that this local and parallelizable property of our algorithm may engender the possibility of substantially more efficient implementations on neuromorphic hardware (Furber et al., 2014; Merolla et al., 2014; Davies et al., 2018), which may ameliorate much of the computational overhead compared to backprop. Future work could also examine whether our method is more capable than backprop of handling the continuously varying inputs the brain is presented with in practice, rather than the artificial paradigm of being presented with a series of i.i.d. datapoints.\nOur work also reveals a close connection between backprop and inference. Namely, the recursive computation of gradients is effectively a by-product of a variational-inference algorithm which infers the values of the vertices of the computation graph under a hierarchical Gaussian generative model. While the deep connections between stochastic gradient descent and inference in terms of Kalman filtering (Ruck et al., 1992; Ollivier, 2019) or MCMC sampling methods (Chen et al., 2014; Mandt et al., 2017) is known, the relation between recursive gradient computation itself and variational inference is underexplored except in the case of a single layer (Amari, 1995). Our method can provide a principled generalisation of backprop through the inverse-variance Σ−1 parameters of the Gaussian generative model. These parameters weight the relative contribution of different factors to the overall gradient by their uncertainty, thus naturally handling the case of backprop with differentially noisy inputs. Moreover, the Σ−1 parameters can be learnt as a gradient descent on F : dΣidt = − dF dΣi = −Σ−1i i Ti Σ −1 i − Σ −1 i . This specific generalisation is afforded by the Gaussian form of the generative model, however, and other generative models may yield novel optimisation algorithms able to quantify and handle uncertainties throughout the entire computational graph."},{"heading":"APPENDIX A: PREDICTIVE CODING CNN IMPLEMENTATION DETAILS","text":"The key concept in a CNN is that of an image convolution, where a small weight matrix is ’slid’ (or convolved) across an image to produce an output image. Each patch of the output image only depends on a relatively small patch of the input image. Moreover, the weights of the filter stay the same during the convolution, so each pixel of the output image is generated using the same weights. The weight sharing implicit in the convolution operation enforces translational invariance, since different image patches are all processed with the same weights.\nThe forward equations of a convolutional layer for a specific output pixel\nvi,j = k=i+f∑ k=i−f l=j+f∑ l=j−f θk,lxi+k,j+l\nWhere vi,j is the (i, j)th element of the output, xi,j is the element of the input image and θk,l is an weight element of a feature map. To setup a predictive coding CNN, we augment each intermediate xi and vi with error units i of the same dimension as the output of the convolutional layer.\nPredictions v̂ are projected forward using the forward equations. Prediction errors also need to be transmitted backwards for the architecture to work. To achieve this we must have that prediction errors are transmitted upwards by a ’backwards convolution’. We thus define the backwards prediction errors ̂j as follows:\n̂i,j = i+f∑ k=i−f j+f∑ l=j−f θj,ĩi,j\nWhere ̃ is an error map zero-padded to ensure the correct convolutional output size. Inference in the predictive coding network then proceeds by updating the intermediate values of each layer as follows:\ndvl dt = l − ̂l+1\nSince the CNN is also parameter-linear, weights can be updated using the simple Hebbian rule of the multiplication of the pre and post synaptic potentials.\ndθl dt = ∑ i,j li,jvl−1 T i,j\nThere is an additional biological implausibility here due to the weight sharing of the CNN. Since the same weights are copied for each position on the image, the weight updates have contributions from all aspects of the image simultaneously which violates the locality condition. A simple fix for this, which makes the network scheme plausible is to simply give each position on the image a filter with separate weights, thus removing the weight sharing implicit in the CNN. In effect this gives each patch of pixels a local receptive field with its own set of weights. The performance and scalability\nof such a locally connected predictive coding architecture would be an interesting avenue for future work, as this architecture has substantial homologies with the structure of the visual cortex.\nIn our experiments we used a relatively simple CNN architecture consisting of one convolutional layer of kernel size 5, and a filter bank of 6 filters. This was followed by a max-pooling layer with a (2,2) kernel and a further convolutional layer with a (5,5) kernel and filter bank of 16 filters. This was then followed by three fully connected layers of 200, 150, and 10 (or 100 for CIFAR100) output units. Each convolutional and fully connected layer used the relu activation function, except the output layer which was linear. Although this architecture is far smaller than state of the art for convolutional networks, the primary point of our paper was to demonstrate the equivalence of predictive coding and backprop. Further work could investigate scaling up predictive coding to more state-of-the-art architectures.\nOur datasets consisted of 32x32 RGB images. We normalised the values of all pixels of each image to lie between 0 and 1, but otherwise performed no other image preprocessing. We did not use data augmentation of any kind. We set the weight learning rate for the predictive coding and backprop networks 0.0001. A minibatch size of 64 was used. These parameters were chosen without any detailed hyperparameter search and so are likely suboptimal. The magnitude of the gradient updates was clamped to lie between -50 and 50 in all of our models. This was done to prevent divergences, as occasionally occurred in the LSTM networks, likely due to exploding gradients.\nThe predictive coding scheme converged to the exact backprop gradients very precisely within 100 inference iterations using an inference learning rate of 0.1. This gives the predictive coding CNN approximately a 100x computational overhead compared to backprop. The divergence between the true and approximate gradients remained approximately constant throughout training, as shown by Figure 5, which shows the mean divergence for each layer of the CNN over the course of an example training run on the CIFAR10 dataset. The training loss of the predictive coding and backprop networks for SVHN, CIFAR10 and CIFAR100 are presented in Figure 4.\nWhile the experiments in the main paper all used the mean-squared-error loss function, it is also possible to use alternative loss functions. In Figure 6, we show performance of the CNN on CIFAR and SVHN datasets is also very close to backprop when trained with a multi-class cross-entropy loss L = ∑ i Ti ln vLi. In this case the output layer used a softmax function as its nonlinearity, to ensure that the logits passed to the cross-entropy loss were valid probabilities. The cross-entropy loss is also straightforward to fit into the predictive coding framework since the gradient with respect to the pre-activations of the output is also just the negative prediction error ∂L∂vL = T − vL, although the softmax function itself may be challenging to implement neurally since it is non-local as its’ normalisation coefficient requires of the exponentiated activities of all neurons in a layer. Nevertheless, this demonstrates that predictive coding can approximate backprop for any given loss function, not simply mean-square-error."},{"heading":"APPENDIX B: PREDICTIVE CODING RNN","text":"The computation graph on RNNs is relatively straightforward. We consider only a single layer RNN here although the architecture can be straightforwardly extended to hierarchically stacked RNNs. An RNN is similar to a feedforward network except that it possesses an additional hidden state h which is maintained and updated over time as a function of both the current input x and the previous hidden\n(a) Conv Layer 1 (b) Conv Layer 2\n(c) FC Layer 1 (d) FC Layer 2\nMean divergence between the true numerical and predictive coding backprops over the course of training. In general, the divergence appeared to follow a largely random walk pattern, and was generally neglible.\nImportantly, the divergence did not grow over time throughout training, implying that errors from slightly incorrect gradients did not appear to compound.\nSVHN training accuracy (a) SVHN test accuracy\n(b) CIFAR training accuracy (c) CIFAR test accuracy\nTraining and test accuracies of the CNN network on the SVHN and CIFAR datasets using the cross-entropy loss. As can be seen performance remains very close to backprop, thus demonstrating that our predictive coding\nalgorithm can be used with different loss functions, not just mean-squared-error.\nstate. The output of the network y is a function of h. By considering the RNN at a single timestep we obtain the following equations.\nht = f(θhht−1 + θxxt) (8) yt = g(θyht)\nWhere f and g are elementwise nonlinear activation functions. And θh, θx, θy are weight matrices for each specific input. To predict a sequence the RNN simply rolls forward the above equations to generate new predictions and hidden states at each timestep.\nRNNs are typically trained through an algorithm called backpropagation through time (BPTT) which essentially just unrolls the RNN into a single feedforward computation graph and then performs backpropagation through this unrolled graph. To train the RNN using predictive coding we take the same approach and simply apply predictive coding to the unrolled graph.\nIt is important to note that this is an additional aspect of biological implausibility that we do not address in this paper. BPTT requires updates to proceed backwards through time from the end of the sequence to the beginning. Ignoring any biological implausibility with the rules themselves, this updating sequence is clearly not biologically plausible as naively it requires maintaining the entire sequence of predictions and prediction errors perfectly in memory until the end of the sequence, and waiting until the sequence ends before making any updates. There is a small literature on trying to produce biologically plausible, or forward-looking approximations to BPTT which does not require updates to be propagated back through time (Williams and Zipser, 1989; Lillicrap and Santoro, 2019; Steil, 2004; Ollivier et al., 2015; Tallec and Ollivier, 2017). While this is a fascinating area, we do not address it in this paper. We are solely concerned with the fact that predictive coding approximates backpropagation on feedforward computation graphs for which the unrolled RNN graph is a sufficient substrate.\nTo learn a predictive coding RNN, we first augment each of the variables ht and yt of the original graph with additional error units ht and yt . Predictions ŷt, ĥt are generated according to the feedforward rules (16). A sequence of true labels {T1...TT } is then presented to the network, and then inference proceeds by recursively applying the following rules backwards through time until convergence.\nyt = L− ŷt ht = ht − ĥt dht dt = ht − ytθTy − ht+1θTh\nUpon convergence the weights are updated according to the following rules.\ndθy dt = T∑ t=0 yt ∂g(θyht) ∂θy hTt\ndθx dt = T∑ t=0 ht ∂f(θhht−1 + θxxt) ∂θx xTt\ndθh dt = T∑ t=0 ht ∂f(θhht−1 + θxxt) ∂θh hTt+1\nSince the RNN feedforward updates are parameter-linear, these rules are Hebbian, only requiring the multiplication of pre and post-synaptic potentials. This means that the predictive coding updates proposed here are biologically plausible and could in theory be implemented in the brain. The only biological implausibility remains the BPTT learning scheme.\nOur RNN was trained on a simple character-level name-origin dataset which can be found here: https://download.pytorch.org/tutorial/data.zip. The RNN was presented with sequences of characters representing names and had to predict the national origin of the name – French, Spanish, Russian, etc. The characters were presented to the network as one-hot-encoded vectors without any embedding. The output categories were also presented as a one-hot vector. The RNN has a hidden size of 256\nunits. A tanh nonlinearity was used between hidden states and the output layer was linear. The network was trained on randomly selected name-category pairs from the dataset. The training loss for the predictive coding and backprop RNNs, averaged over 5 seeds is presented below (Figure 7).\nAPPENDIX C: PREDICTIVE CODING LSTM IMPLEMENTATION DETAILS\nUnlike the other two models, the LSTM possesses a complex and branching internal computation graph, and is thus a good opportunity to make explicit the predictive coding ’recipe’ for approximating backprop on arbitrary computation graphs. The computation graph for a single LSTM cell is shown (with backprop updates) in Figure 8. Prediction for the LSTM occurs by simply rolling forward a copy of the LSTM cell for each timestep. The LSTM cell receives its hidden state ht and cell state ct from the previous timestep. During training we compute derivatives on the unrolled computation graph and receive backwards derivatives (or prediction errors) from the LSTM cell at time t+ 1.\nThe equations that specify the computation graph of the LSTM cell are as follows.\nv1 = ht ⊕ xt v2 = σ(θiv1)\nv3 = ctv2\nv4 = σ(θinpv1)\nv5 = tanh(θcv1)\nv6 = v4v5\nv7 = v3 + v6\nv8 = σ(θov1)\nv9 = tanh(v7)\nv10 = v8v9\ny = σ(θyv10)\nThe recipe to convert this computation graph into a predictive coding algorithm is straightforward. We first rewire the connectivity so that the predictions are set to the forward functions of their parents. We then compute the errors between the vertices and the predictions.\nv̂1 = ht ⊕ xt v̂2 = σ(θiv1)\nv̂3 = ctv2\nv̂4 = σ(θinpv1)\nv̂5 = tanh(θcv1)\nv̂6 = v4v5\nv̂7 = v3 + v6\nv̂ = σ(θov1)\nv̂9 = tanh(v7)\nv̂10 = v8v9\nv̂y = σ(θyv10)\n1 = v1 − v̂1 2 = v2 − v̂2 3 = v3 − v̂3 4 = v4 − v̂4 5 = v5 − v̂5 6 = v6 − v̂6 7 = v7 − v̂7 8 = v8 − v̂8 9 = v9 − v̂9 10 = v10 − v̂10\nDuring inference, the inputs ht,xt and the output yt are fixed. The vertices and then the prediction errors are updated according to Equation 2. This recipe is straightforward and can easily be extended to other more complex machine learning architectures. The full augmented computation graph, including the vertex update rules, is presented in Figure 9.\nEmpirically, we observed rapid convergence to the exact backprop gradients even in the case of very deep computation graphs (as is an unrolled LSTM with a sequence length of 100). Although convergence was slower than was the case for CNNs or lesser sequence lengths, it was still straightforward to achieve convergence to the exact numerical gradients with sufficient iterations.\nBelow we plot the mean divergence between the predictive coding and true numerical gradients as a function of sequence length (and hence depth of graph) for a fixed computational budget of 200 iterations with an inference learning rate of 0.05. As can be seen, the divergence increases roughly\nlinearly with sequence length. Importantly, even with long sequences, the divergence is not especially large, and can be decreased further by increasing the computational budget. As the increase is linear, we believe that predictive coding approaches should be scalable even for backpropagating through very deep and complex graphs.\nWe also plot the number of iterations required to reach a given convergence threshold (here taken to be 0.005) as a function of sequence length (Figure 11). We see that the number of iterations required increases sublinearly with the sequence length, and likely asymptotes at about 300 iterations. Although this is a lot of iterations, the sublinear convergence nevertheless shows that the method can scale to even extremely deep graphs.\nOur architecture consisted of a single LSTM layer (more complex architectures would consist of multiple stacked LSTM layers). The LSTM was trained on a next-character character-level prediction task. The dataset was the full works of Shakespeare, downloadable from Tensorflow. The text was shuffled and split into sequences of 50 characters, which were fed to the LSTM one character at a time. The LSTM was trained then to predict the next character, so as to ultimately be able to generate text. The characters were presented as one-hot-encoded vectors. The LSTM had a hidden size and a cell-size of 1056 units. A minibatch size of 64 was used and a weight learning rate of 0.0001 was used for both predictive coding and backprop networks. To achieve sufficient numerical convergence to the correct gradient, we used 200 variational iterations with an inference learning rate of 0.1. This rendered the predictive LSTM approximately 200x as costly as the backprop LSTM to run. A graph of the LSTM training loss for both predictive coding and backprop LSTMs, averaged over 5 random seeds, can be found below (Figure 12)."},{"heading":"APPENDIX D: DERIVATION OF THE FREE ENERGY FUNCTIONAL","text":"Here we derive in detail the form of the free-energy functional used in sections 2 and 4. We also expand upon the assumptions required and the precise form of the generative model and variational density. Much of this material is presented with considerably more detail in Buckley et al. (2017), and more approachably in Bogacz (2017).\nGiven an arbitrary computation graph with vertices {yi}, which we treat as random variables. Here we treat explicitly an important fact that we glossed over for notational convenience in the introduction. The vis which are optimized in the free-energy functional are technically the mean parameters of\nthe variational density Q(yi; vi, σi) – i.e. they represent the mean (variational) belief of the value of the vertex. The vertex values in the model, which we here denote as {yi}, are technically separate. However, due to our Gaussian assumptions, and the expectation under the variational density, in effect we end up replacing the yi with the vi and optimizing the vis, so in the interests of space and notational simplicity we began as if the vis were variables in the generative model, but they are not. They are parameters of the variational distribution.\nGiven an input y0 and a target yN (the multiple input and/or output case is a straightforward generalization). We wish to infer the posterior p(y1:N−1|y0, yN ). We approximate this intractable posterior with variational inference. Variational inference proceeds by defining an approximate posterior Q(y1:N−1;φ) with some arbitrary parameters φ. We then wish to minimize the KL divergence between the true and approximate posterior.\nargmin φ\nKL[Q(y1:N−1;φ)‖p(y1:N−1|y0, yN )]\nAlthough this KL is itself intractable, since it includes the intractable posterior, we can derive a tractable bound on this KL called the variational free-energy.\nKL[Q(y1:N−1;φ)‖p(y1:N |y0, yN )] = KL[Q(y1:N−1)‖ p(y1:N , y0, yN )\np(y0, yN ) ]\n= KL[Q(y1:N ;φ)‖p(y1:N , y0)] + ln p(y0, yN ) ⇒ KL[Q(y1:N ;φ)‖p(y1:N−1, y0, yN )]︸ ︷︷ ︸\n−F\n≤ KL[Q(y1:N−1;φ)‖p(y1:N−1|y0, yN )]\n(9)\nWe define the negative free-energy−F = KL[Q(y1:N−1)‖p(y1:N−1, y0, yN )] which is a lower bound on the divergence between the true and approximate posteriors. By thus maximizing the negative free-energy (which is identical to the ELBO (Beal et al., 2003; Blei et al., 2017)), or equivalently minimizing the free-energy, we decrease this divergence and make the variational distribution a better approximation to the true posterior.\nTo proceed further, it is necessary to define an explicit form of the generative model p(y0, y1:N−1, yN ) and the approximate posterior Q(y1:N−1;φ). In predictive coding, we define a hierarchical Gaussian generative model which mirrors the exact structure of the computation graph\np(y0:N ) = N (y0; ȳ0,Σ0) N∏ i=1 N (yi; f(P(yi); θyj∈P(yi)),Σi);\nWhere essentially each vertex yi is a Gaussian with a mean which is a function of the prediction of all the parents of the vertex, and the parameters of their edge-functions. ȳ0 is effectively an ”input-prior” which is set to 0 throughout and ignored. The output vertices yN = T are set to the target T .\nWe also define the variational density to be Gaussian with mean v1:N−1 and variance σ1:N−1, but under a mean field approximation, so that the approximation at each node is independent of all others\n(note the variational variance is denoted σ while the variance of the generative model is denoted Σ. The lower-case σ is not used to denote a scalar variable – both variances can be multivariate – but to distinguish between variational and generative variances)\nQ(y1:N−1; v1:N−1, σ1:N−1) = N−1∏ i=1 N (yi; vi, σi)\nWe now can express the free-energy functional concretely. First we decompose it as the sum of an energy and an entropy −F = KL[Q(y1:N−1; v1:N−1, σ1:N−1)‖p(y0, y1:N−1, yN )]\n= −EQ(y1:N−1;v1:N−1,σ1:N−1)[ln p(y0, y1:N−1, yN )]︸ ︷︷ ︸ Energy +EQ(y1:N−1;v1:N−1,σ1:N−1)[lnQ(y1:N−1; v1:N−1, σ1:N−1)]︸ ︷︷ ︸ Entropy\nThen, taking the entropy term first, we can express it concretely in terms of normal distributions. EQ(y1:N−1;v1:N−1,σ1:N−1)[lnQ(y1:N−1; v1:N−1, σ1:N−1)] = EQ(y1:N−1;v1:N−1,σ1:N−1)[ N−1∑ i=1 lnN (yi; vi, σi)]\n= N−1∑ i=1 EQ(yi;vi,σi)[lnN (yi; vi, σi)]\n= N−1∑ i=1 EQ(yi;vi,σi)[− 1 2 ln det(2πσi]) + EQ(yi;vi,σi)[ (yi − vi)2 2σi ]\n= N−1∑ i=1 −1 2 ln det(2πσi]) + σi 2σi\n= N\n2 + N−1∑ i=1 −1 2 ln det(2πσi)\nThe entropy of a multivariate gaussian has a simple analytical form depending only on the variance. Next we turn to the energy term, which is more complex. To derive a clean analytical result, we must make a further assumption, the Laplace approximation, which requires the variational density to be tightly peaked around the mean so the only non-negligible contribution to the expectation is from regions around the mean. This means that we can successfully approximate the approximate posterior with a second-order Taylor expansion around the mean. From the first line onwards we ignore the ln p(y0) and ln p(yN |P(yN )) which lie outside the expectation. EQ(y1:N−1;v1:N−1,σ1:N−1)[ln p(y0:N )] = ln p(y0) + ln p(yN |P(yN )) + N−1∑ i=1 EQ(yi;vi,σi)[ln p(yi|P(yi))]\n= N∑ i=1 EQ[ln p(vi|P(yi))] + EQ[ ∂ ln p(yi|P(yi)) ∂yi (vi − yi)]\n+ EQ[ d2 ln p(vi|P(yi))\ndy2i (vi − yi)2]\n= N∑ i=1 ln p(vi|P(yi)) + ∂2 ln p(vi|P(yi)) ∂y2i σi\nWhere the second term in the Taylor expansion evaluates to 0 since EQ[yi − vi] = (vi − vi) = 0 and the third term contains the expression for the variance EQ[(yi − vi)2] = σi. We can then write out the full Laplace-encoded free-energy as:\n−F = N∑ i=1 ln p(vi|P(yi)) + ∂2 ln p(vi|P(yi)) ∂y2i σi −− 1 2 ln det(2πσi)\nWe wish to minimize F with respect to the variational parameters vi and σi. There is in fact a closedform expression for the optimal variational variance which can be obtained simply by differentiating and setting the derivative to 0.\n∂F ∂σi = ∂2 ln p(vi|P(yi)) ∂y2i − σ−1i\n∂F ∂σi = 0⇒ σ∗i = ∂2 ln p(vi|P(yi)) ∂y2i\n−1\nBecause of this analytical result for the variational variance, we do not need to consider it further in the optimisation problem, and only consider minimizing the variational means vi. This renders all the terms in the free-energy except the ln p(vi|P(yi)) terms constant with respect to the variational parameters. This allows us to write:\n−F ≈ ln p(yN |P(yN )) + N∑ i=1 ln p(vi|P(yi)) (10)\nas presented in section 2. The first term ln p(yN |P(yN )) is effectively the loss at the output (yN = T ) so becomes an additional prediction error ln p(yN |P(yN )) ∝ (T − v̂N )TΣ−1N (T − v̂N ) which can be absorbed into the sum over other prediction errors. Crucially, although the variational variances have an analytical form, the variances of the generative model (the precisions Σi) do not and can be optimised directly to improve the log model-evidence. These precisions allow for a kind of ’uncertainty-aware’ backprop.\nDERIVATION OF VARIATIONAL UPDATE RULES AND FIXED POINTS\nHere, starting from Equation 10, we show how to obtain the variational update rule for the vi’s (Equation 2), and the fixed point equations (Equation 5) (Friston, 2008; 2005; Bogacz, 2017). We first reduce the free-energy to a sum of prediction errors.\n−F ≈ N∑ i=1 ln p(vi|P(vi))\n≈ N∑ i=1 (vi − f(P(v1))TΣ−1i (vi − f(P(v1)) T + ln 2πΣ−1i\n= N∑ i=1 Ti i + ln 2πΣ −1 i\nWhere i = vi − f(P(v1)), and we have utilized the assumption made in section 2 that Σ−1 = I. By setting all precisions to the identity, we are implicitly assuming that all datapoints and vertices of the computational graph have equal variance. Next we assume that the dynamics of each vertex vi follow a gradient descent on the free-energy.\n−dvi dt = ∂F ∂vi = ∂ ∂vi [ N∑ j=1 Tj j ]\n= i ∂ i ∂vi\n+ ∑\nj∈C(vi)\nj ∂ j ∂vi\n= i − ∑\nj∈C(vi)\nj ∂v̂j ∂vi\nWhere we have used the fact that ∂ i∂vi = 1 and ∂ j ∂vj = −∂v̂j∂vi . To obtain the fixed point of the dynamics, we simply solve for dvidt = 0.\ndvi dt = ∂F ∂vi = 0\n⇒ 0 = i − ∑\nj∈C(vi)\nj ∂v̂j ∂vi\n⇒ ∗i = ∑\nj∈C(vi)\nj ∂v̂∗j ∂v∗i\nSimilarly, since ∗i = v ∗ i − v̂∗i then v∗i = ∗i + v̂∗i . So:\nv∗i = ∗ i + v̂ ∗ i\n= v̂∗i − ∑\nj∈C(vi)\nj ∂v̂∗j ∂v∗i"}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Deep learning has seen stunning successes in the last decade in computer vision (Krizhevsky et al., 2012; Szegedy et al., 2015), natural language processing and translation (Vaswani et al., 2017; Radford et al., 2019; Kaplan et al., 2020), and computer game playing (Mnih et al., 2015; Silver et al., 2017; Schrittwieser et al., 2019; Vinyals et al., 2019). While there is a great variety of architectures and models, they are all trained by gradient descent using gradients computed by automatic differentiation (AD). The key insight of AD is that it suffices to define a forward model which maps inputs to predictions according to some parameters. Then, using the chain rule of calculus, it is possible, as long as every operation of the forward model is differentiable, to differentiate back through the computation graph of the model so as to compute the sensitivity of every parameter in the model to the error at the output, and thus adjust every single parameter to best minimize the total loss. Early models were typically simple artificial neural networks where the computation graph is simply a composition of matrix multiplications and elementwise nonlinearities, and for which the implementation of automatic differentation has become known as ‘backpropagation’ (or ’backprop’). However, automatic differentiation allows for substantially more complicated graphs to be differentiated through, up to, and including, arbitrary programs (Griewank et al., 1989; Baydin et al., 2017; Paszke et al., 2017; Revels et al., 2016; Innes et al., 2019; Werbos, 1982; Rumelhart and Zipser, 1985; Linnainmaa, 1970). In recent years this has enabled the differentiation through differential equation solvers (Chen et al., 2018; Tzen and Raginsky, 2019; Rackauckas et al., 2019), physics engines (Degrave et al., 2019; Heiden et al., 2019), raytracers (Pal, 2019), and planning algorithms (Amos and Yarats, 2019; Okada et al., 2017). These advances allow the straightforward training of models which intrinsically embody complex processes and which can encode significantly more prior knowledge and structure about a given problem domain than previously possible.\\nModern deep learning has also been closely intertwined with neuroscience (Hassabis et al., 2017; Hawkins and Blakeslee, 2007; Richards et al., 2019). The backpropagation algorithm itself arose\\nas a technique for training multi-layer perceptrons – simple hierarchical models of neurons inspired by the brain (Werbos, 1982). Despite this origin, and its empirical successes, a consensus has emerged that the brain cannot directly implement backprop, since to do so would require biologically implausible connection rules (Crick, 1989). There are two principal problems. Firstly, backprop in the brain appears to require non-local information (since the activity of any specific neuron affects all subsequent neurons down to the final output neuron). It is difficult to see how this information could be transmitted ’backwards’ throughout the brain with the required fidelity without precise connectivity constraints. The second problem – the ‘weight transport problem’ is that backprop through MLP style networks requires identical forward and backwards weights. In recent years, however, a succession of models have been introduced which claim to implement backprop in MLP-style models using only biologically plausible connectivity schemes, and Hebbian learning rules (Liao et al., 2016; Guerguiev et al., 2017; Sacramento et al., 2018; Bengio and Fischer, 2015; Bengio et al., 2017; Ororbia et al., 2020; Whittington and Bogacz, 2019). Of particular significance is Whittington and Bogacz (2017) who show that predictive coding networks – a type of biologically plausible network which learn through a hierarchical process of prediction error minimization – are mathematically equivalent to backprop in MLP models. In this paper we extend this work, showing that predictive coding can not only approximate backprop in MLPs, but can approximate automatic differentiation along arbitrary computation graphs. This means that in theory there exist potentially biologically plausible algorithms for differentiating through arbitrary programs, utilizing only local connectivity. Moreover, in a class of models which we call parameter-linear, which includes many current machine learning models, the required update rules are Hebbian, raising the possibility that a wide range of current machine learning architectures may be faithfully implemented in the brain, or in neuromorphic hardware.\\nIn this paper we provide two main contributions. (i) We show that predictive coding converges to automatic differentiation across arbitrary computation graphs. (ii) We showcase this result by implementing three core machine learning architectures (CNNs, RNNs, and LSTMs) in a predictive coding framework which utilises only local learning rules and mostly Hebbian plasticity.\"\n },\n {\n \"heading\": \"2 PREDICTIVE CODING ON ARBITRARY COMPUTATION GRAPHS\",\n \"text\": \"Predictive coding is an influential theory of cortical function in theoretical and computational neuroscience. Central to the theory is the idea that the core function of the brain is to minimize prediction errors between what is expected to happen and what actually happens. Predictive coding\\nviews the brain as composed of multiple hierarchical layers which predict the activities of the layers below. Unpredicted activity is registered as prediction error which is then transmitted upwards for a higher layer to process. Over time, synaptic connections are adjusted so that the system improves at minimizing prediction error. Predictive coding possesses a wealth of empirical support (Friston, 2003; 2005; Bogacz, 2017; Whittington and Bogacz, 2019) and offers a single mechanism that accounts for diverse perceptual phenomena such as repetition-suppression (Auksztulewicz and Friston, 2016), endstopping (Rao and Ballard, 1999), bistable perception (Hohwy et al., 2008; Weilnhammer et al., 2017) and illusory motions (Lotter et al., 2016; Watanabe et al., 2018), and even attentional modulation of neural activity (Feldman and Friston, 2010; Kanai et al., 2015). Moreover, the central role of top-down predictions is consistent with the ubiquity, and importance of, top-down diffuse connections between cortical areas. Predictive coding is consistent with many known aspects of neurophysiology, and has been translated into biologically plausible process theories which define candidate cortical microcircuits which can implement the algorithm. (Spratling, 2008; Bastos et al., 2012; Kanai et al., 2015; Shipp, 2016).\\nIn previous work, predictive coding has always been conceptualised as operating on hierarchies of layers (Bogacz, 2017; Whittington and Bogacz, 2017). Here we present a generalized form of predictive coding applied to arbitrary computation graphs. A computation graph G = {E,V} is a directed acyclic graph (DAG) which can represent the computational flow of essentially any program or computable function as a composition of elementary functions. Each edge ei ∈ E of the graph corresponds to an intermediate step – the application of an elementary function – while each vertex vi ∈ V is an intermediate variable computed by applying the functions of the edges to the values of their originating vertices. In this paper, vi denotes the vector of activations within a layer and we denote the set of all vertices as {vi}. Effectively, computation flows ’forward’ from parent nodes to all their children through the edge functions until the leaf nodes give the final output of the program as a whole (see Figure 1 and 2 for an example). Given a target T and a loss function L = g(T, vout), the graph’s output can be evaluated and, and if every edge function is differentiable, automatic differentiation can be performed on the computation graph.\\nPredictive coding can be derived elegantly as a variational inference algorithm under a hierarchical Gaussian generative model (Friston, 2005; Buckley et al., 2017). We extend this approach to arbitrary computation graphs in a supervised setting by defining the inference problem to be solved as that of inferring the vertex value vi of each node in the graph given fixed start nodes v0 (the data), and end nodes vN (the targets). We define a generative model which parametrises the value of each vertex given the feedforward prediction of its parents, p({vi}) = p(v0 . . . vN ) = ∏N i p(vi|P(vi)) 1, and a factorised,\\nvariational posterior Q({vi}|v0, vN ) = Q(v1 . . . vN−1|v0, vN ) = ∏N i Q(vi|P(vi), C(vi)), where P(vi) denotes the set of parents and C(vi) denotes the set of children of a given node vi. From this, we can define a suitable objective functional, the variational free-energy F (VFE), which acts as an upper bound on the divergence between the true and variational posteriors.\\nF = KL[(Q(v1 . . . vN−1|v0, vN )‖p(v0 . . . vN )] ≥ KL[(Q(v1 . . . vN−1)|v0, vN )‖p(v1 . . . vN−1|v0, vN )]\\n≈ N∑ i=0 Ti i\\n(1) Under Gaussian assumptions for the generative model p({vi}) = ∏N i N (vi; v̂i,Σi), and the vari-\\national posterior Q({vi}) = ∏N i N (vi), where the ‘predictions’ v̂i = f(P(vi); θi) are defined as the feedforward value of the vertex produced by running the graph forward, and all the precisions, or inverse variances, Σ−1i are fixed at the identity, we can write F as simply a sum of prediction errors (see Appendix D or (Friston, 2003; Bogacz, 2017; Buckley et al., 2017) for full derivations), with the prediction errors defined as i = vi − v̂i. These prediction errors play a core role in the framework and, in the biological process theories (Friston, 2005; Bastos et al., 2012), are generally considered to be represented by a distinct population of ‘error units’. Since F is an upper bound on the divergence between true and approximate posteriors, by minimizing F , we reduce this divergence, thus improving the quality of the variational posterior and approximating exact Bayesian inference. Predictive coding minimizes F by employing the Cauchy method of steepest descent to set the\\n1This includes the prior p(v0), which simply has no parents.\\ndynamics of the vertex variables vi as a gradient descent directly on F (Bogacz, 2017). dvi dt = ∂F ∂vi = i − ∑\\nj∈C(vi)\\nj ∂v̂j ∂vi\\n(2)\\nThe dynamics of the parameters of the edge functions θ such that v̂i = f(P(vi); θ), can also be derived as a gradient descent on F . Importantly these dynamics require only information (the current vertex value, prediction error, and prediction errors of child vertices) locally available at the vertex.\\ndθi dt = ∂F ∂θi = i ∂v̂i ∂θi\\n(3)\\nTo run generalized predictive coding in practice on a given computation graph G = {E,V}, we augment the graph with error units ∈ E to obtain an augumented computation graph G̃ = {E,V, E}. The predictive coding algorithm then operates in two phases – a feedforward sweep and a backwards iteration phase. In the feedforward sweep, the augmented computation graph is run forward to obtain the set of predictions {v̂i}, and prediction errors { i} = {vi − v̂i} for every vertex. Following Whittington and Bogacz (2017), to achieve exact equivalence with the backprop gradients computed on the original computation graph, we initialize vi = v̂i in the initial feedforward sweep so that the output error computed by the predictive coding network and the original graph are identical.\\nIn the backwards iteration phase, the vertex activities {vi} and prediction errors { i} are updated with Equation 2 for all vertices in parallel until the vertex values converge to a minimum of F . After convergence the parameters are updated according to Equation 3. Note we also assume, following Whittington and Bogacz (2017), that the predictions at each layer are fixed at the values assigned during the feedforward pass throughout the optimisation of the vs. We call this the fixed-prediction assumption. In effect, by removing the coupling between the vertex activities of the parents and the prediction at the child, this assumption separates the global optimisation problem into a local one for each vertex. We implement these dynamics with a simple forward Euler integration scheme so that the update rule for the vertices became vt+1i ← vti−η dFdvti where η is the step-size parameter. Importantly, if the edge function linearly combines the activities and the parameters followed by an elementwise nonlinearity – a condition which we call ‘parameter-linear’ – then both the update rule for the vertices (Equation 2) and the parameters (Equation 3) become Hebbian. Specifically, the update rules for the vertices and weights become dvidt = i − ∑ j jf ′(θj v̂j)θ T j and dθi dt = if ′(θiv̂i)v̂i T , respectively.\"\n },\n {\n \"heading\": \"2.1 APPROXIMATION TO BACKPROP\",\n \"text\": \"Here we show that at the equilibrium of the dynamics, the prediction errors ∗i converge to the correct backpropagated gradients ∂L∂vi , and consequently the parameter updates (Equation 3) become precisely those of a backprop trained network. Standard backprop works by computing the gradient of a vertex as the sum of the gradients of the child vertices. Beginning with the gradient of the output vertex ∂L ∂vL\\n, it recursively computes the gradients of vertices deeper in the graph by the chain rule: ∂L\\n∂vi = ∑ j=C(vi) ∂L ∂vj ∂vj ∂vi\\n(4)\\nIn comparison, in our predictive coding framework, at the equilibrium point (dvidt = 0) the prediction errors ∗i become,\\n∗i = ∑\\nj∈C(vi)\\n∗j ∂v̂i ∂vj\\n(5)\\nImportantly, this means that the equilibrium value of the prediction error at a given vertex (Equation 5) satisfies the same recursive structure as the chain rule of backprop (Equation 4). Since this relationship is recursive, all that is needed for the prediction errors throughout the graph to converge to the backpropagated derivatives is for the prediction errors at the final layer to be equal to the output gradient: ∗L = ∂L ∂v̂L . To see this explicitly, consider a mean-squared-error loss function 2. at the\\n2While the mean-squared-error loss function fits most nicely with the Gaussian generative model, other loss functions can be used in practice. If the loss function can be represented as a log probability distribution, then the generative model can be amended to simply set the output distribution to that distribution. If not, then there is no fully consistent generative model (although all nodes except the output remain Gaussian), but the algorithm will still work in practice. See (Figure 6) in Appendix A for results for CNNs trained with a crossentropy loss.\\nAlgorithm 1: Generalized Predictive Coding Data: Dataset D = {X,L}, Augmented Computation Graph G̃ = {E,V, E}, inference learning rate ηv , weight learning rate ηθ begin\\n/* For each minibatch in the dataset */ for (x, L) ∈ D do\\n/* Fix start of graph to inputs */ v̂0 ← x /* Forward pass to compute predictions */ for v̂i ∈ V do\\nv̂i ← f({P(v̂i); θ) /* Compute output error */ L ← L− v̂L /* Begin backwards iteration phase of the descent on the\\nfree energy */ while not converged do\\nfor (vi, i) ∈ G̃ do /* Compute prediction errors */ i ← vi − v̂i /* Update the vertex values */ vt+1i ← vti + ηv dFdvti\\n/* Update weights at equilibrium */ for θi ∈ E do\\nθt+1i ← θti + ηθ dFdθti\\noutput layer L = 12 (T − v̂L) 2 with T as a vector of targets, and defining L = T − v̂L. We then consider the equilibrium value of the prediction error unit at a penultimate vertex L−1. By Equation 5, we can see that at equilibrium,\\n∗L−1 = ∗ L ∂v̂L ∂vL−1 = (T − v̂∗L) ∂v̂L ∂vL−1\\nsince, (T − v̂L) = ∂L∂v̂L , we can then write,\\n∗L−1 = ∂L\\n∂v̂L ∂v̂L ∂vL−1 = ∂L ∂vL−1 (6)\\nThus the prediction errors of the penultimate nodes converge to the correct backpropagated gradient. Furthermore, recursing through the graph from children to parents allows the correct gradients to be computed3. Thus, by induction, we have shown that the fixed points of the prediction errors of the global optimization correspond exactly to the backpropagated gradients. Intuitively, if we imagine the computation-graph as a chain and the error as ’tension’ in the chain, backprop loads all the tension at the end (the output) and then systematically propagates it backwards. Predictive coding, however, spreads the tension throughout the entire chain until it reaches an equilibrium where the amount of tension at each link is precisely the backpropagated gradient. The full algorithm for training the predictive coding network is explicitly set out in Algorithm 1. Inference is just a forward pass through the network, and is identical to the corresponding ANN.\\n3Some subtlety is needed here since vL−1 may have many children which each contribute to the loss. However, these different paths sum together at the node vL−1, thus propagating the correct gradient backwards.\\nBy a similar argument, it is apparent that the dynamics of the parameters θi as a gradient descent on F also exactly match the backpropagated parameter gradients.\\ndθi dt = dF dθi = ∗i d ∗i dθi\\n= dL\\ndv̂i dv̂i dθi = dL dθi (7)\\nWhich follows from the fact that ∗i = dL dv̂i\\nand that d ∗ i\\ndθ = dv̂i dθi .\"\n },\n {\n \"heading\": \"3 RELATED WORK\",\n \"text\": \"A number of recent works have tried to provide biologically plausible approximations to backprop. The requirement of symmetry between the forwards and backwards weights has been questioned by Lillicrap et al. (2016) who show that random fixed feedback weights suffice for effective learning. Recent additional work has shown that learning the backwards weights also helps (Amit, 2019; Akrout et al., 2019). Several schemes have also been proposed to approximate backprop using only local learning rules and/or Hebbian connectivity. These include target-prop (Lee et al., 2015) which approximate the backward gradients with trained inverse functions, but which fails to asymptotically compute the exact backprop gradients, and contrastive Hebbian (Seung, 2003; Scellier and Bengio, 2017; Scellier et al., 2018) approaches which do exactly approximate backprop, but which require two separate learning phases and the storing of information across successive phases. There are also dendritic error theories (Guerguiev et al., 2017; Sacramento et al., 2018) which are computationally similar to predictive coding (Whittington and Bogacz, 2019; Lillicrap et al., 2020). Whittington and Bogacz (2017) showed that predictive coding can approximate backprop in MLP models, and demonstrated comparable performance on MNIST. We advance upon this work by extending the proof to arbitrary computation graphs, enabling the design of predictive coding variants of a range of standard machine learning architectures, which we show perform comparably to backprop on considerably more difficult tasks than MNIST. Our algorithm evinces asymptotic (and in practice rapid) convergence to the exact backprop gradients, does not require separate learning phases, and utilises only local information and largely Hebbian plasticity.\"\n },\n {\n \"heading\": \"4 RESULTS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"4.1 NUMERICAL RESULTS\",\n \"text\": \"To demonstrate the correctness of our derivation and empirical convergence to the true gradients, we present a numerical test in the simple scalar case, where we use predictive coding to derive the gradients of an arbitrary, highly nonlinear test function vL = tan( √ θv0) + sin(v 2 0) where θ is an arbitrary parameter. For our tests, we set v0 to 5 and θ to 2. The computation graph for this function is presented in Figure 2. Although simple, this is a good test of predictive coding because the function is highly nonlinear, and its computation graph does not follow a simple layer structure but includes some branching. An arbitrary target of T = 3 was set at the output and the gradient of the loss L = (vL − T )2 with respect to the input v0 was computed by predictive coding. We show (Figure 2) that the predictive coding optimisation rapidly converges to the exact numerical gradients computed by automatic differentiation, and that moreover this optimization is very robust and can handle even exceptionally high learning rates (up to 0.5) without divergence.\\nIn summary, we have shown and numerically verified that at the equilibrium point of the global free-energy F on an arbitrary computation graph, the error units exactly equal the backpropagated gradients, and that this descent requires only local connectivity, does not require a separate phases or a sequential backwards sweep, and in the case of parameter-linear functions, requires only Hebbian plasticity. Our results provide a straightforward recipe for the direct implementation of predictive coding algorithms to approximate certain computation graphs, such as those found in common machine learning algorithms, in a potentially biologically plausible manner. Next, we showcase this capability by developing predictive coding variants of core machine learning architectures - convolutional neural networks (CNNs) recurrent neural networks (RNNs) and LSTMs (Hochreiter and Schmidhuber, 1997), and show performance comparable with backprop on tasks substantially more challenging than MNIST.\\nθ\\n* ( ⋅ , ⋅ ) ⋅ tan( ⋅ )\\n+( ⋅ , ⋅ )\\n( ⋅ )2 sin( ⋅ )\\nμ1 = θ * v0 μ2 = ( ̂v1) μ3 = tan( ̂v2)\\nμ4 = v20 μ5 = sin( ̂v4) μL = ̂v3 + ̂v5\\n̂v1 ̂v2 ̂v3\\n̂v4 ̂v5\\nϵ1 ϵ2 ϵ3\\nϵ5ϵ4\\nT ϵL\\nϵL dμL d ̂v3\\nϵL dμL d ̂v5\\nϵ5 dμ5 d ̂v4\\nϵ3 dμ3 d ̂v2ϵ2 dμ2 d ̂v1ϵ1\\ndμ1 dθ\\nϵ1 dμ1 dv0\\nϵ4 dμ4 dv0\\nv0\"\n },\n {\n \"heading\": \"4.2 PREDICTIVE CODING CNN, RNN, AND LSTM\",\n \"text\": \"First, we constructed predictive coding CNN models (see Appendix B for full implementation details). In the predictive coding CNN, each filter kernel was augmented with ‘error maps’ which measured the difference between the forward convolutional predictions and the backwards messages. Our CNN was composed of a convolutional layer, followed by a max-pooling layer, then two further convolutional layers followed by 3 fully-connected layers. We compared our predictive coding CNN to a backprop-trained CNN with the exact same architecture and hyperparameters. We tested our models on three image classification datasets significantly more challenging than MNIST – SVHN, CIFAR10, and CIFAR100. SVHN is a digit recognition task like MNIST, but has more naturalistic backgrounds, is in colour with continuously varying inputs and contains distractor digits. CIFAR10 and CIFAR100 are large image datasets composed of RGB 32x32 images. CIFAR10 has\\n10 classes of image, while CIFAR100 is substantially more challenging with 100 possible classes. In general (Figure 3), performance was identical between the predictive coding and backprop CNNs and comparable to the standard performance of basic CNN models on these datasets, Moreover, the predictive coding gradient remained close to the true numerical gradient throughout training.\\nWe also constructed predictive coding RNN and LSTM models, thus demonstrating the ability of predictive coding to scale to non-parameter-linear, branching, computation graphs. The RNN was trained on a character-level name classification task, while the LSTM was trained on a next-character prediction task on the full works of Shakespeare. Full implementation details can be found in Appendices B and C. LSTMs and RNNs are recurrent networks which are trained through backpropagation through time (BPTT). BPTT simply unrolls the network through time and backpropagates through the unrolled graph. Analogously we trained the predictive coding RNN and LSTM by applying predictive coding to the unrolled computation graph. The depth of the unrolled graph depends heavily on the sequence length, and in our tasks using a sequence length of 100 we still found that predictive coding evinced rapid convergence to the correct numerical gradient, and that the performance was approximately identical to the equivalent backprop-trained networks (Figure 3), thus showing that the algorithm is scalable even to very deep computation graphs.\"\n },\n {\n \"heading\": \"5 DISCUSSION\",\n \"text\": \"We have shown that predictive coding provides a local and potentially biologically plausible approximation to backprop on arbitrary, deep, and branching computation graphs. Moreover, convergence to the exact backprop gradients is rapid and robust, even in extremely deep graphs such as the unrolled LSTM. Our algorithm is fully parallelizable, does not require separate phases, and can produce equivalent performance to backprop in core machine-learning architectures. These results broaden the horizon of local approximations to backprop by demonstrating that they can be implemented on arbitrary computation graphs, not only simple MLP architectures. Our work prescribes a straightforward recipe for backpropagating through any computation graph with predictive coding using only local learning rules. In the future, this process could potentially be made fully automatic and translated onto neuromorphic hardware. Our results also raise the possibility that the brain may implement machine-learning type architectures much more directly than often considered. Many lines of work suggest a close correspondence between the representations and activations of CNNs and activity in higher visual areas (Yamins et al., 2014; Tacchetti et al., 2017; Eickenberg et al., 2017; Khaligh-Razavi and Kriegeskorte, 2014; Lindsay, 2020), for instance, and this similarity may be found to extend to other machine learning architectures.\\nIt is important to note that predictive coding, as advanced here, still retains some biologically implausible features. Although using only local and Hebbian updates, the predictive coding algorithm still requires identical forward and backwards weights, as well as mandating a very precise oneto-one connectivity structure between value neurons vi and error neurons i. However, recent work (Millidge et al., 2020) has begun to show that these implausibilities can be relaxed using learnable backwards weights instead of requiring weight symmetry, and allowing for learnable dense connectivity between value and error neurons, without harm to performance in simple MLP settings. An additional limitation to the biological plausibility of our method is the fixed-prediction assumption, which requires that the feedforward pass values be somehow stored during the backwards\\niteration phase. In biological neurons this could potentially be implemented by utilizing synaptic mechanisms for maintaining information over short periods, such as eligibility traces, or alternatively through synchronised phase locking (Buzsaki, 2006). Alternatively, it is important to note that this fixed-prediction assumption is only required for exact convergence to backprop, and predictive coding networks have been shown to be able to attain strong discriminative classification performance without it (Whittington and Bogacz, 2017).\\nAlthough we have implemented three core machine learning architectures as predictive coding networks, we have nevertheless focused on relatively small and straightforward networks and thus both our backprop and predictive coding networks perform below the state of the art on the presented tasks. This is primarily because our focus was on demonstrating the theoretical convergence between the two algorithms. Nevertheless, we believe that due to the generality of our theoretical results, ’scaling up’ the existing architectures to implement performance-matched predictive coding versions of more advanced machine learning architectures such as resnets (He et al., 2016), GANs (Goodfellow et al., 2014), and transformers (Vaswani et al., 2017) should be relatively straightforward.\\nIn terms of computational cost, one inference iteration in the predictive coding network is about as costly as a backprop backwards pass. Thus, due to using 100-200 iterations for full convergence, our algorithm is substantially more expensive than backprop which limits the scalability of our method. However, this serial cost is misleading when talking about highly parallel neural architectures. In the brain, neurons cannot wait for a sequential forward and backward sweep. By phrasing our algorithm as a global descent, our algorithm is fully parallel across layers. There is no waiting and no phases to be coordinated. Each neuron need only respond to its local driving inputs and downwards error signals. We believe that this local and parallelizable property of our algorithm may engender the possibility of substantially more efficient implementations on neuromorphic hardware (Furber et al., 2014; Merolla et al., 2014; Davies et al., 2018), which may ameliorate much of the computational overhead compared to backprop. Future work could also examine whether our method is more capable than backprop of handling the continuously varying inputs the brain is presented with in practice, rather than the artificial paradigm of being presented with a series of i.i.d. datapoints.\\nOur work also reveals a close connection between backprop and inference. Namely, the recursive computation of gradients is effectively a by-product of a variational-inference algorithm which infers the values of the vertices of the computation graph under a hierarchical Gaussian generative model. While the deep connections between stochastic gradient descent and inference in terms of Kalman filtering (Ruck et al., 1992; Ollivier, 2019) or MCMC sampling methods (Chen et al., 2014; Mandt et al., 2017) is known, the relation between recursive gradient computation itself and variational inference is underexplored except in the case of a single layer (Amari, 1995). Our method can provide a principled generalisation of backprop through the inverse-variance Σ−1 parameters of the Gaussian generative model. These parameters weight the relative contribution of different factors to the overall gradient by their uncertainty, thus naturally handling the case of backprop with differentially noisy inputs. Moreover, the Σ−1 parameters can be learnt as a gradient descent on F : dΣidt = − dF dΣi = −Σ−1i i Ti Σ −1 i − Σ −1 i . This specific generalisation is afforded by the Gaussian form of the generative model, however, and other generative models may yield novel optimisation algorithms able to quantify and handle uncertainties throughout the entire computational graph.\"\n },\n {\n \"heading\": \"APPENDIX A: PREDICTIVE CODING CNN IMPLEMENTATION DETAILS\",\n \"text\": \"The key concept in a CNN is that of an image convolution, where a small weight matrix is ’slid’ (or convolved) across an image to produce an output image. Each patch of the output image only depends on a relatively small patch of the input image. Moreover, the weights of the filter stay the same during the convolution, so each pixel of the output image is generated using the same weights. The weight sharing implicit in the convolution operation enforces translational invariance, since different image patches are all processed with the same weights.\\nThe forward equations of a convolutional layer for a specific output pixel\\nvi,j = k=i+f∑ k=i−f l=j+f∑ l=j−f θk,lxi+k,j+l\\nWhere vi,j is the (i, j)th element of the output, xi,j is the element of the input image and θk,l is an weight element of a feature map. To setup a predictive coding CNN, we augment each intermediate xi and vi with error units i of the same dimension as the output of the convolutional layer.\\nPredictions v̂ are projected forward using the forward equations. Prediction errors also need to be transmitted backwards for the architecture to work. To achieve this we must have that prediction errors are transmitted upwards by a ’backwards convolution’. We thus define the backwards prediction errors ̂j as follows:\\n̂i,j = i+f∑ k=i−f j+f∑ l=j−f θj,ĩi,j\\nWhere ̃ is an error map zero-padded to ensure the correct convolutional output size. Inference in the predictive coding network then proceeds by updating the intermediate values of each layer as follows:\\ndvl dt = l − ̂l+1\\nSince the CNN is also parameter-linear, weights can be updated using the simple Hebbian rule of the multiplication of the pre and post synaptic potentials.\\ndθl dt = ∑ i,j li,jvl−1 T i,j\\nThere is an additional biological implausibility here due to the weight sharing of the CNN. Since the same weights are copied for each position on the image, the weight updates have contributions from all aspects of the image simultaneously which violates the locality condition. A simple fix for this, which makes the network scheme plausible is to simply give each position on the image a filter with separate weights, thus removing the weight sharing implicit in the CNN. In effect this gives each patch of pixels a local receptive field with its own set of weights. The performance and scalability\\nof such a locally connected predictive coding architecture would be an interesting avenue for future work, as this architecture has substantial homologies with the structure of the visual cortex.\\nIn our experiments we used a relatively simple CNN architecture consisting of one convolutional layer of kernel size 5, and a filter bank of 6 filters. This was followed by a max-pooling layer with a (2,2) kernel and a further convolutional layer with a (5,5) kernel and filter bank of 16 filters. This was then followed by three fully connected layers of 200, 150, and 10 (or 100 for CIFAR100) output units. Each convolutional and fully connected layer used the relu activation function, except the output layer which was linear. Although this architecture is far smaller than state of the art for convolutional networks, the primary point of our paper was to demonstrate the equivalence of predictive coding and backprop. Further work could investigate scaling up predictive coding to more state-of-the-art architectures.\\nOur datasets consisted of 32x32 RGB images. We normalised the values of all pixels of each image to lie between 0 and 1, but otherwise performed no other image preprocessing. We did not use data augmentation of any kind. We set the weight learning rate for the predictive coding and backprop networks 0.0001. A minibatch size of 64 was used. These parameters were chosen without any detailed hyperparameter search and so are likely suboptimal. The magnitude of the gradient updates was clamped to lie between -50 and 50 in all of our models. This was done to prevent divergences, as occasionally occurred in the LSTM networks, likely due to exploding gradients.\\nThe predictive coding scheme converged to the exact backprop gradients very precisely within 100 inference iterations using an inference learning rate of 0.1. This gives the predictive coding CNN approximately a 100x computational overhead compared to backprop. The divergence between the true and approximate gradients remained approximately constant throughout training, as shown by Figure 5, which shows the mean divergence for each layer of the CNN over the course of an example training run on the CIFAR10 dataset. The training loss of the predictive coding and backprop networks for SVHN, CIFAR10 and CIFAR100 are presented in Figure 4.\\nWhile the experiments in the main paper all used the mean-squared-error loss function, it is also possible to use alternative loss functions. In Figure 6, we show performance of the CNN on CIFAR and SVHN datasets is also very close to backprop when trained with a multi-class cross-entropy loss L = ∑ i Ti ln vLi. In this case the output layer used a softmax function as its nonlinearity, to ensure that the logits passed to the cross-entropy loss were valid probabilities. The cross-entropy loss is also straightforward to fit into the predictive coding framework since the gradient with respect to the pre-activations of the output is also just the negative prediction error ∂L∂vL = T − vL, although the softmax function itself may be challenging to implement neurally since it is non-local as its’ normalisation coefficient requires of the exponentiated activities of all neurons in a layer. Nevertheless, this demonstrates that predictive coding can approximate backprop for any given loss function, not simply mean-square-error.\"\n },\n {\n \"heading\": \"APPENDIX B: PREDICTIVE CODING RNN\",\n \"text\": \"The computation graph on RNNs is relatively straightforward. We consider only a single layer RNN here although the architecture can be straightforwardly extended to hierarchically stacked RNNs. An RNN is similar to a feedforward network except that it possesses an additional hidden state h which is maintained and updated over time as a function of both the current input x and the previous hidden\\n(a) Conv Layer 1 (b) Conv Layer 2\\n(c) FC Layer 1 (d) FC Layer 2\\nMean divergence between the true numerical and predictive coding backprops over the course of training. In general, the divergence appeared to follow a largely random walk pattern, and was generally neglible.\\nImportantly, the divergence did not grow over time throughout training, implying that errors from slightly incorrect gradients did not appear to compound.\\nSVHN training accuracy (a) SVHN test accuracy\\n(b) CIFAR training accuracy (c) CIFAR test accuracy\\nTraining and test accuracies of the CNN network on the SVHN and CIFAR datasets using the cross-entropy loss. As can be seen performance remains very close to backprop, thus demonstrating that our predictive coding\\nalgorithm can be used with different loss functions, not just mean-squared-error.\\nstate. The output of the network y is a function of h. By considering the RNN at a single timestep we obtain the following equations.\\nht = f(θhht−1 + θxxt) (8) yt = g(θyht)\\nWhere f and g are elementwise nonlinear activation functions. And θh, θx, θy are weight matrices for each specific input. To predict a sequence the RNN simply rolls forward the above equations to generate new predictions and hidden states at each timestep.\\nRNNs are typically trained through an algorithm called backpropagation through time (BPTT) which essentially just unrolls the RNN into a single feedforward computation graph and then performs backpropagation through this unrolled graph. To train the RNN using predictive coding we take the same approach and simply apply predictive coding to the unrolled graph.\\nIt is important to note that this is an additional aspect of biological implausibility that we do not address in this paper. BPTT requires updates to proceed backwards through time from the end of the sequence to the beginning. Ignoring any biological implausibility with the rules themselves, this updating sequence is clearly not biologically plausible as naively it requires maintaining the entire sequence of predictions and prediction errors perfectly in memory until the end of the sequence, and waiting until the sequence ends before making any updates. There is a small literature on trying to produce biologically plausible, or forward-looking approximations to BPTT which does not require updates to be propagated back through time (Williams and Zipser, 1989; Lillicrap and Santoro, 2019; Steil, 2004; Ollivier et al., 2015; Tallec and Ollivier, 2017). While this is a fascinating area, we do not address it in this paper. We are solely concerned with the fact that predictive coding approximates backpropagation on feedforward computation graphs for which the unrolled RNN graph is a sufficient substrate.\\nTo learn a predictive coding RNN, we first augment each of the variables ht and yt of the original graph with additional error units ht and yt . Predictions ŷt, ĥt are generated according to the feedforward rules (16). A sequence of true labels {T1...TT } is then presented to the network, and then inference proceeds by recursively applying the following rules backwards through time until convergence.\\nyt = L− ŷt ht = ht − ĥt dht dt = ht − ytθTy − ht+1θTh\\nUpon convergence the weights are updated according to the following rules.\\ndθy dt = T∑ t=0 yt ∂g(θyht) ∂θy hTt\\ndθx dt = T∑ t=0 ht ∂f(θhht−1 + θxxt) ∂θx xTt\\ndθh dt = T∑ t=0 ht ∂f(θhht−1 + θxxt) ∂θh hTt+1\\nSince the RNN feedforward updates are parameter-linear, these rules are Hebbian, only requiring the multiplication of pre and post-synaptic potentials. This means that the predictive coding updates proposed here are biologically plausible and could in theory be implemented in the brain. The only biological implausibility remains the BPTT learning scheme.\\nOur RNN was trained on a simple character-level name-origin dataset which can be found here: https://download.pytorch.org/tutorial/data.zip. The RNN was presented with sequences of characters representing names and had to predict the national origin of the name – French, Spanish, Russian, etc. The characters were presented to the network as one-hot-encoded vectors without any embedding. The output categories were also presented as a one-hot vector. The RNN has a hidden size of 256\\nunits. A tanh nonlinearity was used between hidden states and the output layer was linear. The network was trained on randomly selected name-category pairs from the dataset. The training loss for the predictive coding and backprop RNNs, averaged over 5 seeds is presented below (Figure 7).\\nAPPENDIX C: PREDICTIVE CODING LSTM IMPLEMENTATION DETAILS\\nUnlike the other two models, the LSTM possesses a complex and branching internal computation graph, and is thus a good opportunity to make explicit the predictive coding ’recipe’ for approximating backprop on arbitrary computation graphs. The computation graph for a single LSTM cell is shown (with backprop updates) in Figure 8. Prediction for the LSTM occurs by simply rolling forward a copy of the LSTM cell for each timestep. The LSTM cell receives its hidden state ht and cell state ct from the previous timestep. During training we compute derivatives on the unrolled computation graph and receive backwards derivatives (or prediction errors) from the LSTM cell at time t+ 1.\\nThe equations that specify the computation graph of the LSTM cell are as follows.\\nv1 = ht ⊕ xt v2 = σ(θiv1)\\nv3 = ctv2\\nv4 = σ(θinpv1)\\nv5 = tanh(θcv1)\\nv6 = v4v5\\nv7 = v3 + v6\\nv8 = σ(θov1)\\nv9 = tanh(v7)\\nv10 = v8v9\\ny = σ(θyv10)\\nThe recipe to convert this computation graph into a predictive coding algorithm is straightforward. We first rewire the connectivity so that the predictions are set to the forward functions of their parents. We then compute the errors between the vertices and the predictions.\\nv̂1 = ht ⊕ xt v̂2 = σ(θiv1)\\nv̂3 = ctv2\\nv̂4 = σ(θinpv1)\\nv̂5 = tanh(θcv1)\\nv̂6 = v4v5\\nv̂7 = v3 + v6\\nv̂ = σ(θov1)\\nv̂9 = tanh(v7)\\nv̂10 = v8v9\\nv̂y = σ(θyv10)\\n1 = v1 − v̂1 2 = v2 − v̂2 3 = v3 − v̂3 4 = v4 − v̂4 5 = v5 − v̂5 6 = v6 − v̂6 7 = v7 − v̂7 8 = v8 − v̂8 9 = v9 − v̂9 10 = v10 − v̂10\\nDuring inference, the inputs ht,xt and the output yt are fixed. The vertices and then the prediction errors are updated according to Equation 2. This recipe is straightforward and can easily be extended to other more complex machine learning architectures. The full augmented computation graph, including the vertex update rules, is presented in Figure 9.\\nEmpirically, we observed rapid convergence to the exact backprop gradients even in the case of very deep computation graphs (as is an unrolled LSTM with a sequence length of 100). Although convergence was slower than was the case for CNNs or lesser sequence lengths, it was still straightforward to achieve convergence to the exact numerical gradients with sufficient iterations.\\nBelow we plot the mean divergence between the predictive coding and true numerical gradients as a function of sequence length (and hence depth of graph) for a fixed computational budget of 200 iterations with an inference learning rate of 0.05. As can be seen, the divergence increases roughly\\nlinearly with sequence length. Importantly, even with long sequences, the divergence is not especially large, and can be decreased further by increasing the computational budget. As the increase is linear, we believe that predictive coding approaches should be scalable even for backpropagating through very deep and complex graphs.\\nWe also plot the number of iterations required to reach a given convergence threshold (here taken to be 0.005) as a function of sequence length (Figure 11). We see that the number of iterations required increases sublinearly with the sequence length, and likely asymptotes at about 300 iterations. Although this is a lot of iterations, the sublinear convergence nevertheless shows that the method can scale to even extremely deep graphs.\\nOur architecture consisted of a single LSTM layer (more complex architectures would consist of multiple stacked LSTM layers). The LSTM was trained on a next-character character-level prediction task. The dataset was the full works of Shakespeare, downloadable from Tensorflow. The text was shuffled and split into sequences of 50 characters, which were fed to the LSTM one character at a time. The LSTM was trained then to predict the next character, so as to ultimately be able to generate text. The characters were presented as one-hot-encoded vectors. The LSTM had a hidden size and a cell-size of 1056 units. A minibatch size of 64 was used and a weight learning rate of 0.0001 was used for both predictive coding and backprop networks. To achieve sufficient numerical convergence to the correct gradient, we used 200 variational iterations with an inference learning rate of 0.1. This rendered the predictive LSTM approximately 200x as costly as the backprop LSTM to run. A graph of the LSTM training loss for both predictive coding and backprop LSTMs, averaged over 5 random seeds, can be found below (Figure 12).\"\n },\n {\n \"heading\": \"APPENDIX D: DERIVATION OF THE FREE ENERGY FUNCTIONAL\",\n \"text\": \"Here we derive in detail the form of the free-energy functional used in sections 2 and 4. We also expand upon the assumptions required and the precise form of the generative model and variational density. Much of this material is presented with considerably more detail in Buckley et al. (2017), and more approachably in Bogacz (2017).\\nGiven an arbitrary computation graph with vertices {yi}, which we treat as random variables. Here we treat explicitly an important fact that we glossed over for notational convenience in the introduction. The vis which are optimized in the free-energy functional are technically the mean parameters of\\nthe variational density Q(yi; vi, σi) – i.e. they represent the mean (variational) belief of the value of the vertex. The vertex values in the model, which we here denote as {yi}, are technically separate. However, due to our Gaussian assumptions, and the expectation under the variational density, in effect we end up replacing the yi with the vi and optimizing the vis, so in the interests of space and notational simplicity we began as if the vis were variables in the generative model, but they are not. They are parameters of the variational distribution.\\nGiven an input y0 and a target yN (the multiple input and/or output case is a straightforward generalization). We wish to infer the posterior p(y1:N−1|y0, yN ). We approximate this intractable posterior with variational inference. Variational inference proceeds by defining an approximate posterior Q(y1:N−1;φ) with some arbitrary parameters φ. We then wish to minimize the KL divergence between the true and approximate posterior.\\nargmin φ\\nKL[Q(y1:N−1;φ)‖p(y1:N−1|y0, yN )]\\nAlthough this KL is itself intractable, since it includes the intractable posterior, we can derive a tractable bound on this KL called the variational free-energy.\\nKL[Q(y1:N−1;φ)‖p(y1:N |y0, yN )] = KL[Q(y1:N−1)‖ p(y1:N , y0, yN )\\np(y0, yN ) ]\\n= KL[Q(y1:N ;φ)‖p(y1:N , y0)] + ln p(y0, yN ) ⇒ KL[Q(y1:N ;φ)‖p(y1:N−1, y0, yN )]︸ ︷︷ ︸\\n−F\\n≤ KL[Q(y1:N−1;φ)‖p(y1:N−1|y0, yN )]\\n(9)\\nWe define the negative free-energy−F = KL[Q(y1:N−1)‖p(y1:N−1, y0, yN )] which is a lower bound on the divergence between the true and approximate posteriors. By thus maximizing the negative free-energy (which is identical to the ELBO (Beal et al., 2003; Blei et al., 2017)), or equivalently minimizing the free-energy, we decrease this divergence and make the variational distribution a better approximation to the true posterior.\\nTo proceed further, it is necessary to define an explicit form of the generative model p(y0, y1:N−1, yN ) and the approximate posterior Q(y1:N−1;φ). In predictive coding, we define a hierarchical Gaussian generative model which mirrors the exact structure of the computation graph\\np(y0:N ) = N (y0; ȳ0,Σ0) N∏ i=1 N (yi; f(P(yi); θyj∈P(yi)),Σi);\\nWhere essentially each vertex yi is a Gaussian with a mean which is a function of the prediction of all the parents of the vertex, and the parameters of their edge-functions. ȳ0 is effectively an ”input-prior” which is set to 0 throughout and ignored. The output vertices yN = T are set to the target T .\\nWe also define the variational density to be Gaussian with mean v1:N−1 and variance σ1:N−1, but under a mean field approximation, so that the approximation at each node is independent of all others\\n(note the variational variance is denoted σ while the variance of the generative model is denoted Σ. The lower-case σ is not used to denote a scalar variable – both variances can be multivariate – but to distinguish between variational and generative variances)\\nQ(y1:N−1; v1:N−1, σ1:N−1) = N−1∏ i=1 N (yi; vi, σi)\\nWe now can express the free-energy functional concretely. First we decompose it as the sum of an energy and an entropy −F = KL[Q(y1:N−1; v1:N−1, σ1:N−1)‖p(y0, y1:N−1, yN )]\\n= −EQ(y1:N−1;v1:N−1,σ1:N−1)[ln p(y0, y1:N−1, yN )]︸ ︷︷ ︸ Energy +EQ(y1:N−1;v1:N−1,σ1:N−1)[lnQ(y1:N−1; v1:N−1, σ1:N−1)]︸ ︷︷ ︸ Entropy\\nThen, taking the entropy term first, we can express it concretely in terms of normal distributions. EQ(y1:N−1;v1:N−1,σ1:N−1)[lnQ(y1:N−1; v1:N−1, σ1:N−1)] = EQ(y1:N−1;v1:N−1,σ1:N−1)[ N−1∑ i=1 lnN (yi; vi, σi)]\\n= N−1∑ i=1 EQ(yi;vi,σi)[lnN (yi; vi, σi)]\\n= N−1∑ i=1 EQ(yi;vi,σi)[− 1 2 ln det(2πσi]) + EQ(yi;vi,σi)[ (yi − vi)2 2σi ]\\n= N−1∑ i=1 −1 2 ln det(2πσi]) + σi 2σi\\n= N\\n2 + N−1∑ i=1 −1 2 ln det(2πσi)\\nThe entropy of a multivariate gaussian has a simple analytical form depending only on the variance. Next we turn to the energy term, which is more complex. To derive a clean analytical result, we must make a further assumption, the Laplace approximation, which requires the variational density to be tightly peaked around the mean so the only non-negligible contribution to the expectation is from regions around the mean. This means that we can successfully approximate the approximate posterior with a second-order Taylor expansion around the mean. From the first line onwards we ignore the ln p(y0) and ln p(yN |P(yN )) which lie outside the expectation. EQ(y1:N−1;v1:N−1,σ1:N−1)[ln p(y0:N )] = ln p(y0) + ln p(yN |P(yN )) + N−1∑ i=1 EQ(yi;vi,σi)[ln p(yi|P(yi))]\\n= N∑ i=1 EQ[ln p(vi|P(yi))] + EQ[ ∂ ln p(yi|P(yi)) ∂yi (vi − yi)]\\n+ EQ[ d2 ln p(vi|P(yi))\\ndy2i (vi − yi)2]\\n= N∑ i=1 ln p(vi|P(yi)) + ∂2 ln p(vi|P(yi)) ∂y2i σi\\nWhere the second term in the Taylor expansion evaluates to 0 since EQ[yi − vi] = (vi − vi) = 0 and the third term contains the expression for the variance EQ[(yi − vi)2] = σi. We can then write out the full Laplace-encoded free-energy as:\\n−F = N∑ i=1 ln p(vi|P(yi)) + ∂2 ln p(vi|P(yi)) ∂y2i σi −− 1 2 ln det(2πσi)\\nWe wish to minimize F with respect to the variational parameters vi and σi. There is in fact a closedform expression for the optimal variational variance which can be obtained simply by differentiating and setting the derivative to 0.\\n∂F ∂σi = ∂2 ln p(vi|P(yi)) ∂y2i − σ−1i\\n∂F ∂σi = 0⇒ σ∗i = ∂2 ln p(vi|P(yi)) ∂y2i\\n−1\\nBecause of this analytical result for the variational variance, we do not need to consider it further in the optimisation problem, and only consider minimizing the variational means vi. This renders all the terms in the free-energy except the ln p(vi|P(yi)) terms constant with respect to the variational parameters. This allows us to write:\\n−F ≈ ln p(yN |P(yN )) + N∑ i=1 ln p(vi|P(yi)) (10)\\nas presented in section 2. The first term ln p(yN |P(yN )) is effectively the loss at the output (yN = T ) so becomes an additional prediction error ln p(yN |P(yN )) ∝ (T − v̂N )TΣ−1N (T − v̂N ) which can be absorbed into the sum over other prediction errors. Crucially, although the variational variances have an analytical form, the variances of the generative model (the precisions Σi) do not and can be optimised directly to improve the log model-evidence. These precisions allow for a kind of ’uncertainty-aware’ backprop.\\nDERIVATION OF VARIATIONAL UPDATE RULES AND FIXED POINTS\\nHere, starting from Equation 10, we show how to obtain the variational update rule for the vi’s (Equation 2), and the fixed point equations (Equation 5) (Friston, 2008; 2005; Bogacz, 2017). We first reduce the free-energy to a sum of prediction errors.\\n−F ≈ N∑ i=1 ln p(vi|P(vi))\\n≈ N∑ i=1 (vi − f(P(v1))TΣ−1i (vi − f(P(v1)) T + ln 2πΣ−1i\\n= N∑ i=1 Ti i + ln 2πΣ −1 i\\nWhere i = vi − f(P(v1)), and we have utilized the assumption made in section 2 that Σ−1 = I. By setting all precisions to the identity, we are implicitly assuming that all datapoints and vertices of the computational graph have equal variance. Next we assume that the dynamics of each vertex vi follow a gradient descent on the free-energy.\\n−dvi dt = ∂F ∂vi = ∂ ∂vi [ N∑ j=1 Tj j ]\\n= i ∂ i ∂vi\\n+ ∑\\nj∈C(vi)\\nj ∂ j ∂vi\\n= i − ∑\\nj∈C(vi)\\nj ∂v̂j ∂vi\\nWhere we have used the fact that ∂ i∂vi = 1 and ∂ j ∂vj = −∂v̂j∂vi . To obtain the fixed point of the dynamics, we simply solve for dvidt = 0.\\ndvi dt = ∂F ∂vi = 0\\n⇒ 0 = i − ∑\\nj∈C(vi)\\nj ∂v̂j ∂vi\\n⇒ ∗i = ∑\\nj∈C(vi)\\nj ∂v̂∗j ∂v∗i\\nSimilarly, since ∗i = v ∗ i − v̂∗i then v∗i = ∗i + v̂∗i . So:\\nv∗i = ∗ i + v̂ ∗ i\\n= v̂∗i − ∑\\nj∈C(vi)\\nj ∂v̂∗j ∂v∗i\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":99,"cells":{"paper_id":{"kind":"string","value":"SP:9e6b5b7d9e7459c015130f4b80f7bc75424de050"},"summaries":{"kind":"list like","value":["This paper proposes a simple scheme for training with multiple augmentations of training data in one iteration and reweighting the instances by their relative loss. As authors note in their related works, the idea of reweighting examples based on their relative loss has been widely studied in a variety of machine learning problems. In contrast, this work proposes the reweighting only within augmentations of a single sample. They derive their particular reweighting scheme by proposing an alternative risk (Eq. 3). The new objective is a function of both model parameters and the distribution of augmentations. They propose to find the model parameters that minimize the alternative risk for the hardest distribution of augmentations that maximizes their alternative risk (Eq. 4). Then they consider the distribution of augmentations that are a function of model parameters and input and show that for fixed model parameters, the optimal distribution on a fixed finite set of augmentations is determined by the softmax on the loss of the model for each augmented input. In section 3.2, they propose two variations of their loss using the ground-truth label to evaluate the loss (hard loss) versus using the prediction of the model for the original raw input (soft loss). In section 3.3, they propose specific considerations for augmenting text data. They provide experiments on image and text data with ablations studies."],"string":"[\n \"This paper proposes a simple scheme for training with multiple augmentations of training data in one iteration and reweighting the instances by their relative loss. As authors note in their related works, the idea of reweighting examples based on their relative loss has been widely studied in a variety of machine learning problems. In contrast, this work proposes the reweighting only within augmentations of a single sample. They derive their particular reweighting scheme by proposing an alternative risk (Eq. 3). The new objective is a function of both model parameters and the distribution of augmentations. They propose to find the model parameters that minimize the alternative risk for the hardest distribution of augmentations that maximizes their alternative risk (Eq. 4). Then they consider the distribution of augmentations that are a function of model parameters and input and show that for fixed model parameters, the optimal distribution on a fixed finite set of augmentations is determined by the softmax on the loss of the model for each augmented input. In section 3.2, they propose two variations of their loss using the ground-truth label to evaluate the loss (hard loss) versus using the prediction of the model for the original raw input (soft loss). In section 3.3, they propose specific considerations for augmenting text data. They provide experiments on image and text data with ablations studies.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Data augmentation is an effective technique to improve the generalization of deep neural networks. However, previous data augmentation methods usually treat the augmented samples equally without considering their individual impacts on the model. To address this, for the augmented samples from the same training example, we propose to assign different weights to them. We construct the maximal expected loss which is the supremum over any reweighted loss on augmented samples. Inspired by adversarial training, we minimize this maximal expected loss (MMEL) and obtain a simple and interpretable closed-form solution: more attention should be paid to augmented samples with large loss values (i.e., harder examples). Minimizing this maximal expected loss enables the model to perform well under any reweighting strategy. The proposed method can generally be applied on top of any data augmentation methods. Experiments are conducted on both natural language understanding tasks with token-level data augmentation, and image classification tasks with commonly-used image augmentation techniques like random crop and horizontal flip. Empirical results show that the proposed method improves the generalization performance of the model."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"ING THE"},{"affiliations":[],"name":"MAXIMAL EXPECTED LOSS"},{"affiliations":[],"name":"Mingyang Yi"},{"affiliations":[],"name":"Lu Hou"},{"affiliations":[],"name":"Lifeng Shang"},{"affiliations":[],"name":"Xin Jiang"},{"affiliations":[],"name":"Qun Liu"},{"affiliations":[],"name":"Zhi-Ming Ma"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"ING THE\"\n },\n {\n \"affiliations\": [],\n \"name\": \"MAXIMAL EXPECTED LOSS\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Mingyang Yi\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Lu Hou\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Lifeng Shang\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Xin Jiang\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Qun Liu\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Zhi-Ming Ma\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["S. Behpour","K. Kitani","B. Ziebart"],"title":"Ada: Adversarial data augmentation for object detection","venue":"In IEEE Winter Conference on Applications of Computer Vision,","year":2019},{"authors":["Y. Bengio","J. Louradour","R. Collobert","J. Weston"],"title":"Curriculum learning","venue":"In International Conference on Machine Learning,","year":2009},{"authors":["Y. Cheng","L. Jiang","W. Macherey"],"title":"Robust neural machine translation with doubly adversarial inputs","venue":"In Annual Meeting of the Association for Computational Linguistics,","year":2019},{"authors":["Y. Cheng","L. Jiang","W. Macherey","J. Eisenstein"],"title":"Advaug: Robust adversarial augmentation for neural machine translation","venue":"In Annual Conference of the Association for Computational Linguistics,","year":2020},{"authors":["D. Csiba","P. Richtárik"],"title":"Importance sampling for minibatches","venue":"The Journal of Machine Learning Research,","year":2018},{"authors":["E.D. Cubuk","B. Zoph","D. Mane","V. Vasudevan","Q.V. Le"],"title":"Autoaugment: Learning augmentation policies from data","venue":"Preprint arXiv:1805.09501,","year":2018},{"authors":["A.P. Dempster","N.M. Laird","D.B. Rubin"],"title":"Maximum likelihood from incomplete data via the em algorithm","venue":"Journal of the Royal Statistical Society: Series B (Methodological),","year":1977},{"authors":["J. Deng","W. Dong","R. Socher","L. Li","K. Li","L. Fei-Fei"],"title":"Imagenet: A large-scale hierarchical image database","venue":"IEEE Conference on Computer Vision and Pattern Recognition,","year":2009},{"authors":["J. Devlin","M. Chang","K. Lee","K. Toutanova. Bert"],"title":"Pre-training of deep bidirectional transformers for language understanding","venue":"In North American Chapter of the Association for Computational Linguistics,","year":2019},{"authors":["T. DeVries","G.W. Taylor"],"title":"Improved regularization of convolutional neural networks with cutout","venue":"Preprint arXiv:1708.04552,","year":2017},{"authors":["Y. Freund","R. Schapire"],"title":"A short introduction to boosting","venue":"Journal-Japanese Society For Artificial Intelligence,","year":1999},{"authors":["P. Goyal","K. He"],"title":"Focal loss for dense object detection","venue":"IEEE Transactions on Pattern Analysis and Machine Intelligence,","year":2018},{"authors":["H. Guo","Y. Mao","R. Zhang"],"title":"Augmenting data with mixup for sentence classification: An empirical study","venue":"Preprint arXiv:1905.08941,","year":2019},{"authors":["K. He","X. Zhang","S. Ren","J. Sun"],"title":"Deep residual learning for image recognition","venue":"In IEEE Conference on Computer Vision and Pattern Recognition,","year":2016},{"authors":["L. Jiang","D. Meng","S. Yu","Z. Lan","S. Shan","A. Hauptmann"],"title":"Self-paced learning with diversity","venue":"In Advances in Neural Information Processing Systems,","year":2014},{"authors":["L. Jiang","Z. Zhou","T. Leung","L. Li","F. Li"],"title":"Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels","venue":"In International Conference on Machine Learning,","year":2018},{"authors":["L. Jiang","D. Huang","M. Liu","W. Yang"],"title":"Beyond synthetic noise: Deep learning on controlled noisy labels","venue":"In International Conference on Machine Learning,","year":2020},{"authors":["H.P. Jiang","He","W. Chen","X. Liu","J. Gao","T. Zhao"],"title":"Smart: Robust and efficient fine-tuning for pre-trained natural language models through principled regularized optimization","venue":"In Annual Conference of the Association for Computational Linguistics,","year":2020},{"authors":["X. Jiao","Y. Yin","L. Shang","X. Jiang","X. Chen","L. Li","F. Wang","Q. 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Berger-Wolf"],"title":"A game-theoretic adversarial approach to dynamic network prediction","venue":"In Pacific-Asia Conference on Knowledge Discovery and Data Mining,","year":2018},{"authors":["S. Lim","I. Kim","T. Kim","C. Kim","S. Kim"],"title":"Fast autoaugment","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["T. Lin","P. Goyal","R. Girshick","K. He","P. Dollár"],"title":"Focal loss for dense object detection","venue":"In IEEE International Conference on Computer Vision,","year":2017},{"authors":["I. Loshchilov","F. Hutter"],"title":"Decoupled weight decay regularization","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["A. Madry","A. Makelov","L. Schmidt","D. Tsipras","A. Vladu"],"title":"Towards deep learning models resistant to adversarial attacks","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["T. Malisiewicz","A. Gupta","A.A. 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Uszkoreit","L. Jones","A. N Gomez","Ł. Kaiser","I. Polosukhin"],"title":"Attention is all you need","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["A. Wang","A. Singh","J. Michael","F. Hill","O. Levy","S.R. Bowman"],"title":"Glue: A multi-task benchmark and analysis platform for natural language understanding","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["W.Y.D. Wang","Yang"],"title":"That’s so annoying!!!: A lexical and frame-semantic embedding based data augmentation approach to automatic categorization of annoying behaviors using# petpeeve tweets","venue":"In Conference on Empirical Methods in Natural Language Processing,","year":2015},{"authors":["J. Wei","K. Zou"],"title":"Eda: Easy data augmentation techniques for boosting performance on text classification tasks","venue":"In Conference on Empirical Methods in Natural Language Processing,","year":2019},{"authors":["Q. Xie","Z. Dai","E. 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Le\"\n ],\n \"title\": \"Autoaugment: Learning augmentation policies from data\",\n \"venue\": \"Preprint arXiv:1805.09501,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"A.P. Dempster\",\n \"N.M. Laird\",\n \"D.B. Rubin\"\n ],\n \"title\": \"Maximum likelihood from incomplete data via the em algorithm\",\n \"venue\": \"Journal of the Royal Statistical Society: Series B (Methodological),\",\n \"year\": 1977\n },\n {\n \"authors\": [\n \"J. Deng\",\n \"W. Dong\",\n \"R. Socher\",\n \"L. Li\",\n \"K. Li\",\n \"L. Fei-Fei\"\n ],\n \"title\": \"Imagenet: A large-scale hierarchical image database\",\n \"venue\": \"IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"J. Devlin\",\n \"M. Chang\",\n \"K. Lee\",\n \"K. Toutanova. Bert\"\n ],\n \"title\": \"Pre-training of deep bidirectional transformers for language understanding\",\n \"venue\": \"In North American Chapter of the Association for Computational Linguistics,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"T. DeVries\",\n \"G.W. Taylor\"\n ],\n \"title\": \"Improved regularization of convolutional neural networks with cutout\",\n \"venue\": \"Preprint arXiv:1708.04552,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Y. Freund\",\n \"R. Schapire\"\n ],\n \"title\": \"A short introduction to boosting\",\n \"venue\": \"Journal-Japanese Society For Artificial Intelligence,\",\n \"year\": 1999\n },\n {\n \"authors\": [\n \"P. Goyal\",\n \"K. He\"\n ],\n \"title\": \"Focal loss for dense object detection\",\n \"venue\": \"IEEE Transactions on Pattern Analysis and Machine Intelligence,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"H. Guo\",\n \"Y. Mao\",\n \"R. Zhang\"\n ],\n \"title\": \"Augmenting data with mixup for sentence classification: An empirical study\",\n \"venue\": \"Preprint arXiv:1905.08941,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"K. He\",\n \"X. Zhang\",\n \"S. Ren\",\n \"J. Sun\"\n ],\n \"title\": \"Deep residual learning for image recognition\",\n \"venue\": \"In IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"L. Jiang\",\n \"D. Meng\",\n \"S. Yu\",\n \"Z. Lan\",\n \"S. Shan\",\n \"A. Hauptmann\"\n ],\n \"title\": \"Self-paced learning with diversity\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"L. Jiang\",\n \"Z. Zhou\",\n \"T. Leung\",\n \"L. Li\",\n \"F. Li\"\n ],\n \"title\": \"Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"L. Jiang\",\n \"D. Huang\",\n \"M. Liu\",\n \"W. Yang\"\n ],\n \"title\": \"Beyond synthetic noise: Deep learning on controlled noisy labels\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"H.P. Jiang\",\n \"He\",\n \"W. Chen\",\n \"X. Liu\",\n \"J. Gao\",\n \"T. Zhao\"\n ],\n \"title\": \"Smart: Robust and efficient fine-tuning for pre-trained natural language models through principled regularized optimization\",\n \"venue\": \"In Annual Conference of the Association for Computational Linguistics,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"X. Jiao\",\n \"Y. Yin\",\n \"L. Shang\",\n \"X. Jiang\",\n \"X. Chen\",\n \"L. Li\",\n \"F. Wang\",\n \"Q. Liu\"\n ],\n \"title\": \"Tinybert: Distilling bert for natural language understanding\",\n \"venue\": null,\n \"year\": 1909\n },\n {\n \"authors\": [\n \"A. Katharopoulos\",\n \"F. Fleuret\"\n ],\n \"title\": \"Not all samples are created equal: Deep learning with importance sampling\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"A. Krizhevsky\",\n \"I. Sutskever\",\n \"G.E. Hinton\"\n ],\n \"title\": \"Imagenet classification with deep convolutional neural networks\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"A. Krizhevsky\",\n \"V. Nair\",\n \"G. Hinton\"\n ],\n \"title\": \"The cifar-10 dataset\",\n \"venue\": \"online: http://www. cs. toronto. edu/kriz/cifar. html,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"V. Kumar\",\n \"A. Choudhary\",\n \"E. Cho\"\n ],\n \"title\": \"Data augmentation using pre-trained transformer models\",\n \"venue\": \"Preprint arXiv:2003.02245,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"J. Li\",\n \"B. Ziebart\",\n \"B. Berger-Wolf\"\n ],\n \"title\": \"A game-theoretic adversarial approach to dynamic network prediction\",\n \"venue\": \"In Pacific-Asia Conference on Knowledge Discovery and Data Mining,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"S. Lim\",\n \"I. Kim\",\n \"T. Kim\",\n \"C. Kim\",\n \"S. Kim\"\n ],\n \"title\": \"Fast autoaugment\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"T. Lin\",\n \"P. Goyal\",\n \"R. Girshick\",\n \"K. He\",\n \"P. Dollár\"\n ],\n \"title\": \"Focal loss for dense object detection\",\n \"venue\": \"In IEEE International Conference on Computer Vision,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"I. Loshchilov\",\n \"F. Hutter\"\n ],\n \"title\": \"Decoupled weight decay regularization\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"A. Madry\",\n \"A. Makelov\",\n \"L. Schmidt\",\n \"D. Tsipras\",\n \"A. Vladu\"\n ],\n \"title\": \"Towards deep learning models resistant to adversarial attacks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"T. Malisiewicz\",\n \"A. Gupta\",\n \"A.A. Efros\"\n ],\n \"title\": \"Ensemble of exemplar-svms for object detection and beyond\",\n \"venue\": \"In International Conference on Computer Vision,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"J. Martens\"\n ],\n \"title\": \"New insights and perspectives on the natural gradient method\",\n \"venue\": \"Preprint arXiv:1412.1193,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"D. Needell\",\n \"R. Ward\",\n \"N. Srebro\"\n ],\n \"title\": \"Stochastic gradient descent, weighted sampling, and the randomized kaczmarz algorithm\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"D.S. Park\",\n \"W. Chan\",\n \"Y. Zhang\",\n \"C. Chiu\",\n \"B. Zoph\",\n \"E.D. Cubuk\",\n \"Q.V. Le\"\n ],\n \"title\": \"Specaugment: A simple data augmentation method for automatic speech recognition\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"A. Radford\",\n \"J. Wu\",\n \"R. Child\",\n \"D. Luan\",\n \"D. Amodei\",\n \"I. Sutskever\"\n ],\n \"title\": \"Language models are unsupervised multitask learners\",\n \"venue\": \"OpenAI Blog,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"A. Raghunathan\",\n \"S.M. Xie\",\n \"F. Yang\",\n \"J. C Duchi\",\n \"P. Liang\"\n ],\n \"title\": \"Adversarial training can hurt generalization\",\n \"venue\": \"Preprint arXiv:1906.06032,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"M. Ren\",\n \"W. Zeng\",\n \"B. Yang\",\n \"R. Urtasun\"\n ],\n \"title\": \"Learning to reweight examples for robust deep learning\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"J. Shu\",\n \"Q. Xie\",\n \"L. Yi\",\n \"Q. Zhao\",\n \"S. Zhou\",\n \"Z. Xu\",\n \"D. Meng\"\n ],\n \"title\": \"Meta-weight-net: Learning an explicit mapping for sample weighting\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"I. Sutskever\",\n \"O. Vinyals\",\n \"Q.V. Le\"\n ],\n \"title\": \"Sequence to sequence learning with neural networks\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"T. Tran\",\n \"T. Pham\",\n \"G. Carneiro\",\n \"L. Palmer\",\n \"I. Reid\"\n ],\n \"title\": \"A bayesian data augmentation approach for learning deep models\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"A. Vaswani\",\n \"N. Shazeer\",\n \"N. Parmar\",\n \"J. Uszkoreit\",\n \"L. Jones\",\n \"A. N Gomez\",\n \"Ł. Kaiser\",\n \"I. Polosukhin\"\n ],\n \"title\": \"Attention is all you need\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"A. Wang\",\n \"A. Singh\",\n \"J. Michael\",\n \"F. Hill\",\n \"O. Levy\",\n \"S.R. Bowman\"\n ],\n \"title\": \"Glue: A multi-task benchmark and analysis platform for natural language understanding\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"W.Y.D. Wang\",\n \"Yang\"\n ],\n \"title\": \"That’s so annoying!!!: A lexical and frame-semantic embedding based data augmentation approach to automatic categorization of annoying behaviors using# petpeeve tweets\",\n \"venue\": \"In Conference on Empirical Methods in Natural Language Processing,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"J. Wei\",\n \"K. Zou\"\n ],\n \"title\": \"Eda: Easy data augmentation techniques for boosting performance on text classification tasks\",\n \"venue\": \"In Conference on Empirical Methods in Natural Language Processing,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Q. Xie\",\n \"Z. Dai\",\n \"E. Hovy\",\n \"M. Luong\",\n \"Q.V. Le\"\n ],\n \"title\": \"Unsupervised data augmentation for consistency training\",\n \"venue\": \"Preprint arXiv:1904.12848,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Z. Xie\",\n \"S.I. Wang\",\n \"J. Li\",\n \"D. Lévy\",\n \"A. Nie\",\n \"D. Jurafsky\",\n \"A.Y. Ng\"\n ],\n \"title\": \"Data noising as smoothing in neural network language models\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Y. Yang\",\n \"L. Huang\",\n \"M. Ma\"\n ],\n \"title\": \"Breaking the beam search curse: A study of (re-) scoring methods and stopping criteria for neural machine translation\",\n \"venue\": \"In Conference on Empirical Methods in Natural Language Processing,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"H. Zhang\",\n \"M. Cisse\",\n \"Y.N. Dauphin\",\n \"D. Lopez-Paz\"\n ],\n \"title\": \"mixup: Beyond empirical risk minimization\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"X. Zhang\",\n \"J. Zhao\",\n \"Y. LeCun\"\n ],\n \"title\": \"Character-level convolutional networks for text classification\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"P. Zhao\",\n \"T. Zhang\"\n ],\n \"title\": \"Accelerating minibatch stochastic gradient descent using stratified sampling\",\n \"venue\": null,\n \"year\": 2014\n },\n {\n \"authors\": [\n \"C. Zhu\",\n \"Y. Cheng\",\n \"Z. Gan\",\n \"S. Sun\",\n \"T. Goldstein\",\n \"J. Liu\"\n ],\n \"title\": \"Freelb: Enhanced adversarial training for natural language understanding\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Deep neural networks have achieved state-of-the-art results in various tasks in natural language processing (NLP) tasks (Sutskever et al., 2014; Vaswani et al., 2017; Devlin et al., 2019) and computer vision (CV) tasks (He et al., 2016; Goodfellow et al., 2016). One approach to improve the generalization performance of deep neural networks is data augmentation (Xie et al., 2019; Jiao et al., 2019; Cheng et al., 2019; 2020). However, there are some problems if we directly incorporate these augmented samples into the training set. Minimizing the average loss on all these samples means treating them equally, without considering their different implicit impacts on the loss.\nTo address this, we propose to minimize a reweighted loss on these augmented samples to make the model utilize them in a cleverer way. Example reweighting has previously been explored extensively in curriculum learning (Bengio et al., 2009; Jiang et al., 2014), boosting algorithms (Freund & Schapire, 1999), focal loss (Lin et al., 2017) and importance sampling (Csiba & Richtárik, 2018). However, none of them focus on the reweighting of augmented samples instead of the original training samples. A recent work (Jiang et al., 2020a) also assigns different weights on augmented samples. But weights in their model are predicted by a mentor network while we obtain the weights from the closed-form solution by minimizing the maximal expected loss (MMEL). In addition, they focus on image samples with noisy labels, while our method can generally be applied to also textual data as well as image data. Tran et al. (2017) propose to minimize the loss on the augmented samples under the framework of Expectation-Maximization algorithm. But they mainly focus on the generation of augmented samples.\n∗This work is done when Mingyang Yi is an intern at Huawei Noah’s Ark Lab.\nUnfortunately, in practise there is no way to directly access the optimal reweighting strategy. Thus, inspired by adversarial training (Madry et al., 2018), we propose to minimize the maximal expected loss (MMEL) on augmented samples from the same training example. Since the maximal expected loss is the supremum over any possible reweighting strategy on augmented samples’ losses, minimizing this supremum makes the model perform well under any reweighting strategy. More importantly, we derive a closed-form solution of the weights, where augmented samples with larger training losses have larger weights. Intuitively, MMEL allows the model to keep focusing on augmented samples that are harder to train.\nThe procedure of our method is summarized as follows. We first generate the augmented samples with commonly-used data augmentation technique, e.g., lexical substitution for textual input (Jiao et al., 2019), random crop and horizontal flip for image data (Krizhevsky et al., 2012). Then we explicitly derive the closed-form solution of the weights on each of the augmented samples. After that, we update the model parameters with respect to the reweighted loss. The proposed method can generally be applied above any data augmentation methods in various domains like natural language processing and computer vision. Empirical results on both natural language understanding tasks and image classification tasks show that the proposed reweighting strategy consistently outperforms the counterpart of without using it, as well as other reweighting strategies like uniform reweighting."},{"heading":"2 RELATED WORK","text":"Data augmentation. Data augmentation is proven to be an effective technique to improve the generalization ability of various tasks, e.g., natural language processing (Xie et al., 2019; Zhu et al., 2020; Jiao et al., 2019), computer vision (Krizhevsky et al., 2014), and speech recognition (Park et al., 2019). For image data, baseline augmentation methods like random crop, flip, scaling, and color augmentation (Krizhevsky et al., 2012) have been widely used. Other heuristic data augmentation techniques like Cutout (DeVries & Taylor, 2017) which masks image patches and Mixup (Zhang et al., 2018) which combines pairs of examples and their labels, are later proposed. Automatically searching for augmentation policies (Cubuk et al., 2018; Lim et al., 2019) have recently proposed to improve the performance further. For textual data, Zhang et al. (2015); Wei & Zou (2019) and Wang (2015) respectively use lexical substitution based on the embedding space. Jiao et al. (2019); Cheng et al. (2019); Kumar et al. (2020) generate augmented samples with a pre-trained language model. Some other techniques like back translation (Xie et al., 2019), random noise injection (Xie et al., 2017) and data mixup (Guo et al., 2019; Cheng et al., 2020) are also proven to be useful.\nAdversarial training. Adversarial learning is used to enhance the robustness of model (Madry et al., 2018), which dynamically constructs the augmented adversarial samples by projected gradient descent across training. Although adversarial training hurts the generalization of model on the task of image classification (Raghunathan et al., 2019), it is shown that adversarial training can be used as data augmentation to help generalization in neural machine translation (Cheng et al., 2019; 2020) and natural language understanding (Zhu et al., 2020; Jiang et al., 2020b). Our proposed method differs from adversarial training in that we adversarially decide the weight on each augmented sample, while traditional adversarial training adversarially generates augmented input samples.\nIn (Behpour et al., 2019), adversarial learning is used as data augmentation in object detection. The adversarial samples (i.e., bounding boxes that are maximally different from the ground truth) are reweighted to form the underlying annotation distribution. However, besides the difference in the model and task, their training objective and the resultant solution are also different from ours.\nSample reweighting. Minimizing a reweighted loss on training samples has been widely explored in literature. Curriculum learning (Bengio et al., 2009; Jiang et al., 2014) feeds first easier and then harder data into the model to accelerate training. Zhao & Zhang (2014); Needell et al. (2014); Csiba & Richtárik (2018); Katharopoulos & Fleuret (2018) use importance sampling to reduce the variance of stochastic gradients to achieve faster convergence rate. Boosting algorithms (Freund & Schapire, 1999) choose harder examples to train subsequent classifiers. Similarly, hard example mining (Malisiewicz et al., 2011) downsamples the majority class and exploits the most difficult examples. Focal loss (Lin et al., 2017; Goyal & He, 2018) focuses on harder examples by reshaping the standard cross-entropy loss in object detection. Ren et al. (2018); Jiang et al. (2018); Shu et al. (2019) use meta-learning method to reweight examples to handle the noisy label problem. Unlike all\nthese existing methods, in this work, we reweight the augmented samples’ losses instead of training samples."},{"heading":"3 MINIMIZE THE MAXIMAL EXPECTED LOSS","text":"In this section, we derive our reweighting strategy on augmented samples from the perspective of maximal expected loss. We first give a derivation of the closed-form solution of the weights on augmented samples. Then we describe two kinds of loss under this formulation. Finally, we give the implementation details using the natural language understanding task as an example."},{"heading":"3.1 WHY MAXIMAL EXPECTED LOSS","text":"Consider a classification task with N training samples. For the i-th training sample xi, its label is denoted as yxi . Let fθ(·) be the model with parameter θ which outputs the classification probabilities. `(·, ·) denotes the loss function, e.g. the cross-entropy loss between outputs fθ(xi) and the groundtruth label yxi . Given an original training sample xi, the set of augmented samples generated by some method isB(xi). Without loss of generality, we assume xi ∈ B(xi). The conventional training objective is to minimize the loss on every augmented sample z in B(xi) as\nmin θ\n1\nN N∑ i=1 1 |B(xi)| ∑ (z,yz)∈B(xi) `(fθ(z), yz) , (1) where yz is the label of z ∈ B(xi), and can be different with yxi . |B(xi)| is the number of augmented samples in B(xi), which is assumed to be finite.\nIn equation (1), for each given xi, the weights on its augmented samples are the same (i.e., 1/|B(xi)|). However, different samples have different implicit impacts on the loss, and we can assign different weights on them to facilitate training. Note that computing the weighted sum of losses of each augmented sample in B(xi) can be viewed as taking expectation of loss on augmented samples z ∈ B(xi) under a certain distribution. When the augmented samples generated from the same training sample are drawn from a uniform distribution, the loss in equation (1) can be rewritten as\nmin θ Rθ(PU ) = min θ\n1\nN N∑ i=1 [ Ez∼PU (·|xi) [`(fθ(z), yz)]− λPKL(PU (· | xi) ‖ PU (· | xi)) ] , (2)\nwhere the Kullback–Leibler (KL) divergence KL(PU (· | xi) ‖ PU (· | xi)) equals zero. Here PU (· | xi) denotes the uniform distribution on B(xi). When the augmented samples are drawn from a more general distribution PB(· | ·)1 instead of the uniform distribution, we can generalize PU (· | ·) here to some other conditional distribution PB .\nmin θ Rθ(PB) = min θ\n1\nN N∑ i=1 [ Ez∼PB(·|xi) [`(fθ(z), yz)]− λPKL(PB(· | xi) ‖ PU (· | xi)) ] . (3)\nRemark 1. When PB(· | xi) reduces to the uniform distribution PU (· | xi) for any xi, since KL(PU (· | xi) ‖ PU (· | xi)) = 0, the objective in equation (3) reduces to the one in equation (1).\nThe KL divergence term in equation (3) is used as a regularizer to encourage PB close to PU (see Remark 2). From equation (3), the conditional distribution PB determines the weights of each augmented sample in B(xi). There may exist an optimal formulation of PB in some regime, e.g. corresponding to the optimal generalization ability of model. Unfortunately, we can not explicitly characterize such an unknown optimal PB . To address this, we borrow the idea from adversarial training (Madry et al., 2018) and minimize the maximal reweighted loss on augmented samples. Then, the model is guaranteed to perform well under any reweighting strategy, including the underlying optimal one. Specifically, let the conditional distribution PB be P∗θ = arg supPB Rθ(PB). Our objective is to minimize the following reweighted loss\nmin θ Rθ(P∗θ) = min\nθ sup PB\nRθ(PB). (4)\nThe following Remark 2 discusses about the KL divergence term in equation (3). 1In the following, we simplify PB(· | ·) as PB if there is no obfuscation.\nRemark 2. Since we take a supremum over PB in equation (4), the regularizer KL(PB ‖ PU ) encourages PB to be close to PU because it reaches the minimal value zero when PB = PU . Thus the regularizer controls the diversity among the augmented samples by constraining the discrepancy between PB and uniform distribution PU , e.g., a larger λP promotes a larger diversity among the augmented samples.\nThe following Theorem 1 gives the explicit formulation of Rθ(P∗θ). Theorem 1. Let Rθ(PB) and Rθ(P∗θ) be defined in equation (1) and (4), then we have\nRθ(P∗θ) = 1\nN N∑ i=1 ∑ z∈B(xi) P∗θ(z | xi)`(fθ(z), yz)− λPP∗θ(z | xi) log (|B(xi)|P∗θ(z | xi)) , (5) where\nP∗θ(z | xi) = exp\n( 1 λP `(fθ(z), yz) ) ∑\nz∈B(xi) exp ( 1 λP `(fθ(z), yz)\n) = Softmaxz ( 1 λP `(fθ(B(xi)), yB(xi)) ) , (6)\nwhere Softmaxz( 1λP `(fθ(B(xi)), yB(xi))) represents the output probability of z for vector ( 1λP `(fθ(z1), yz1), · · · , 1 λP `(fθ(z|B(xi)|), y|B(xi)|)). Remark 3. If we ignore the KL divergence term in equation (3), due to the equivalence of minimizing cross-entropy loss and MLE loss (Martens, 2019), the proposed MMEL also falls into the generalized Expectation-Maximization (GEM) framework (Dempster et al., 1977). Specifically, given a training example, the augmented samples of it can be viewed as latent variable, and any reweighting on these augmented samples corresponds to a specific conditional distribution of these augmented samples given the training sample. In the expectation step (E-step), we explicitly derive the closed-form solution of the weights on each of these augmented samples according to (6). In the maximization step, since there is no analytical solution for deep neural networks, following (Tran et al., 2017), we update the model parameters with respect to the reweighted loss by one step of gradient descent.\nThe proof of this theorem can be found in Appendix A. From Theorem 1, the loss of it decides the weight on each augmented sample z ∈ Bxi , and the weight is normalized by Softmax over all augmented samples in Bxi . The reweighting strategy allows more attention paid to augmented samples with higher loss values. The strategy is similar to those in (Lin et al., 2017; Zhao & Zhang, 2014) but they apply it on training samples."},{"heading":"3.2 TWO TYPES OF LOSS","text":"For augmented sample z ∈ B(xi), instead of computing the discrepancy between the output probability fθ(z) and the hard label yz as in equation (5), one can also compute the discrepancy between fθ(z) and the “soft” probability fθ(xi) in the absence of ground-truth label on augmented samples as in (Xie et al., 2019). In the following, We use superscript “hard\" for the loss in equation (5) as\nRhardθ (P∗θ ,xi) = ∑\nz∈B(xi)\nP∗θ(z | xi)`(fθ(z), yz))− λPP∗θ(z | xi) log (|B(xi)|P∗θ(z | xi)), (7)\nto distinguish with the following objective which uses the “soft probability”: Rsoftθ (P∗θ ,xi) = `(fθ(xi), yxi) + λT ∑\nz∈B(xi);z 6=xi\n( P∗θ(z | xi)`(fθ(z), fθ(xi))\n− λPP∗θ(z | xi) log (|B(xi)| − 1)P∗θ(z | xi) ) .\n(8)\nThe two terms in Rsoftθ (P∗θ,xi) respectively correspond to the loss on original training samples xi and the reweighted loss on the augmented samples. The reweighted loss promotes a small discrepancy between the augmented samples and the original training sample. λT > 0 is the coefficient used to balance the two loss terms, and P∗θ(z | xi) is defined similar to (6) as\nP∗θ(z | xi) = exp\n( 1 λP `(fθ(z), fθ(xi)) ) ∑\nz∈B(xi);z 6=xi exp ( 1 λP `(fθ(z), fθ(xi)) ) . (9)\nAlgorithm 1 Minimize the Maximal Expected Loss (MMEL) Input: Training set {(x1, yx1), · · · , (xN , yxN )}, batch size S, learning rate η, number of training iterations T , Rθ equals Rhardθ or R soft θ .\n1: for i in {1, 2, · · · , N} do . generate augmented samples 2: Generating B(xi) using some data augmentation method. 3: end for 4: for t = 1, · · · , T do . minimize the maximal expected loss 5: Randomly sample a mini-batch S = {(xi1 , yxi1 ), · · · , (xiS , yxiS )} from training set. 6: Fetch the augmented samples B(xi1), B(xi2), · · · , B(xiS ). 7: Compute P∗θ according to (6) or (9). 8: Update model parameters θt+1 = θt − ηS ∑ x∈S ∇θRθ(P∗θ,x). 9: end for\nThe two losses are shown in Figure 1. Summing over all the training samples, we get the two kinds of reweighted training objectives.\nRemark 4. The proposed MMEL-S tries to reduce the discrepancy between fθ(z) and fθ(xi) for z ∈ B(xi). However, if the prediction fθ(xi) is inaccurate, such misleading supervision for z may lead to the degraded performance of MMEL-S. More details are in Appendix B."},{"heading":"3.3 EXAMPLE: MMEL IMPLEMENTATION ON NATURAL LANGUAGE UNDERSTANDING TASKS","text":"In this section, we elaborate on implementing the proposed method using textual data in natural language understanding tasks as an example. Our method is separated into two phases. In the first phase, we generate augmented samples. Then in the second phase, with these augmented samples, we update the model parameters under these augmented samples with respect to the hard reweighted loss (7) or the soft counterpart (8). The generation and training procedure can be decoupled, and the augmented samples are offline generated in the first phase by only once. On the other hand, in the second phase, since we have the explicit solution of weights on augmented samples and the multiple forward and backward passes on these augmented samples can be computed in parallel, the whole training time is similar to the regular training counterpart for an appropriate number of augmented samples. The whole training process is shown in Algorithm 1.\nGeneration of Textual Augmented Data. Various methods have been proposed to generate augmented samples for textual data. Recently, large-scale pre-trained language models like BERT (Devlin et al., 2019) and GPT-2 (Radford et al., 2019) learn contextualized representations and have been used widely in generating high-quality augmented sentences (Jiao et al., 2019; Kumar et al., 2020). In this paper, we use a pre-trained BERT trained from masked language modeling to generate augmented samples. For each original input sentence, we randomly mask k tokens. Then we do a forward propagation of the BERT to predict the tokens in those masked positions by greedy search. Details can be found in Algorithm 2 in Appendix C.\nMismatching Label. For Rhardθ in equation (7), the loss term `(fθ(z), yz) on augmented sample z ∈ B(xi) for some xi relies on its label yz . Unlike image data, where conventional augmentation methods like random crop and horizontal flip of an image do not change its label, substituting even one word in a sentence can drastically change its meaning. For instance, suppose the original sentence is “She is my daughter”, and the word “She” is masked. The top 5 words predicted by the pre-trained BERT are “This, She, That, It, He”. Apparently, for the task of linguistic acceptability task, replacing “She” with “He” can change the label from linguistically “acceptable” to “non-acceptable”. Thus for textual input, for the term `(fθ(z), yz) in hard loss (7), instead of directly setting yz as yxi (Zhu et al., 2020), we replace yz with the output probability of a trained teacher model. On the other hand, for the soft loss in equation (8), if an augmented sample z ∈ B(xi) is predicted to a different class from xi by the teacher model, it is unreasonable to still minimize the discrepancy between fθ(z) and fθ(xi). In this case, we replace fθ(xi) in the loss term λT ∑ z∈B(xi);z 6=xi P ∗ θ(z | xi)`(fθ(z), fθ(xi)) with the output probability from the teacher model."},{"heading":"4 EXPERIMENTS","text":"In this section, we evaluate the efficacy of the proposed MMEL algorithm with both hard loss (MMEL-H) and soft loss (MMEL-S). Experiments are conducted on both the image classification tasks CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2014) with the ResNet Model (He et al., 2016), and the General Language Understanding Evaluation (GLUE) tasks (Wang et al., 2019) with the BERT model (Devlin et al., 2019)."},{"heading":"4.1 EXPERIMENTS ON IMAGE CLASSIFICATION TASKS.","text":"Data. CIFAR (Krizhevsky et al., 2014) is a benchmark dataset for image classification. We use both CIFAR-10 and CIFAR-100 in our experiments, which are colorful images with 50000 training samples and 10000 validation samples, but from 10 and 100 object classes, respectively.\nSetup. The model we used is ResNet (He et al., 2016) with different depths. We use random crop and horizontal flip (Krizhevsky et al., 2012) to augment the original training images. Since these operations do not change the augmented sample label, we directly adopt the original training sample label for all its augmented samples. Following (He et al., 2016), we use the SGD with momentum optimizer to train each model for 200 epochs. The learning rate starts from 0.1 and decays by a factor of 0.2 at epochs 60, 120 and 160. The batch size is 128, and weight decay is 5e-4. For each xi, |B(xi)| = 10. The λP of the KL regularization coefficient is 1.0 for both MMEL-H and MMEL-S. The λT in equation (8) for MMEL-S is selected from {0.5, 1.0, 2.0}.\nWe compare our proposed MMEL with conventional training with data augmentation (abbreviated as “Baseline(DA)”) under the same number of epochs. Though MMEL can be computed efficiently in parallel, the proposed MMEL encounters |B(xi)| = 10 times more training data. For fair comparison, we also compare with two other baselines that also use 10 times more data: (i) naive training with data augmentation but with 10 times more training epochs compared with MMEL (abbreviated as “Baseline(DA+Long)”). In this case, the learning rate accordingly decays at epochs 600, 1200 and 1600; (ii) training with data augmentation under the framework of MMEL but with uniform weights on the augmented samples (abbreviated as “Baseline(DA+UNI)”).\nMain Results. The results are shown in Table 1. As can be seen, for both CIFAR-10 and CIFAR-100, MMEL-H and MMEL-S significantly outperform the Baseline(DA), with over 0.5 points higher accuracy on all four architectures. Compared to Baseline(DA+Long), the proposed MMEL-H and MMEL-S also have comparable or better performance, while being much more efficient in training. This is because our backward pass only computes the gradient of the weighted loss instead of the separate loss of each example. Compared to Baseline(DA+UNI) which has the same computational cost as MMEL-H and MMEL-S, the proposed methods also have better performance. This indicates the efficacy of the proposed maximal expected loss based reweighting strategy.\nWe further evaluate the proposed method on larege-scale dataset ImageNet(Deng et al., 2009). The detailed results are in Appendix B.\nVarying the Number of Augmented Samples. One hyperparameter of the proposed method is the number of augmented samples |B(xi)|. In Table 2, we evaluate the effect of |B(xi)| on the CIFAR dataset. We vary |B(xi)| in {2, 5, 10, 20} for both MMEL-H and MMEL-S with other settings unchanged. As can be seen, the performance of MMEL improves with more augmented samples for small |B(xi)|. However, the performance gain begins to saturate when |B(xi)| reaches 5 or 10 for some cases. Since a larger |B(xi)| also brings more training cost, we should choose a proper number of augmented samples rather than continually increasing it."},{"heading":"4.2 RESULTS ON NATURAL LANGUAGE UNDERSTANDING TASKS","text":"Data. GLUE is a benchmark containing various natural language understanding tasks, including textual entailment (RTE and MNLI), question answering (QNLI), similarity and paraphrase (MRPC, QQP, STS-B), sentiment analysis (SST-2) and linguistic acceptability (CoLA). Among them, STS-B is a regression task, CoLA and SST-2 are single sentence classification tasks, while the rest are sentence-pair classification tasks. Following (Devlin et al., 2019), for the development set, we report Spearman correlation for STS-B, Matthews correlation for CoLA and accuracy for the other tasks. For the test set for QQP and MRPC, we report “F1”.\nSetup. The backbone model is BERTBASE (Devlin et al., 2019). We use the method in Section 3.3 to generate augmented samples. For the problem of mismatching label as described in Section 3.3, we use a BERTBASE model fine-tuned on the downstream task as teacher model to predict the label of each generated sample z in B(xi). For each xi, |B(xi)| = 5. The fraction of masked tokens for each sentence is 0.4. The λP of the KL regularization coefficient is 1.0 for both MMEL-H and MMEL-S. The λT in equation (8) for MMEL-S is 1.0. The other detailed hyperparameters in training can be found in Appendix D.\nThe derivation of MMEL in Section 3 is based on the classification task, while STS-B is a regression task. Hence, we generalize our loss function accordingly for regression tasks as follows. For the hard loss in equation (7), we directly replace yz ∈ R with the prediction of teacher model on z. For the soft loss (8), for each entry of fθ(xi) in loss term λT ∑ z∈B(xi);z 6=xi P ∗ θ(z | xi)MSE(fθ(z), fθ(xi)), we replace it with the prediction of teacher model if the difference between them is larger than 0.5.\nSimilar to Section 4.1, We compare with three baselines. However, we change the first baseline to naive training without data augmentation (abbreviated as “Baseline”) since data augmentation is not used by default in NLP tasks. The other two baselines are similar to those in Section 4.1: (i) “Baseline(DA+Long)” which fine-tunes BERT with data augmentation with the same batch size; and\n(ii)“Baseline(DA+UNI)” which fine-tunes BERT with augmented samples by using average loss. We also compare with another recent data augmentation technique SMART (Jiang et al., 2020b).\nMain Results. The development and test set results on the GLUE benchmark are shown in Table 3. The development set results for the BERT baseline are from our re-implementation, which is comparable or better than the reported results in the original paper (Devlin et al., 2019). The results for SMART are taken from (Jiang et al., 2020b), and there are no test set results in (Jiang et al., 2020b). As can be seen, data augmentation significantly improves the generalization of GLUE tasks. Compared to the baseline without data augmentation (Baseline), MMEL-H or MMEL-S consistently achieves better performance, especially on small datasets like CoLA and RTE. Similar to the observation in the image classification task in Section 4.1, the proposed MMEL-H and MMEL-S are more efficient and have better performance than Baseline(DA+Long). MMEL-H and MMEL-S also outperform Baseline(DA+UNI), indicating the superiority of using the proposed reweighting strategy. In addition, our proposed method also beats SMART in both accuracy and efficiency because they use PGD-k (Madry et al., 2018) to construct adversarial augmented samples which requires nearly k times more training cost. Figure 2 shows the development set accuracy across over the training procedure. As can be seen, training with MMEL-H or MMEL-S converges faster and has better accuracy except SST-2 and RTE where the performance is similar.\nEffect of Predicted Labels. For the augmented samples from same origin, we use a fine-tuned task-specific BERTBASE teacher model to predict their labels as mentioned in Section 3.3 to handle the problem of mismatching label. In Table 4, we show the comparison between using the label of the original sample and using predicted labels. As can be seen, using the predicted label significantly improves the performance. By comparing with the results in Table 3, using the label of the original sample even hurts the performance."},{"heading":"5 CONCLUSION","text":"In this work, we propose to minimize a reweighted loss over the augmented samples which directly considers their implicit impacts on the loss. Since we can not access the optimal reweighting strategy, we propose to minimize the supremum of the loss under all reweighting strategies, and give a closedform solution of the optimal weights. Our method can be applied on top of any data augmentation methods. Experiments on both image classification tasks and natural language understanding tasks show that the proposed method improves the generalization performance of the model, while being efficient in training."},{"heading":"A PROOF OF THEOREM 1","text":"Proof. For any given xi and B(xi), we aim to find Pθ(· | xi) on B(xi) such that\nmax Pθ(·|xi) ∑ z∈B(xi) Pθ(z | xi)`(fθ(z), yz))− λPPθ(z | xi) log (|B(xi)|Pθ(z | xi))\ns.t. ∑\nz∈B(xi)\nPθ(z | xi) = 1. (10)\nSince the objective is convex, by Lagrange multiplier method, let\nL(Pθ, λ) = ∑\nz∈B(xi)\nPθ(z | xi)`(fθ(z), yz))− λPPθ(z | xi) log (|B(xi)|Pθ(z | xi))\n+ λ ∑ z∈B(xi) Pθ(z | xi)− 1 . (11)\nFrom ∇PθL(Pθ, λ) = ∇λL(Pθ, λ) = 0, for any pairs of zu, zv ∈ B(xi), we have\n`(fθ(zu), yzu)− λP (log |B(xi)|+ 1 + log Pθ(zu | xi)) = `(fθ(zv), yzv )− λP (log |B(xi)|+ 1 + log Pθ(zv | xi)).\n(12)\nHence we have Pθ(zv | xi) = Pθ(zu | xi) exp ( `(fθ(zv), yzv )− `(fθ(zu), yzu)\nλP\n) . (13)\nSumming over zv ∈ B(xi), we have\nPθ(zu | xi) ∑\nzv∈B(xi)\nexp\n( `(fθ(zv), yzu)− `(fθ(zu), yzu)\nλP\n) = 1. (14)\nThe proof completes."},{"heading":"B MMEL ON LARGE-SCALE DATASET","text":"In this section, we evaluate the proposed method MMEL on large-scale image classification task ImageNet(Deng et al., 2009).\nData. ImageNet is a benchmark dataset which contains colorful images with over 1 million training samples and 50000 validation samples from 1000 categories.\nSetup. The model we used is ResNet for ImageNet with three different depths (He et al., 2016). All these experiments are conducted for 100 epochs, and the learning rate decays at epochs 30, 60, and 90. We set batch size as 256, and |B(xi)| = 10 for each xi. The other experimental settings follow Section 4.1, expect for the following hyperparameters. We compare the proposed method with “Baseline(DA)”.\nMain Results. The results are shown in Table 5. From the results, the proposed MMEL-H improves the performance of the model for all three depths. However, the proposed MMEL-S is beaten by the baseline method. We speculate this is due to the relatively larger proportion of inaccurate prediction of original training samples on the large-scale dataset. More specifically, as in equation (8), for each augmented sample z ∈ B(xi), the proposed MMEL-S encourages the model to fit the output of original training sample fθ(xi). However, the accuracy of the original training samples in the ImageNet dataset can not reach 100% e.g., about 80% for ResNet50 on ImageNet. The inaccurate prediction fθ(xi) can be a misleading supervision for augmented sample z ∈ B(xi), leading to degraded performance of the proposed MMEL-S. Thus, we suggest using the MMEL-H if the accuracy of the original training samples is relative low."},{"heading":"C GENERATING AUGMENTED SAMPLES FOR TEXTUAL SAMPLES","text":"In this section, we elaborate the procedure of generating augmented sentences using greedy-based and beam-based method for a sequence. For each original input sentence, we randomly mask k tokens (which is obtained by rounding the product of masking ratio and length of the sequence to the nearest number) and then we do a forward propagation of the BERT to predict the tokens in those masked positions using greedy search. The detailed procedure is shown in Algorithm 2. We also use beam search (Yang et al., 2018) to generate augmented data. The details of beam search can be referred to (Yang et al., 2018). For sentence-pair tasks, we treat the two sentences separately and generate augmented samples for each of them.\nAlgorithm 2 Augmented Sample Generation by Greedy Search Input: Pre-trained language model BertModel, original sentence x, number of augmented samples |B(x)| − 1, number of masked tokens k. Output: Augmented samples B(x) = {z1, z2, · · · , z|B(x)|−1}.\n1: Randomly sample k positions {p1, · · · , pk} and get xmask. 2: for i = 1, 2, · · · |B(x)| − 1 do . Generate the i-th augmented sample 3: zi ← xmask. 4: zi[p1]← the ith most likely word predicted by BertModel(zi[p1]|zi). 5: for j in {2, 3. · · · , k} do 6: zi[pj ]← the most likely word predicted by BertModel(zi[pj ]|zi). 7: end for 8: end for\nIn the following, we vary the factors that may affect the quality of the generated augmented samples. These factors include\n1. The number of masked tokens, which equals the replacement proportion multiplied with the sentence length. This affects the diversity of augmented samples, i.e., replacing a large proportion of tokens makes the augmented sample less similar to the original one.\n2. Treating the two sentences separately in sentence-pair tasks when generating augmented examples, or concatenate them as a single sentence;\n3. Different generation methods like greedy search (Algorithm 2) and beam search.\nThe results are shown in Table 6. As can be seen, compared with Baseline without data augmentation, MMEL-H and MMEL-S under all hyperparameter configurations have higher accuracy, showing the efficacy of data augmentation and the proposed reweighting strategy. There is no significant difference in using greedy search or beam search to generate the augmented samples. In this natural understanding task, training with augmented samples generated with proper larger replacement proportion (i.e., larger diversity) has slightly better performance. For sentence-pair tasks, treating the two sentences separately and generate augmented samples for each of them has slightly better performance. In the experiments in Section 4.2, we use Greedy search, masking proportion 0.4, and generate augmented sentence for each sentence in sentence-pair tasks."},{"heading":"D HYPERPARAMETERS FOR THE EXPERIMENT ON THE GLUE BENCHMARK.","text":"The optimizer we used is AdamW (Loshchilov & Hutter, 2018). The hyperparameters of BERTBASE model are listed in Table 7."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Deep neural networks have achieved state-of-the-art results in various tasks in natural language processing (NLP) tasks (Sutskever et al., 2014; Vaswani et al., 2017; Devlin et al., 2019) and computer vision (CV) tasks (He et al., 2016; Goodfellow et al., 2016). One approach to improve the generalization performance of deep neural networks is data augmentation (Xie et al., 2019; Jiao et al., 2019; Cheng et al., 2019; 2020). However, there are some problems if we directly incorporate these augmented samples into the training set. Minimizing the average loss on all these samples means treating them equally, without considering their different implicit impacts on the loss.\\nTo address this, we propose to minimize a reweighted loss on these augmented samples to make the model utilize them in a cleverer way. Example reweighting has previously been explored extensively in curriculum learning (Bengio et al., 2009; Jiang et al., 2014), boosting algorithms (Freund & Schapire, 1999), focal loss (Lin et al., 2017) and importance sampling (Csiba & Richtárik, 2018). However, none of them focus on the reweighting of augmented samples instead of the original training samples. A recent work (Jiang et al., 2020a) also assigns different weights on augmented samples. But weights in their model are predicted by a mentor network while we obtain the weights from the closed-form solution by minimizing the maximal expected loss (MMEL). In addition, they focus on image samples with noisy labels, while our method can generally be applied to also textual data as well as image data. Tran et al. (2017) propose to minimize the loss on the augmented samples under the framework of Expectation-Maximization algorithm. But they mainly focus on the generation of augmented samples.\\n∗This work is done when Mingyang Yi is an intern at Huawei Noah’s Ark Lab.\\nUnfortunately, in practise there is no way to directly access the optimal reweighting strategy. Thus, inspired by adversarial training (Madry et al., 2018), we propose to minimize the maximal expected loss (MMEL) on augmented samples from the same training example. Since the maximal expected loss is the supremum over any possible reweighting strategy on augmented samples’ losses, minimizing this supremum makes the model perform well under any reweighting strategy. More importantly, we derive a closed-form solution of the weights, where augmented samples with larger training losses have larger weights. Intuitively, MMEL allows the model to keep focusing on augmented samples that are harder to train.\\nThe procedure of our method is summarized as follows. We first generate the augmented samples with commonly-used data augmentation technique, e.g., lexical substitution for textual input (Jiao et al., 2019), random crop and horizontal flip for image data (Krizhevsky et al., 2012). Then we explicitly derive the closed-form solution of the weights on each of the augmented samples. After that, we update the model parameters with respect to the reweighted loss. The proposed method can generally be applied above any data augmentation methods in various domains like natural language processing and computer vision. Empirical results on both natural language understanding tasks and image classification tasks show that the proposed reweighting strategy consistently outperforms the counterpart of without using it, as well as other reweighting strategies like uniform reweighting.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Data augmentation. Data augmentation is proven to be an effective technique to improve the generalization ability of various tasks, e.g., natural language processing (Xie et al., 2019; Zhu et al., 2020; Jiao et al., 2019), computer vision (Krizhevsky et al., 2014), and speech recognition (Park et al., 2019). For image data, baseline augmentation methods like random crop, flip, scaling, and color augmentation (Krizhevsky et al., 2012) have been widely used. Other heuristic data augmentation techniques like Cutout (DeVries & Taylor, 2017) which masks image patches and Mixup (Zhang et al., 2018) which combines pairs of examples and their labels, are later proposed. Automatically searching for augmentation policies (Cubuk et al., 2018; Lim et al., 2019) have recently proposed to improve the performance further. For textual data, Zhang et al. (2015); Wei & Zou (2019) and Wang (2015) respectively use lexical substitution based on the embedding space. Jiao et al. (2019); Cheng et al. (2019); Kumar et al. (2020) generate augmented samples with a pre-trained language model. Some other techniques like back translation (Xie et al., 2019), random noise injection (Xie et al., 2017) and data mixup (Guo et al., 2019; Cheng et al., 2020) are also proven to be useful.\\nAdversarial training. Adversarial learning is used to enhance the robustness of model (Madry et al., 2018), which dynamically constructs the augmented adversarial samples by projected gradient descent across training. Although adversarial training hurts the generalization of model on the task of image classification (Raghunathan et al., 2019), it is shown that adversarial training can be used as data augmentation to help generalization in neural machine translation (Cheng et al., 2019; 2020) and natural language understanding (Zhu et al., 2020; Jiang et al., 2020b). Our proposed method differs from adversarial training in that we adversarially decide the weight on each augmented sample, while traditional adversarial training adversarially generates augmented input samples.\\nIn (Behpour et al., 2019), adversarial learning is used as data augmentation in object detection. The adversarial samples (i.e., bounding boxes that are maximally different from the ground truth) are reweighted to form the underlying annotation distribution. However, besides the difference in the model and task, their training objective and the resultant solution are also different from ours.\\nSample reweighting. Minimizing a reweighted loss on training samples has been widely explored in literature. Curriculum learning (Bengio et al., 2009; Jiang et al., 2014) feeds first easier and then harder data into the model to accelerate training. Zhao & Zhang (2014); Needell et al. (2014); Csiba & Richtárik (2018); Katharopoulos & Fleuret (2018) use importance sampling to reduce the variance of stochastic gradients to achieve faster convergence rate. Boosting algorithms (Freund & Schapire, 1999) choose harder examples to train subsequent classifiers. Similarly, hard example mining (Malisiewicz et al., 2011) downsamples the majority class and exploits the most difficult examples. Focal loss (Lin et al., 2017; Goyal & He, 2018) focuses on harder examples by reshaping the standard cross-entropy loss in object detection. Ren et al. (2018); Jiang et al. (2018); Shu et al. (2019) use meta-learning method to reweight examples to handle the noisy label problem. Unlike all\\nthese existing methods, in this work, we reweight the augmented samples’ losses instead of training samples.\"\n },\n {\n \"heading\": \"3 MINIMIZE THE MAXIMAL EXPECTED LOSS\",\n \"text\": \"In this section, we derive our reweighting strategy on augmented samples from the perspective of maximal expected loss. We first give a derivation of the closed-form solution of the weights on augmented samples. Then we describe two kinds of loss under this formulation. Finally, we give the implementation details using the natural language understanding task as an example.\"\n },\n {\n \"heading\": \"3.1 WHY MAXIMAL EXPECTED LOSS\",\n \"text\": \"Consider a classification task with N training samples. For the i-th training sample xi, its label is denoted as yxi . Let fθ(·) be the model with parameter θ which outputs the classification probabilities. `(·, ·) denotes the loss function, e.g. the cross-entropy loss between outputs fθ(xi) and the groundtruth label yxi . Given an original training sample xi, the set of augmented samples generated by some method isB(xi). Without loss of generality, we assume xi ∈ B(xi). The conventional training objective is to minimize the loss on every augmented sample z in B(xi) as\\nmin θ\\n1\\nN N∑ i=1 1 |B(xi)| ∑ (z,yz)∈B(xi) `(fθ(z), yz) , (1) where yz is the label of z ∈ B(xi), and can be different with yxi . |B(xi)| is the number of augmented samples in B(xi), which is assumed to be finite.\\nIn equation (1), for each given xi, the weights on its augmented samples are the same (i.e., 1/|B(xi)|). However, different samples have different implicit impacts on the loss, and we can assign different weights on them to facilitate training. Note that computing the weighted sum of losses of each augmented sample in B(xi) can be viewed as taking expectation of loss on augmented samples z ∈ B(xi) under a certain distribution. When the augmented samples generated from the same training sample are drawn from a uniform distribution, the loss in equation (1) can be rewritten as\\nmin θ Rθ(PU ) = min θ\\n1\\nN N∑ i=1 [ Ez∼PU (·|xi) [`(fθ(z), yz)]− λPKL(PU (· | xi) ‖ PU (· | xi)) ] , (2)\\nwhere the Kullback–Leibler (KL) divergence KL(PU (· | xi) ‖ PU (· | xi)) equals zero. Here PU (· | xi) denotes the uniform distribution on B(xi). When the augmented samples are drawn from a more general distribution PB(· | ·)1 instead of the uniform distribution, we can generalize PU (· | ·) here to some other conditional distribution PB .\\nmin θ Rθ(PB) = min θ\\n1\\nN N∑ i=1 [ Ez∼PB(·|xi) [`(fθ(z), yz)]− λPKL(PB(· | xi) ‖ PU (· | xi)) ] . (3)\\nRemark 1. When PB(· | xi) reduces to the uniform distribution PU (· | xi) for any xi, since KL(PU (· | xi) ‖ PU (· | xi)) = 0, the objective in equation (3) reduces to the one in equation (1).\\nThe KL divergence term in equation (3) is used as a regularizer to encourage PB close to PU (see Remark 2). From equation (3), the conditional distribution PB determines the weights of each augmented sample in B(xi). There may exist an optimal formulation of PB in some regime, e.g. corresponding to the optimal generalization ability of model. Unfortunately, we can not explicitly characterize such an unknown optimal PB . To address this, we borrow the idea from adversarial training (Madry et al., 2018) and minimize the maximal reweighted loss on augmented samples. Then, the model is guaranteed to perform well under any reweighting strategy, including the underlying optimal one. Specifically, let the conditional distribution PB be P∗θ = arg supPB Rθ(PB). Our objective is to minimize the following reweighted loss\\nmin θ Rθ(P∗θ) = min\\nθ sup PB\\nRθ(PB). (4)\\nThe following Remark 2 discusses about the KL divergence term in equation (3). 1In the following, we simplify PB(· | ·) as PB if there is no obfuscation.\\nRemark 2. Since we take a supremum over PB in equation (4), the regularizer KL(PB ‖ PU ) encourages PB to be close to PU because it reaches the minimal value zero when PB = PU . Thus the regularizer controls the diversity among the augmented samples by constraining the discrepancy between PB and uniform distribution PU , e.g., a larger λP promotes a larger diversity among the augmented samples.\\nThe following Theorem 1 gives the explicit formulation of Rθ(P∗θ). Theorem 1. Let Rθ(PB) and Rθ(P∗θ) be defined in equation (1) and (4), then we have\\nRθ(P∗θ) = 1\\nN N∑ i=1 ∑ z∈B(xi) P∗θ(z | xi)`(fθ(z), yz)− λPP∗θ(z | xi) log (|B(xi)|P∗θ(z | xi)) , (5) where\\nP∗θ(z | xi) = exp\\n( 1 λP `(fθ(z), yz) ) ∑\\nz∈B(xi) exp ( 1 λP `(fθ(z), yz)\\n) = Softmaxz ( 1 λP `(fθ(B(xi)), yB(xi)) ) , (6)\\nwhere Softmaxz( 1λP `(fθ(B(xi)), yB(xi))) represents the output probability of z for vector ( 1λP `(fθ(z1), yz1), · · · , 1 λP `(fθ(z|B(xi)|), y|B(xi)|)). Remark 3. If we ignore the KL divergence term in equation (3), due to the equivalence of minimizing cross-entropy loss and MLE loss (Martens, 2019), the proposed MMEL also falls into the generalized Expectation-Maximization (GEM) framework (Dempster et al., 1977). Specifically, given a training example, the augmented samples of it can be viewed as latent variable, and any reweighting on these augmented samples corresponds to a specific conditional distribution of these augmented samples given the training sample. In the expectation step (E-step), we explicitly derive the closed-form solution of the weights on each of these augmented samples according to (6). In the maximization step, since there is no analytical solution for deep neural networks, following (Tran et al., 2017), we update the model parameters with respect to the reweighted loss by one step of gradient descent.\\nThe proof of this theorem can be found in Appendix A. From Theorem 1, the loss of it decides the weight on each augmented sample z ∈ Bxi , and the weight is normalized by Softmax over all augmented samples in Bxi . The reweighting strategy allows more attention paid to augmented samples with higher loss values. The strategy is similar to those in (Lin et al., 2017; Zhao & Zhang, 2014) but they apply it on training samples.\"\n },\n {\n \"heading\": \"3.2 TWO TYPES OF LOSS\",\n \"text\": \"For augmented sample z ∈ B(xi), instead of computing the discrepancy between the output probability fθ(z) and the hard label yz as in equation (5), one can also compute the discrepancy between fθ(z) and the “soft” probability fθ(xi) in the absence of ground-truth label on augmented samples as in (Xie et al., 2019). In the following, We use superscript “hard\\\" for the loss in equation (5) as\\nRhardθ (P∗θ ,xi) = ∑\\nz∈B(xi)\\nP∗θ(z | xi)`(fθ(z), yz))− λPP∗θ(z | xi) log (|B(xi)|P∗θ(z | xi)), (7)\\nto distinguish with the following objective which uses the “soft probability”: Rsoftθ (P∗θ ,xi) = `(fθ(xi), yxi) + λT ∑\\nz∈B(xi);z 6=xi\\n( P∗θ(z | xi)`(fθ(z), fθ(xi))\\n− λPP∗θ(z | xi) log (|B(xi)| − 1)P∗θ(z | xi) ) .\\n(8)\\nThe two terms in Rsoftθ (P∗θ,xi) respectively correspond to the loss on original training samples xi and the reweighted loss on the augmented samples. The reweighted loss promotes a small discrepancy between the augmented samples and the original training sample. λT > 0 is the coefficient used to balance the two loss terms, and P∗θ(z | xi) is defined similar to (6) as\\nP∗θ(z | xi) = exp\\n( 1 λP `(fθ(z), fθ(xi)) ) ∑\\nz∈B(xi);z 6=xi exp ( 1 λP `(fθ(z), fθ(xi)) ) . (9)\\nAlgorithm 1 Minimize the Maximal Expected Loss (MMEL) Input: Training set {(x1, yx1), · · · , (xN , yxN )}, batch size S, learning rate η, number of training iterations T , Rθ equals Rhardθ or R soft θ .\\n1: for i in {1, 2, · · · , N} do . generate augmented samples 2: Generating B(xi) using some data augmentation method. 3: end for 4: for t = 1, · · · , T do . minimize the maximal expected loss 5: Randomly sample a mini-batch S = {(xi1 , yxi1 ), · · · , (xiS , yxiS )} from training set. 6: Fetch the augmented samples B(xi1), B(xi2), · · · , B(xiS ). 7: Compute P∗θ according to (6) or (9). 8: Update model parameters θt+1 = θt − ηS ∑ x∈S ∇θRθ(P∗θ,x). 9: end for\\nThe two losses are shown in Figure 1. Summing over all the training samples, we get the two kinds of reweighted training objectives.\\nRemark 4. The proposed MMEL-S tries to reduce the discrepancy between fθ(z) and fθ(xi) for z ∈ B(xi). However, if the prediction fθ(xi) is inaccurate, such misleading supervision for z may lead to the degraded performance of MMEL-S. More details are in Appendix B.\"\n },\n {\n \"heading\": \"3.3 EXAMPLE: MMEL IMPLEMENTATION ON NATURAL LANGUAGE UNDERSTANDING TASKS\",\n \"text\": \"In this section, we elaborate on implementing the proposed method using textual data in natural language understanding tasks as an example. Our method is separated into two phases. In the first phase, we generate augmented samples. Then in the second phase, with these augmented samples, we update the model parameters under these augmented samples with respect to the hard reweighted loss (7) or the soft counterpart (8). The generation and training procedure can be decoupled, and the augmented samples are offline generated in the first phase by only once. On the other hand, in the second phase, since we have the explicit solution of weights on augmented samples and the multiple forward and backward passes on these augmented samples can be computed in parallel, the whole training time is similar to the regular training counterpart for an appropriate number of augmented samples. The whole training process is shown in Algorithm 1.\\nGeneration of Textual Augmented Data. Various methods have been proposed to generate augmented samples for textual data. Recently, large-scale pre-trained language models like BERT (Devlin et al., 2019) and GPT-2 (Radford et al., 2019) learn contextualized representations and have been used widely in generating high-quality augmented sentences (Jiao et al., 2019; Kumar et al., 2020). In this paper, we use a pre-trained BERT trained from masked language modeling to generate augmented samples. For each original input sentence, we randomly mask k tokens. Then we do a forward propagation of the BERT to predict the tokens in those masked positions by greedy search. Details can be found in Algorithm 2 in Appendix C.\\nMismatching Label. For Rhardθ in equation (7), the loss term `(fθ(z), yz) on augmented sample z ∈ B(xi) for some xi relies on its label yz . Unlike image data, where conventional augmentation methods like random crop and horizontal flip of an image do not change its label, substituting even one word in a sentence can drastically change its meaning. For instance, suppose the original sentence is “She is my daughter”, and the word “She” is masked. The top 5 words predicted by the pre-trained BERT are “This, She, That, It, He”. Apparently, for the task of linguistic acceptability task, replacing “She” with “He” can change the label from linguistically “acceptable” to “non-acceptable”. Thus for textual input, for the term `(fθ(z), yz) in hard loss (7), instead of directly setting yz as yxi (Zhu et al., 2020), we replace yz with the output probability of a trained teacher model. On the other hand, for the soft loss in equation (8), if an augmented sample z ∈ B(xi) is predicted to a different class from xi by the teacher model, it is unreasonable to still minimize the discrepancy between fθ(z) and fθ(xi). In this case, we replace fθ(xi) in the loss term λT ∑ z∈B(xi);z 6=xi P ∗ θ(z | xi)`(fθ(z), fθ(xi)) with the output probability from the teacher model.\"\n },\n {\n \"heading\": \"4 EXPERIMENTS\",\n \"text\": \"In this section, we evaluate the efficacy of the proposed MMEL algorithm with both hard loss (MMEL-H) and soft loss (MMEL-S). Experiments are conducted on both the image classification tasks CIFAR-10 and CIFAR-100 (Krizhevsky et al., 2014) with the ResNet Model (He et al., 2016), and the General Language Understanding Evaluation (GLUE) tasks (Wang et al., 2019) with the BERT model (Devlin et al., 2019).\"\n },\n {\n \"heading\": \"4.1 EXPERIMENTS ON IMAGE CLASSIFICATION TASKS.\",\n \"text\": \"Data. CIFAR (Krizhevsky et al., 2014) is a benchmark dataset for image classification. We use both CIFAR-10 and CIFAR-100 in our experiments, which are colorful images with 50000 training samples and 10000 validation samples, but from 10 and 100 object classes, respectively.\\nSetup. The model we used is ResNet (He et al., 2016) with different depths. We use random crop and horizontal flip (Krizhevsky et al., 2012) to augment the original training images. Since these operations do not change the augmented sample label, we directly adopt the original training sample label for all its augmented samples. Following (He et al., 2016), we use the SGD with momentum optimizer to train each model for 200 epochs. The learning rate starts from 0.1 and decays by a factor of 0.2 at epochs 60, 120 and 160. The batch size is 128, and weight decay is 5e-4. For each xi, |B(xi)| = 10. The λP of the KL regularization coefficient is 1.0 for both MMEL-H and MMEL-S. The λT in equation (8) for MMEL-S is selected from {0.5, 1.0, 2.0}.\\nWe compare our proposed MMEL with conventional training with data augmentation (abbreviated as “Baseline(DA)”) under the same number of epochs. Though MMEL can be computed efficiently in parallel, the proposed MMEL encounters |B(xi)| = 10 times more training data. For fair comparison, we also compare with two other baselines that also use 10 times more data: (i) naive training with data augmentation but with 10 times more training epochs compared with MMEL (abbreviated as “Baseline(DA+Long)”). In this case, the learning rate accordingly decays at epochs 600, 1200 and 1600; (ii) training with data augmentation under the framework of MMEL but with uniform weights on the augmented samples (abbreviated as “Baseline(DA+UNI)”).\\nMain Results. The results are shown in Table 1. As can be seen, for both CIFAR-10 and CIFAR-100, MMEL-H and MMEL-S significantly outperform the Baseline(DA), with over 0.5 points higher accuracy on all four architectures. Compared to Baseline(DA+Long), the proposed MMEL-H and MMEL-S also have comparable or better performance, while being much more efficient in training. This is because our backward pass only computes the gradient of the weighted loss instead of the separate loss of each example. Compared to Baseline(DA+UNI) which has the same computational cost as MMEL-H and MMEL-S, the proposed methods also have better performance. This indicates the efficacy of the proposed maximal expected loss based reweighting strategy.\\nWe further evaluate the proposed method on larege-scale dataset ImageNet(Deng et al., 2009). The detailed results are in Appendix B.\\nVarying the Number of Augmented Samples. One hyperparameter of the proposed method is the number of augmented samples |B(xi)|. In Table 2, we evaluate the effect of |B(xi)| on the CIFAR dataset. We vary |B(xi)| in {2, 5, 10, 20} for both MMEL-H and MMEL-S with other settings unchanged. As can be seen, the performance of MMEL improves with more augmented samples for small |B(xi)|. However, the performance gain begins to saturate when |B(xi)| reaches 5 or 10 for some cases. Since a larger |B(xi)| also brings more training cost, we should choose a proper number of augmented samples rather than continually increasing it.\"\n },\n {\n \"heading\": \"4.2 RESULTS ON NATURAL LANGUAGE UNDERSTANDING TASKS\",\n \"text\": \"Data. GLUE is a benchmark containing various natural language understanding tasks, including textual entailment (RTE and MNLI), question answering (QNLI), similarity and paraphrase (MRPC, QQP, STS-B), sentiment analysis (SST-2) and linguistic acceptability (CoLA). Among them, STS-B is a regression task, CoLA and SST-2 are single sentence classification tasks, while the rest are sentence-pair classification tasks. Following (Devlin et al., 2019), for the development set, we report Spearman correlation for STS-B, Matthews correlation for CoLA and accuracy for the other tasks. For the test set for QQP and MRPC, we report “F1”.\\nSetup. The backbone model is BERTBASE (Devlin et al., 2019). We use the method in Section 3.3 to generate augmented samples. For the problem of mismatching label as described in Section 3.3, we use a BERTBASE model fine-tuned on the downstream task as teacher model to predict the label of each generated sample z in B(xi). For each xi, |B(xi)| = 5. The fraction of masked tokens for each sentence is 0.4. The λP of the KL regularization coefficient is 1.0 for both MMEL-H and MMEL-S. The λT in equation (8) for MMEL-S is 1.0. The other detailed hyperparameters in training can be found in Appendix D.\\nThe derivation of MMEL in Section 3 is based on the classification task, while STS-B is a regression task. Hence, we generalize our loss function accordingly for regression tasks as follows. For the hard loss in equation (7), we directly replace yz ∈ R with the prediction of teacher model on z. For the soft loss (8), for each entry of fθ(xi) in loss term λT ∑ z∈B(xi);z 6=xi P ∗ θ(z | xi)MSE(fθ(z), fθ(xi)), we replace it with the prediction of teacher model if the difference between them is larger than 0.5.\\nSimilar to Section 4.1, We compare with three baselines. However, we change the first baseline to naive training without data augmentation (abbreviated as “Baseline”) since data augmentation is not used by default in NLP tasks. The other two baselines are similar to those in Section 4.1: (i) “Baseline(DA+Long)” which fine-tunes BERT with data augmentation with the same batch size; and\\n(ii)“Baseline(DA+UNI)” which fine-tunes BERT with augmented samples by using average loss. We also compare with another recent data augmentation technique SMART (Jiang et al., 2020b).\\nMain Results. The development and test set results on the GLUE benchmark are shown in Table 3. The development set results for the BERT baseline are from our re-implementation, which is comparable or better than the reported results in the original paper (Devlin et al., 2019). The results for SMART are taken from (Jiang et al., 2020b), and there are no test set results in (Jiang et al., 2020b). As can be seen, data augmentation significantly improves the generalization of GLUE tasks. Compared to the baseline without data augmentation (Baseline), MMEL-H or MMEL-S consistently achieves better performance, especially on small datasets like CoLA and RTE. Similar to the observation in the image classification task in Section 4.1, the proposed MMEL-H and MMEL-S are more efficient and have better performance than Baseline(DA+Long). MMEL-H and MMEL-S also outperform Baseline(DA+UNI), indicating the superiority of using the proposed reweighting strategy. In addition, our proposed method also beats SMART in both accuracy and efficiency because they use PGD-k (Madry et al., 2018) to construct adversarial augmented samples which requires nearly k times more training cost. Figure 2 shows the development set accuracy across over the training procedure. As can be seen, training with MMEL-H or MMEL-S converges faster and has better accuracy except SST-2 and RTE where the performance is similar.\\nEffect of Predicted Labels. For the augmented samples from same origin, we use a fine-tuned task-specific BERTBASE teacher model to predict their labels as mentioned in Section 3.3 to handle the problem of mismatching label. In Table 4, we show the comparison between using the label of the original sample and using predicted labels. As can be seen, using the predicted label significantly improves the performance. By comparing with the results in Table 3, using the label of the original sample even hurts the performance.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"In this work, we propose to minimize a reweighted loss over the augmented samples which directly considers their implicit impacts on the loss. Since we can not access the optimal reweighting strategy, we propose to minimize the supremum of the loss under all reweighting strategies, and give a closedform solution of the optimal weights. Our method can be applied on top of any data augmentation methods. Experiments on both image classification tasks and natural language understanding tasks show that the proposed method improves the generalization performance of the model, while being efficient in training.\"\n },\n {\n \"heading\": \"A PROOF OF THEOREM 1\",\n \"text\": \"Proof. For any given xi and B(xi), we aim to find Pθ(· | xi) on B(xi) such that\\nmax Pθ(·|xi) ∑ z∈B(xi) Pθ(z | xi)`(fθ(z), yz))− λPPθ(z | xi) log (|B(xi)|Pθ(z | xi))\\ns.t. ∑\\nz∈B(xi)\\nPθ(z | xi) = 1. (10)\\nSince the objective is convex, by Lagrange multiplier method, let\\nL(Pθ, λ) = ∑\\nz∈B(xi)\\nPθ(z | xi)`(fθ(z), yz))− λPPθ(z | xi) log (|B(xi)|Pθ(z | xi))\\n+ λ ∑ z∈B(xi) Pθ(z | xi)− 1 . (11)\\nFrom ∇PθL(Pθ, λ) = ∇λL(Pθ, λ) = 0, for any pairs of zu, zv ∈ B(xi), we have\\n`(fθ(zu), yzu)− λP (log |B(xi)|+ 1 + log Pθ(zu | xi)) = `(fθ(zv), yzv )− λP (log |B(xi)|+ 1 + log Pθ(zv | xi)).\\n(12)\\nHence we have Pθ(zv | xi) = Pθ(zu | xi) exp ( `(fθ(zv), yzv )− `(fθ(zu), yzu)\\nλP\\n) . (13)\\nSumming over zv ∈ B(xi), we have\\nPθ(zu | xi) ∑\\nzv∈B(xi)\\nexp\\n( `(fθ(zv), yzu)− `(fθ(zu), yzu)\\nλP\\n) = 1. (14)\\nThe proof completes.\"\n },\n {\n \"heading\": \"B MMEL ON LARGE-SCALE DATASET\",\n \"text\": \"In this section, we evaluate the proposed method MMEL on large-scale image classification task ImageNet(Deng et al., 2009).\\nData. ImageNet is a benchmark dataset which contains colorful images with over 1 million training samples and 50000 validation samples from 1000 categories.\\nSetup. The model we used is ResNet for ImageNet with three different depths (He et al., 2016). All these experiments are conducted for 100 epochs, and the learning rate decays at epochs 30, 60, and 90. We set batch size as 256, and |B(xi)| = 10 for each xi. The other experimental settings follow Section 4.1, expect for the following hyperparameters. We compare the proposed method with “Baseline(DA)”.\\nMain Results. The results are shown in Table 5. From the results, the proposed MMEL-H improves the performance of the model for all three depths. However, the proposed MMEL-S is beaten by the baseline method. We speculate this is due to the relatively larger proportion of inaccurate prediction of original training samples on the large-scale dataset. More specifically, as in equation (8), for each augmented sample z ∈ B(xi), the proposed MMEL-S encourages the model to fit the output of original training sample fθ(xi). However, the accuracy of the original training samples in the ImageNet dataset can not reach 100% e.g., about 80% for ResNet50 on ImageNet. The inaccurate prediction fθ(xi) can be a misleading supervision for augmented sample z ∈ B(xi), leading to degraded performance of the proposed MMEL-S. Thus, we suggest using the MMEL-H if the accuracy of the original training samples is relative low.\"\n },\n {\n \"heading\": \"C GENERATING AUGMENTED SAMPLES FOR TEXTUAL SAMPLES\",\n \"text\": \"In this section, we elaborate the procedure of generating augmented sentences using greedy-based and beam-based method for a sequence. For each original input sentence, we randomly mask k tokens (which is obtained by rounding the product of masking ratio and length of the sequence to the nearest number) and then we do a forward propagation of the BERT to predict the tokens in those masked positions using greedy search. The detailed procedure is shown in Algorithm 2. We also use beam search (Yang et al., 2018) to generate augmented data. The details of beam search can be referred to (Yang et al., 2018). For sentence-pair tasks, we treat the two sentences separately and generate augmented samples for each of them.\\nAlgorithm 2 Augmented Sample Generation by Greedy Search Input: Pre-trained language model BertModel, original sentence x, number of augmented samples |B(x)| − 1, number of masked tokens k. Output: Augmented samples B(x) = {z1, z2, · · · , z|B(x)|−1}.\\n1: Randomly sample k positions {p1, · · · , pk} and get xmask. 2: for i = 1, 2, · · · |B(x)| − 1 do . Generate the i-th augmented sample 3: zi ← xmask. 4: zi[p1]← the ith most likely word predicted by BertModel(zi[p1]|zi). 5: for j in {2, 3. · · · , k} do 6: zi[pj ]← the most likely word predicted by BertModel(zi[pj ]|zi). 7: end for 8: end for\\nIn the following, we vary the factors that may affect the quality of the generated augmented samples. These factors include\\n1. The number of masked tokens, which equals the replacement proportion multiplied with the sentence length. This affects the diversity of augmented samples, i.e., replacing a large proportion of tokens makes the augmented sample less similar to the original one.\\n2. Treating the two sentences separately in sentence-pair tasks when generating augmented examples, or concatenate them as a single sentence;\\n3. Different generation methods like greedy search (Algorithm 2) and beam search.\\nThe results are shown in Table 6. As can be seen, compared with Baseline without data augmentation, MMEL-H and MMEL-S under all hyperparameter configurations have higher accuracy, showing the efficacy of data augmentation and the proposed reweighting strategy. There is no significant difference in using greedy search or beam search to generate the augmented samples. In this natural understanding task, training with augmented samples generated with proper larger replacement proportion (i.e., larger diversity) has slightly better performance. For sentence-pair tasks, treating the two sentences separately and generate augmented samples for each of them has slightly better performance. In the experiments in Section 4.2, we use Greedy search, masking proportion 0.4, and generate augmented sentence for each sentence in sentence-pair tasks.\"\n },\n {\n \"heading\": \"D HYPERPARAMETERS FOR THE EXPERIMENT ON THE GLUE BENCHMARK.\",\n \"text\": \"The optimizer we used is AdamW (Loshchilov & Hutter, 2018). The hyperparameters of BERTBASE model are listed in Table 7.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2021,"string":"2,021"},"title":{"kind":"string","value":"REWEIGHTING AUGMENTED SAMPLES BY MINIMIZ-"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":8379,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NjU1ODY0Niwic3ViIjoiL2RhdGFzZXRzL2FsbGVuYWkvbXVwLWZ1bGwiLCJleHAiOjE3NzY1NjIyNDYsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.S19R5LAD2ws-gwD-K3TlRiU0LXMHlx6IIXSlMMgMZBuL-7yu3SHBgeY0bjXNbk268dBafiG1NI9ximAxq9jbAQ","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":8379,"numBytesParquet":213262732},{"config":"default","split":"validation","numRows":1060,"numBytesParquet":27794829}]},"discussionsStats":{"closed":1,"open":0,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
paper_id stringlengths 43 43 | summaries list | abstractText stringlengths 98 40k | authors list | references list | sections list | year int64 1.98k 2.02k ⌀ | title stringlengths 4 183 ⌀ |
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SP:b80bc890180934092cde037b49d94d6e4e06fad9 | [
"This paper presents a novel way of making full use of compact episodic memory to alleviate catastrophic forgetting in continual learning. This is done by adding the proposed discriminative representation loss to regularize the gradients produced by new samples. Authors gave insightful analysis on the influence of ... | The use of episodic memories in continual learning has been shown to be effective in terms of alleviating catastrophic forgetting. In recent studies, several gradientbased approaches have been developed to make more efficient use of compact episodic memories, which constrain the gradients resulting from new samples wit... | [] | [
{
"authors": [
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{
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{
"heading": "1 INTRODUCTION",
"text": "In the real world, we are often faced with situations where data distributions are changing over time, and we would like to update our models by new data in time, with bounded growth in system size. These situations fall under the umbrella of “continual learning”, whi... | 2,020 | null |
SP:09f2fe6a482bbd6f9bd2c62aa841f995171ba939 | [
"This paper proposes a new framework that computes the task-specific representations to modulate the model parameters during the multi-task learning (MTL). This framework uses a single model with shared representations for learning multiple tasks together. Also, explicit task information may not be always available... | Existing Multi-Task Learning(MTL) strategies like joint or meta-learning focus more on shared learning and have little to no scope for task-specific learning. This creates the need for a distinct shared pretraining phase and a task-specific finetuning phase. The finetuning phase creates separate models for each task, w... | [] | [
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"Francis M. Tyers",
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"title": "Common voice: A massivelymultilingual speec... | [
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"text": "The process of Multi-Task Learning (MTL) on a set of related tasks is inspired by the patterns displayed by human learning. It involves a pretraining phase over all the tasks, followed by a finetuning phase. During pretraining, the model tries to grasp the shared know... | 2,020 | null |
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"This work proposes a deep reinforcement learning-based optimization strategy to the fuel optimization problem for the hybrid electric vehicle. The problem has been formulated as a fully observed stochastic Markov Decision Process (MDP). A deep neural network is used to parameterize the policy and value function. A... | This paper deals with the fuel optimization problem for hybrid electric vehicles in reinforcement learning framework. Firstly, considering the hybrid electric vehicle as a completely observable non-linear system with uncertain dynamics, we solve an open-loop deterministic optimization problem to determine a nominal opt... | [] | [
{
"authors": [
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"This paper proposes 3 deep generative models based on VAEs (with different encoding schemes for RNA secondary structure) for the generation of RNA secondary structures. They test each model on 3 benchmark tasks: unsupervised generation, semi-supervised learning and targeted generation. This paper has many interes... | Our work is concerned with the generation and targeted design of RNA, a type of genetic macromolecule that can adopt complex structures which influence their cellular activities and functions. The design of large scale and complex biological structures spurs dedicated graph-based deep generative modeling techniques, wh... | [
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"affiliations": [],
"name": "Zichao Yan"
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"Denise Carvalho-Silva",
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],
"title": "Ensembl 2017",
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{
"heading": "1 INTRODUCTION",
"text": "There is an increasing interest in developing deep generative models for biochemical data, especially in the context of generating drug-like molecules. Learning generative models of biochemical molecules can facilitate the development and discovery of novel treatments... | 2,021 | RNA SECONDARY STRUCTURES |
SP:0bd749fe44c37b521bd40f701e1428890aaa9c95 | [
"This paper presents a benchmark for discourse phenomena in machine translation. Its main novelty lies in the relatively large scale, spanning three translation directions, four discourse phenomena, and 150-5000 data points per language and phenomenon. A relatively large number of systems from previous work is benc... | Despite increasing instances of machine translation (MT) systems including extrasentential context information, the evidence for translation quality improvement is sparse, especially for discourse phenomena. Popular metrics like BLEU are not expressive or sensitive enough to capture quality improvements or drops that a... | [
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"year": 2018
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"heading": "1 INTRODUCTION AND RELATED WORK",
"text": "The advances in neural machine translation (NMT) systems have led to great achievements in terms of state-of-the-art performance in automatic translation tasks. There have even been claims that their translations are no worse than what an average bili... | 2,020 | DIP BENCHMARK TESTS: EVALUATION BENCH- |
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"The authors present a framework that uses a combination of VAE and GAN to recover private user images using Side channel analysis of memory access . A VAE-LP model first reconstructs a coarse image from side channel information which is reshaped and processed using a convolutional network. The output of the VAE-... | System side channels denote effects imposed on the underlying system and hardware when running a program, such as its accessed CPU cache lines. Side channel analysis (SCA) allows attackers to infer program secrets based on observed side channel logs. Given the ever-growing adoption of machine learning as a service (MLa... | [
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"text": "Side channel analysis (SCA) recovers program secrets based on the victim program’s nonfunctional characteristics (e.g., its execution time) that depend on the values of program secrets. SCA constitutes a major threat in today’s system and hardware security landscape. ... | 2,021 | null |
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"text": "Deep ensembles perform better than a single network thanks to the diversity among their members. Recent approaches regularize predictions to increase diversity; however, they also drastically decrease individual members’ performances. In this paper, we argue that learning strateg... | 2,021 | null |
SP:5561773ab024b083be4e362db079e371abf79653 | [
"The paper proposed a new training framework, namely GSL, for novel content synthesis. And GSL enables learning of disentangled representations of tangible attributes and achieve novel image synthesis by recombining those swappable components under a zero-shot setting. The framework leverages the underlying semanti... | Visual cognition of primates is superior to that of artificial neural networks in its ability to “envision” a visual object, even a newly-introduced one, in different attributes including pose, position, color, texture, etc. To aid neural networks to envision objects with different attributes, we propose a family of ob... | [
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"name": "Sami Abu-El-Haija"
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"This paper presents an approach to learn goal conditioned policies by relying on self-play which sets the goals and discovers a curriculum of tasks for learning. Alice and Bob are the agents. Alice's task is to set a goal by following a number of steps in the environment and she is rewarded when the goal is too ch... | We train a single, goal-conditioned policy that can solve many robotic manipulation tasks, including tasks with previously unseen goals and objects. We rely on asymmetric self-play for goal discovery, where two agents, Alice and Bob, play a game. Alice is asked to propose challenging goals and Bob aims to solve them. W... | [] | [
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"heading": "1 INTRODUCTION",
"text": "We are motivated to train a single goal-conditioned policy (Kaelbling, 1993) that can solve any robotic manipulation task that a human may request in a given environment. In this work, we make progress towards this goal by solving a robotic manipulation problem in a t... | 2,020 | null |
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"text": "Graph neural networks (GNNs) have been intensively studied recently [29, 26, 39, 68], due to their established performance towards various real-world tasks [15, 69, 53], as well as close connections to spectral graph theory [12, 9, 16]. While most GNN architectures ar... | 2,022 | Transfer Learning of Graph Neural Networks with Ego-graph Information Maximization |
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"text": "With rapid progress in AI, there is an increasing desire for general AI (Goertzel & Pennachin, 2007; Chollet, 2019) and explainable AI (Adadi & Berrada, 2018; Molnar, 2019), which exhibit broad, human-like cognitive capacities. One common pursuit is to move away from ... | 2,020 | null |
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"This paper proposes searching for an architecture generator that outputs good student architectures for a given teacher. The authors claim that by learning the parameters of the generator instead of relying directly on the search space, it is possible to explore the search space of architectures more effectively, ... | State-of-the-art results in deep learning have been improving steadily, in good part due to the use of larger models. However, widespread use is constrained by device hardware limitations, resulting in a substantial performance gap between state-ofthe-art models and those that can be effectively deployed on small devic... | [] | [
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{
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"text": "Debugging machine learning (ML) models is a critical part of the ML development life cycle. Uncovering bugs helps ML developers make important decisions about both development and deployment. In practice, much of debugging uses aggregate test statistics (like those in... | 2,020 | DEFUSE: DEBUGGING CLASSIFIERS THROUGH DIS- |
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"text": "Mixed Integer Programming (MIP) has been applied widely in many real-world problems, such as scheduling (Barnhart et al., 2003) and transportation (Melo & Wolsey, 2012). Branch and Bound (B&B) is a general and widely used paradigm for solving MIP problems (Wolsey & Ne... | 2,020 | null |
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SP:ba25b5b02701e01998e9dd22e4230c4e095f4542 | [
"The paper deals with the problem of credit assignment and synchronous estimation in cooperative multi-agent reinforcement learning problems. The authors introduce marginal advantage functions and use them for the estimation of the counterfactual advantage function. These functions permit to decompose the Multi-Age... | Cooperative multi-agent tasks require agents to deduce their own contributions with shared global rewards, known as the challenge of credit assignment. General methods for policy based multi-agent reinforcement learning to solve the challenge introduce differentiate value functions or advantage functions for individual... | [] | [
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{
"heading": "1 INTRODUCTION",
"text": "Reinforcement learning(RL) algorithms have shown amazing performance on many singleagent(SA) environment tasks (Mnih et al., 2013)(Jaderberg et al., 2016)(Oh et al., 2018). However, for many real-world problems, the environment is much more complex where RL agents oft... | 2,020 | null |
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"Amol Kapoor",
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"Hrayr Harutyunyan",
"Greg Ver Steeg",
"Aram Galstyan"
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"title": "Mixhop: Higher-order graph convolutional architectures via sparsified neighbor... | [
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"heading": null,
"text": "We study Graph Convolutional Networks (GCN) from the graph signal processing viewpoint by addressing a difference between learning graph filters with fullyconnected weights versus trainable polynomial coefficients. We find that by stacking graph filters with learnable polynomial ... | 2,020 | null |
SP:f19be0fdce321827638f91d57607ba340b1c3e4b | [
"The main objective of this paper is to reduce the model stability, in particular, the prediction churn of neural networks. The prediction churn is defined as the changed prediction w.r.t. model randomness, e.g. multiple runs of networks. The paper proposed to use a interpolated version of global label smoothing an... | Training modern neural networks is an inherently noisy process that can lead to high prediction churn– disagreements between re-trainings of the same model due to factors such as randomization in the parameter initialization and mini-batches– even when the trained models all attain high accuracies. Such prediction chur... | [] | [
{
"authors": [
"Ehsan Amid",
"Manfred KK Warmuth",
"Rohan Anil",
"Tomer Koren"
],
"title": "Robust bi-tempered logistic loss based on bregman divergences",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"Roha... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep neural networks (DNNs) have proved to be immensely successful at solving complex classification tasks across a range of problems. Much of the effort has been spent towards improving their predictive performance (i.e. accuracy), while comparatively little has been... | 2,020 | DEEP k-NN LABEL SMOOTHING IMPROVES STABIL- |
SP:5751b2abad772e44e69e125a769f25892c2a2e30 | [
"This paper proposes Adversarial Feature Desensitization (AFD) as a defense against adversarial examples. AFD employs a min-max adversarial learning framework where the classifier learns to encode features of both clean and adversarial images as the same distribution, thereby desensitizing adversarial features. Wit... | Neural networks are known to be vulnerable to adversarial attacks – slight but carefully constructed perturbations of the inputs which can drastically impair the network’s performance. Many defense methods have been proposed for improving robustness of deep networks by training them on adversarially perturbed inputs. H... | [
{
"affiliations": [],
"name": "Pouya Bashivan"
},
{
"affiliations": [],
"name": "Reza Bayat"
},
{
"affiliations": [],
"name": "Adam Ibrahim"
},
{
"affiliations": [],
"name": "Kartik Ahuja"
},
{
"affiliations": [],
"name": "Mojtaba Faramarzi"
},
{
"affi... | [
{
"authors": [
"Anish Athalye",
"Nicholas Carlini",
"David Wagner"
],
"title": "Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples",
"venue": "35th International Conference on Machine Learning,",
"year": 2018
},
{
"author... | [
{
"heading": "1 Introduction",
"text": "When training a classifier, it is common to assume that the training and test samples are drawn from the same underlying distribution. In adversarial machine learning, however, this assumption is intentionally violated by using the classifier itself to perturb the sam... | 2,021 | Adversarial Feature Desensitization |
SP:95ba9ad102adafaabf9671737e6549728d104629 | [
"This paper derives various types of graph embeddings to encode aspects of syntactic information that the brain may be processing during real-time sentence comprehension. These embeddings, along with indicators of punctuation, POS and dependency tags, and BERT embeddings, are used to predict brain activity recorded... | We are far from having a complete mechanistic understanding of the brain computations involved in language processing and of the role that syntax plays in those computations. Most language studies do not computationally model syntactic structure and most studies that do model syntactic processing use effort-based metri... | [] | [
{
"authors": [
"Link"
],
"title": "BERT-Large, Cased: 24-layer, 1024-hidden, 16-heads, 340M parameters. URL https://storage.googleapis.com/bert_models/2018_10_18/cased_ L-24_H-1024_A-16.zip",
"venue": null,
"year": 2018
},
{
"authors": [
"Bijaya Adhikari",
"Yao Zhang",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Neuroscientists have long been interested in how the brain processes syntax. To date, there is no consensus on which brain regions are involved in processing it. Classically, only a small number of regions in the left hemisphere were thought to be involved in language... | 2,020 | null |
SP:7327dc440b5c193c1dda156276860f89594721fa | [
"This paper explores the problem of generalizing to novel combinations of verbs and nouns in a task for captioning video stills from videos about cooking. The paper introduces a new dataset based off of EPIC-Kitchens (Damen et al. 2018) which masks out verbs and nouns and splits the evaluation data into seen combin... | Children acquire language subconsciously by observing the surrounding world and listening to descriptions. They can discover the meaning of words even without explicit language knowledge, and generalize to novel compositions effortlessly. In this paper, we bring this ability to AI, by studying the task of multimodal co... | [] | [
{
"authors": [
"Chris Baber"
],
"title": "Designing smart objects to support affording situations: Exploiting affordance through an understanding of forms of engagement",
"venue": "Frontiers in psychology,",
"year": 2018
},
{
"authors": [
"Fabien Baradel",
"Natalia Neve... | [
{
"heading": null,
"text": "Children acquire language subconsciously by observing the surrounding world and listening to descriptions. They can discover the meaning of words even without explicit language knowledge, and generalize to novel compositions effortlessly. In this paper, we bring this ability to A... | 2,020 | null |
SP:5be9a3c39234c10c226c42eec95e29cbddbaf8c0 | [
"This paper presents a unified framework for graph convolutional neural networks based on regularized optimization, connecting different variants of graph neural networks including vanilla, attention-based, and topology-based approaches. The authors also propose a novel regularization technique to approach the ov... | Graph Convolutional Networks (GCNs) have attracted a lot of research interest in the machine learning community in recent years. Although many variants have been proposed, we still lack a systematic view of different GCN models and deep understanding of the relations among them. In this paper, we take a step forward to... | [] | [
{
"authors": [
"S. Bai",
"F. Zhang",
"P. Torr"
],
"title": "Hypergraph convolution and hypergraph attention",
"venue": "ArXiv, abs/1901.08150,",
"year": 2019
},
{
"authors": [
"Xavier Bresson",
"Thomas Laurent"
],
"title": "Residual gated graph convn... | [
{
"heading": "1 INTRODUCTION",
"text": "Recent years have witnessed a fast development in graph processing by generalizing convolution operation to graph-structured data, which is known as Graph Convolutional Networks (GCNs) (Kipf & Welling, 2017). Due to the great success, numerous variants of GCNs have be... | 2,020 | null |
SP:dd2a50abff85d2b52b02dfe27cd42e443ea265cf | [
"This article proposes a benchmark of off-policy evaluation, which provides different metrics for policy ranking, evaluation and selection. Offline metrics are provided by evaluating the value function of logged data, and then evaluating absolute error, rank correlation and regret. Verify the effectiveness of diffe... | Off-policy evaluation (OPE) holds the promise of being able to leverage large, offline datasets for both evaluating and selecting complex policies for decision making. The ability to learn offline is particularly important in many real-world domains, such as in healthcare, recommender systems, or robotics, where online... | [
{
"affiliations": [],
"name": "Justin Fu"
},
{
"affiliations": [],
"name": "Mohammad Norouzi"
},
{
"affiliations": [],
"name": "Ofir Nachum"
},
{
"affiliations": [],
"name": "George Tucker"
},
{
"affiliations": [],
"name": "Ziyu Wang"
},
{
"affiliation... | [
{
"authors": [
"Gabriel Barth-Maron",
"Matthew W. Hoffman",
"David Budden",
"Will Dabney",
"Dan Horgan",
"Dhruva TB",
"Alistair Muldal",
"Nicolas Heess",
"Timothy Lillicrap"
],
"title": "Distributional policy gradients",
"venue": "In Internationa... | [
{
"heading": "1 INTRODUCTION",
"text": "Reinforcement learning algorithms can acquire effective policies for a wide range of problems through active online interaction, such as in robotics (Kober et al., 2013), board games and video games (Tesauro, 1995; Mnih et al., 2013; Vinyals et al., 2019), and recomme... | 2,021 | null |
SP:1037f94ce6eae4a42ea7913c76007f5f3c26aeaf | [
"This paper proposes Triple-Search (TRIPS), a differentiable framework of jointly searching for network architecture, quantization precision, and accelerator parameters. To address the dilemma between exploding training memory and biased search, the proposed framework leverages heterogeneous sampling where soft Gum... | The record-breaking performance and prohibitive complexity of deep neural networks (DNNs) have ignited a substantial need for customized DNN accelerators which have the potential to boost DNN acceleration efficiency by orders-ofmagnitude. While it has been recognized that maximizing DNNs’ acceleration efficiency requir... | [] | [
{
"authors": [
"Mohamed S Abdelfattah",
"Łukasz Dudziak",
"Thomas Chau",
"Royson Lee",
"Hyeji Kim",
"Nicholas D Lane"
],
"title": "Best of both worlds: Automl codesign of a cnn and its hardware accelerator",
"venue": null,
"year": 2002
},
{
"authors": ... | [
{
"heading": "1 INTRODUCTION",
"text": "The powerful performance and prohibitive complexity of deep neural networks (DNNs) have fueled a tremendous demand for efficient DNN accelerators which could boost DNN acceleration efficiency by orders-of-magnitude (Chen et al., 2016). In response, extensive research ... | 2,020 | null |
SP:d850572819200f79545616fc92e789ce958b30d4 | [
"This paper deals with continual learning. Specifically, given a stream of tasks we want to maximise performance across all tasks. Typically neural networks suffer from catastrophic forgetting which results in worse performance on tasks seen earlier in training. There are many proposed solutions to this problem. On... | Continual learning often assumes a knowledge of (strict) task boundaries and identities for the instances in a data stream—i.e., a “task-aware” setting. However, in practice it is rarely the case that practitioners can expose task information to the model; thus needing “task-free” continual learning methods. Recent att... | [] | [
{
"authors": [
"Tameem Adel",
"Han Zhao",
"Richard E. Turner"
],
"title": "Continual learning with adaptive weights (claw)",
"venue": "In International Conference on Learning Representations,",
"year": 2020
},
{
"authors": [
"Rahaf Aljundi",
"Klaas Kelchterm... | [
{
"heading": "1 INTRODUCTION",
"text": "Accumulating past knowledge and adapting to evolving environments are one of the key traits in human intelligence (McClelland et al., 1995). While contemporary deep neural networks have achieved impressive results in a range of machine learning tasks Goodfellow et al.... | 2,020 | null |
SP:a692e1e43991839e08a02e9122757224e1582cfd | [
"Given one image, the paper first generates different views which are controlled by differentiable parameter \\alpha, and then minimizes the additional \"conditional variance\" term~(expectation of these views' squared differences). Therefore, the paper encourages representations of the same image remain similar un... | We propose methods to strengthen the invariance properties of representations obtained by contrastive learning. While existing approaches implicitly induce a degree of invariance as representations are learned, we look to more directly enforce invariance in the encoding process. To this end, we first introduce a traini... | [
{
"affiliations": [],
"name": "Adam Foster"
},
{
"affiliations": [],
"name": "Rattana Pukdee"
},
{
"affiliations": [],
"name": "Tom Rainforth"
}
] | [
{
"authors": [
"Philip Bachman",
"R Devon Hjelm",
"William Buchwalter"
],
"title": "Learning representations by maximizing mutual information across views",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"Ben Barre... | [
{
"heading": "1 INTRODUCTION",
"text": "Learning meaningful representations of data is a central endeavour in artificial intelligence. Such representations should retain important information about the original input whilst using fewer bits to store it (van der Maaten et al., 2009; Gregor et al., 2016). Sem... | 2,021 | CONTRASTIVE REPRESENTATION LEARNING |
SP:a24603a5dbc07070aeba98e1206511799111bec6 | [
"This paper studies the potential bias in deep semi-supervised anomaly detection. The bias is evaluated in terms of TPR rate given a fixed FPR rate. It uses the anomaly scores output by unsupervised anomaly detectors as a benchmark to examine the relative scoring bias in deep semi-supervised anomaly detectors. It f... | Anomaly detection presents a unique challenge in machine learning, due to the scarcity of labeled anomaly data. Recent work attempts to mitigate such problems by augmenting training of deep anomaly detection models with additional labeled anomaly samples. However, the labeled data often does not align with the target d... | [] | [
{
"authors": [
"Charu C Aggarwal",
"Saket Sathe"
],
"title": "Theoretical foundations and algorithms for outlier ensembles",
"venue": "Acm Sigkdd Explorations Newsletter,",
"year": 2015
},
{
"authors": [
"Varun Chandola",
"Arindam Banerjee",
"Vipin Kumar"
... | [
{
"heading": "1 INTRODUCTION",
"text": "Anomaly detection (Chandola et al., 2009; Pimentel et al., 2014) trains a formal model to identify unexpected or anomalous instances in incoming data, whose behaviors differ from normal instances. It is particularly useful for detecting problematic events such as digi... | 2,020 | null |
SP:cf6c9061542bf9c43a968faa574ce03ad71a859a | [
"The authors present an approach for testing calibration in conditional probability estimation models. They build on a line of work in the kernel estimation literature assessing whether the conditional distributions are well calibrated (i.e. P(Y | f(X)) = f(X), where f is some predictive model). They develop an MMD... | Most supervised machine learning tasks are subject to irreducible prediction errors. Probabilistic predictive models address this limitation by providing probability distributions that represent a belief over plausible targets, rather than point estimates. Such models can be a valuable tool in decision-making under unc... | [
{
"affiliations": [],
"name": "David Widmann"
},
{
"affiliations": [],
"name": "Fredrik Lindsten"
}
] | [
{
"authors": [
"M.A. Arcones",
"E. Giné"
],
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"year": 1992
},
{
"authors": [
"C. Berg",
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"P. Ressel"
],
"title": "Harmonic Analysis on Semigro... | [
{
"heading": "1 INTRODUCTION",
"text": "We consider the general problem of modelling the relationship between a featureX and a target Y in a probabilistic setting, i.e., we focus on models that approximate the conditional probability distribution P(Y |X) of target Y for given feature X . The use of probabil... | 1,980 | CALIBRATION TESTS BEYOND CLASSIFICATION |
SP:becb496310e88c1e2e7d03131093b9ebcf075c1d | [
"The authors consider the problem of learning a hash function such that semantically similar elements have high collision probability. They modify the approach Deep Hashing Networks (Zhu et al., 2016) with a new loss function. Rather than use a sigmoid based loss function, the authors argue that a loss function ba... | Semantic hashing methods have been explored for learning transformations into binary vector spaces. These learned binary representations may then be used in hashing based retrieval methods, typically by retrieving all neighboring elements in the Hamming ball with radius 1 or 2. Prior studies focus on tasks with a few d... | [] | [
{
"authors": [
"Stanley C Ahalt",
"Ashok K Krishnamurthy",
"Prakoon Chen",
"Douglas E Melton"
],
"title": "Competitive learning algorithms for vector quantization",
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},
{
"authors": [
"Sunil Arya",
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{
"heading": "1 INTRODUCTION",
"text": "One of most challenging aspects in many Information Retrieval (IR) systems is the discovery and identification of the nearest neighbors of a query element in an vector space. This is typically solved using Approximate Nearest Neighbors (ANN) methods as exact solutions... | 2,020 | null |
SP:7611ee6b9dfabf7ec6a65da58cb6e3892705e1c9 | [
"This paper introduces a new method for leveraging auxiliary information and unlabelled data to improve out-of-distribution model performance. Theoretically, in a linear model with latent variables, they demonstrate using auxiliary data as inputs helps in-distribution test-error, but can hurt out-of-distribution er... | Consider a prediction setting with few in-distribution labeled examples and many unlabeled examples both inand out-of-distribution (OOD). The goal is to learn a model which performs well both in-distribution and OOD. In these settings, auxiliary information is often cheaply available for every input. How should we best... | [
{
"affiliations": [],
"name": "DISTRIBUTION ROBUSTNESS"
},
{
"affiliations": [],
"name": "Sang Michael Xie"
},
{
"affiliations": [],
"name": "Ananya Kumar"
},
{
"affiliations": [],
"name": "Robbie Jones"
},
{
"affiliations": [],
"name": "Fereshte Khani"
},
... | [
{
"authors": [
"Sajjad Ahmad",
"Ajay Kalra",
"Haroon Stephen"
],
"title": "Estimating soil moisture using remote sensing data: A machine learning approach",
"venue": "Advances in Water Resources,",
"year": 2010
},
{
"authors": [
"EA AlBadawy",
"A Saha",
... | [
{
"heading": "1 INTRODUCTION",
"text": "When models are tested on distributions that are different from the training distribution, they typically suffer large drops in performance (Blitzer and Pereira, 2007; Szegedy et al., 2014; Jia and Liang, 2017; AlBadawy et al., 2018; Hendrycks et al., 2019a). For exam... | null | null |
SP:b6dd62914f7464efb601c6d9f8a4d35e047447d5 | [
"This paper studies the training of deep hierarchical VAEs and focuses on the problem of posterior collapse. It is argued that reducing the variance of the gradient estimate may help to overcome posterior collapse. The authors focus on reducing the variance of the functions parameterizing the variational distributi... | Variational autoencoders with deep hierarchies of stochastic layers have been known to suffer from the problem of posterior collapse, where the top layers fall back to the prior and become independent of input. We suggest that the hierarchical VAE objective explicitly includes the variance of the function parameterizin... | [] | [
{
"authors": [
"Samuel R. Bowman",
"Luke Vilnis",
"Oriol Vinyals",
"Andrew M. Dai",
"Rafal Józefowicz",
"Samy Bengio"
],
"title": "Generating Sentences from a Continuous Space",
"venue": "In CoNLL",
"year": 2016
},
{
"authors": [
"Xi Chen",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Variational autoencoders (VAE) [10] are a popular latent variable model for unsupervised learning that simplifies learning by the introduction of a learned approximate posterior. Given data x and latent variables z, we specify the conditional distribution p(x|z) by pa... | 2,020 | null |
SP:2d25eeb93ba90f9c4064bf794f9a132a6859c8e4 | [
"The paper proposes an approximation method, called NEMO (Normalized maximum likelihood Estimation for model-based optimization) to compute the conditional normalized maximum log-likelihood of a query data point as a way to quantify the uncertainty in a forward prediction model in offline model-based optimization ... | In this work we consider data-driven optimization problems where one must maximize a function given only queries at a fixed set of points. This problem setting emerges in many domains where function evaluation is a complex and expensive process, such as in the design of materials, vehicles, or neural network architectu... | [
{
"affiliations": [],
"name": "Justin Fu"
},
{
"affiliations": [],
"name": "Sergey Levine"
}
] | [
{
"authors": [
"Andrew Barron",
"Jorma Rissanen",
"Bin Yu"
],
"title": "The minimum description length principle in coding and modeling",
"venue": "IEEE Transactions on Information Theory,",
"year": 1998
},
{
"authors": [
"Endika Bengoetxea",
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{
"heading": "1 INTRODUCTION",
"text": "Many real-world optimization problems involve function evaluations that are the result of expensive or time-consuming process. Examples occur in the design of materials (Mansouri Tehrani et al., 2018), proteins (Brookes et al., 2019; Kumar & Levine, 2019), neural netw... | 2,021 | MALIZED MAXIMUM LIKELIHOOD ESTIMATION |
SP:ce75f565c3c17363695c9e39f28b49a66e3731b8 | [
"This paper proposes a simple approach to discover interpretable latent manipulations in trained text VAEs. The method essentially involves performing PCA on the latent representations to find directions that maximize variance. The authors argue that this results in more interpretable directions. The method is appl... | Language generation models are attracting more and more attention due to their constantly increasing quality and remarkable generation results. State-of-the-art NLG models like BART/T5/GPT-3 do not have latent spaces, therefore there is no natural way to perform controlled generation. In contrast, less popular models w... | [
{
"affiliations": [],
"name": "LANGUAGE VAES"
}
] | [
{
"authors": [
"Samuel R. Bowman",
"Gabor Angeli",
"Christopher Potts",
"Christopher D. Manning"
],
"title": "A large annotated corpus for learning natural language inference",
"venue": "In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing... | [
{
"heading": "1 INTRODUCTION",
"text": "Transformer-based models yield state-of-the-art results on a number of tasks, including representation learning (Devlin et al., 2019; Liu et al., 2019; Clark et al., 2020) and generation (Radford et al.; Raffel et al., 2019; Lewis et al., 2020). Notably, large languag... | 2,020 | null |
SP:b9d78677e836fddeab78615ad35e9545d9c1d08f | [
"This paper extends results of prior work by Steinke and Zakynthinou, by providing generalization bounds in the PAC-Bayesian and single-draw settings that depend on the conditional mutual information. The emphasis in this work is on obtaining fast rates ($1/n$ vs. $1/\\sqrt{n}$). The authors also conduct empirical ... | We present a framework to derive bounds on the test loss of randomized learning algorithms for the case of bounded loss functions. This framework leads to bounds that depend on the conditional information density between the the output hypothesis and the choice of the training set, given a larger set of data samples fr... | [] | [
{
"authors": [
"A.R. Asadi",
"E. Abbe",
"S. Verdú"
],
"title": "Chaining mutual information and tightening generalization bounds",
"venue": "In Proc. Conf. Neural Inf. Process. Syst. (NeurIPS),",
"year": 2018
},
{
"authors": [
"R. Bassily",
"S. Moran",
... | [
{
"heading": null,
"text": "√ n dependence. We demonstrate\nthe usefulness of our tail bounds by showing that they lead to estimates of the test loss achievable with several neural network architectures trained on MNIST and Fashion-MNIST that match the state-of-the-art bounds available in the literature."
... | 2,020 | null |
SP:29a7b851d3edc2176467adc75ba67cc973a11a37 | [
"This work proposes a sequence-to-sequence approach for learning the time evolution of PDEs. The method employs a bi-directional LSTM to predict solutions of a PDE-based formulation for a chosen number of time steps. By itself this is an interesting, and important goal, but the method does not seem to contain any n... | Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or highdimensional PDEs is frequently a challenging task. Inspired by the traditional finite difference and finite elements methods and emerging advancements in m... | [] | [
{
"authors": [
"Uri M Ascher",
"Steven J Ruuth",
"Raymond J Spiteri"
],
"title": "Implicit-explicit runge-kutta methods for time-dependent partial differential equations",
"venue": "Applied Numerical Mathematics,",
"year": 1997
},
{
"authors": [
"Fischer Black",
... | [
{
"heading": "1 INTRODUCTION",
"text": "The research of time-dependent partial differential equations (PDEs) is regarded as one of the most important disciplines in applied mathematics. PDEs appear ubiquitously in a broad spectrum of fields including physics, biology, chemistry, and finance, to name a few. ... | 2,020 | null |
SP:797b07cd8142a35333037bb573db0dfe5dde65ac | [
"In this paper, the authors develop a data selection scheme aimed to minimize a notion of Bayes excess risk for overparametrized linear models. The excess Bayes risk is the expected squared error between the prediction and the target. The authors note that solutions such as V-optimality exist for the underparametri... | The impressive performance exhibited by modern machine learning models hinges on the ability to train such models on a very large amounts of labeled data. However, since access to large volumes of labeled data is often limited or expensive, it is desirable to alleviate this bottleneck by carefully curating the training... | [] | [
{
"authors": [
"Sanjeev Arora",
"Simon S Du",
"Wei Hu",
"Zhiyuan Li",
"Russ R Salakhutdinov",
"Ruosong Wang"
],
"title": "On exact computation with an infinitely wide neural net",
"venue": "In Advances in Neural Information Processing Systems, pp. 8141–8150,",
... | [
{
"heading": "1 INTRODUCTION",
"text": "The impressive performance exhibited by modern machine learning models hinges on the ability to train the aforementioned models on a very large amounts of labeled data. In practice, in many real world scenarios, even when raw data exists aplenty, acquiring labels migh... | 2,020 | null |
SP:4989f7703e106a20401cec0a5058d440720b0379 | [
"This paper proposes a novel algorithm for offline policy optimization. The main idea is to prevent overestimation bias by regularizing against the variance of the importance weighted value estimate. There are two key modifications: (1) using an importance weight from the stationary distribution and (2) using Fench... | Learning policies from fixed offline datasets is a key challenge to scale up reinforcement learning (RL) algorithms towards practical applications. This is often because off-policy RL algorithms suffer from distributional shift, due to mismatch between dataset and the target policy, leading to high variance and over-es... | [] | [
{
"authors": [
"Prashanth L. A",
"Michael C. Fu"
],
"title": "Risk-sensitive reinforcement learning: A constrained optimization",
"venue": "viewpoint. CoRR,",
"year": 2018
},
{
"authors": [
"Prashanth L. A",
"Mohammad Ghavamzadeh"
],
"title": "Variance-con... | [
{
"heading": "1 INTRODUCTION",
"text": "Offline batch reinforcement learning (RL) algoithms are key towards scaling up RL for real world applications, such as robotics (Levine et al., 2016) and medical problems . This is because offline RL provides the appealing ability for agents to learn from fixed datase... | 2,020 | null |
SP:4e77d43eb99688600f6c2115e1882e0b1e11a751 | [
"This paper proposed a novel method which to quantify the reliability of DNN-driven hypotheses in a statistical hypothesis testing framework. Naive statistical testings are not appropriate for the DNN-driven hypotheses, where the hypotheses are selected by looking at the data(i.e. The selection bias exists). To add... | In the past few years, various approaches have been developed to explain and interpret deep neural network (DNN) representations, but it has been pointed out that these representations are sometimes unstable and not reproducible. In this paper, we interpret these representations as hypotheses driven by DNN (called DNN-... | [] | [
{
"authors": [
"Sebastian Bach",
"Alexander Binder",
"Grégoire Montavon",
"Frederick Klauschen",
"Klaus-Robert Müller",
"Wojciech Samek"
],
"title": "On pixel-wise explanations for non-linear classifier decisions by layer-wise relevance propagation",
"venue": "Plo... | [
{
"heading": "1 INTRODUCTION",
"text": "The remarkable predictive performance of deep neural networks (DNNs) stems from their ability to learn appropriate representations from data. In order to understand the decision-making process of DNNs, it is thus important to be able to explain and interpret DNN repre... | 2,020 | null |
SP:8a32dfc80f31fd3da97e15ce98193144d03836b5 | [
"This paper proposes a variant of the GTD2 algorithm by adding an additional regularization term to the objective function, and the new algorithm is named as Gradient-DD (GDD). The regularization ensures that the value function does not change drastically between consecutive iterations. The authors show that the up... | Off-policy algorithms, in which a behavior policy differs from the target policy and is used to gain experience for learning, have proven to be of great practical value in reinforcement learning. However, even for simple convex problems such as linear value function approximation, these algorithms are not guaranteed to... | [] | [
{
"authors": [
"K. Atkinson",
"W. Han",
"D. Stewart"
],
"title": "Numerical Solution of Ordinary Differential Equations",
"venue": "JOHN WILEY & SONS,",
"year": 2008
},
{
"authors": [
"L.C. Baird"
],
"title": "Residual algorithms: Reinforcement learning wi... | [
{
"heading": "1 INTRODUCTION",
"text": "Off-policy algorithms for value function learning enable an agent to use a behavior policy that differs from the target policy in order to gain experience for learning. However, because off-policy methods learn a value function for a target policy given data due to a ... | 2,020 | null |
SP:dcb62a0cc1b03e9ea24b2ed167f14255d9386f95 | [
"This paper presents a methodology for incorporating factor-graphs into model-based and model-free RL methods. The work starts by assuming access to a correct and factor graph showing the relationship between individual state factors, actions, and rewards. The authors propose to make use of this factor graph by usi... | We propose a simple class of deep reinforcement learning (RL) methods, called FactoredRL, that can leverage factored environment structures to improve the sample efficiency of existing model-based and model-free RL algorithms. In tabular and linear approximation settings, the factored Markov decision process literature... | [] | [
{
"authors": [
"Marcin Andrychowicz",
"Filip Wolski",
"Alex Ray",
"Jonas Schneider",
"Rachel Fong",
"Peter Welinder",
"Bob McGrew",
"Josh Tobin",
"OpenAI Pieter Abbeel",
"Wojciech Zaremba"
],
"title": "Hindsight experience replay",
"venue":... | [
{
"heading": null,
"text": "We propose a simple class of deep reinforcement learning (RL) methods, called FactoredRL, that can leverage factored environment structures to improve the sample efficiency of existing model-based and model-free RL algorithms. In tabular and linear approximation settings, the fac... | 2,020 | FACTOREDRL: LEVERAGING FACTORED GRAPHS FOR DEEP REINFORCEMENT LEARNING |
SP:ad7eb2bcb3a83153f140e5e8bfaa8b76110e62ab | [
"It is a very poorly written paper. Basic idea of finding a way to not have to wait for full forward pass is not new. Multiple research papers have been published from the extreme of using stale weight to some form of sub-network backdrop as a proxy for the full network. This paper proposed no new idea for local up... | Deep learning models trained on large data sets have been widely successful in both vision and language domains. As state-of-the-art deep learning architectures have continued to grow in parameter count so have the compute budgets and times required to train them, increasing the need for compute-efficient methods that ... | [] | [
{
"authors": [
"David H. Ackley",
"Geoffrey E. Hinton",
"Terrence J. Sejnowski"
],
"title": "A learning algorithm for Boltzmann machines",
"venue": "Cognitive Science,",
"year": 1985
},
{
"authors": [
"Eugene Belilovsky",
"Michael Eickenberg",
"Edouard... | [
{
"heading": "1 INTRODUCTION",
"text": "Backpropagation (Rumelhart et al., 1985) is by far the most common method used to train neural networks. Alternatives to backpropagation are typically used only when backpropagation is impractical due to a non-differentiable loss (Schulman et al., 2015), non-smooth lo... | 2,020 | null |
SP:a3e5acdd322677d019a4582db78dab2dc1102818 | [
"This paper discusses a well-known problem of VAE training that decoder produces blurry reconstruction with constant variance. While much existing work addressed this problem by introducing independent variance training (as of the original VAE model) or additional hyper-parameters, those approaches usually come wit... | Variational autoencoders (VAEs) provide an effective and simple method for modeling complex distributions. However, training VAEs often requires considerable hyperparameter tuning to determine the optimal amount of information retained by the latent variable. We study the impact of calibrated decoders, which learn the ... | [] | [
{
"authors": [
"Martı́n Abadi",
"Paul Barham",
"Jianmin Chen",
"Zhifeng Chen",
"Andy Davis",
"Jeffrey Dean",
"Matthieu Devin",
"Sanjay Ghemawat",
"Geoffrey Irving",
"Michael Isard"
],
"title": "Tensorflow: A system for large-scale machine learn... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep density models based on the variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014) have found ubiquitous use in probabilistic modeling and representation learning as they are both conceptually simple and are able to scale to very complex dis... | 2,020 | null |
SP:3a1d7f7165762299ba2d9bab4144576660b9a784 | [
"This paper proposes a sampling free technique based on variance propagation to model predictive distributions of deep learning models. Estimating uncertainty of deep learning models is an important line of research for understanding the reliability of predictions and ensuring robustness to out-of-distribution data... | Uncertainty evaluation is a core technique when deep neural networks (DNNs) are used in real-world problems. In practical applications, we often encounter unexpected samples that have not seen in the training process. Not only achieving the high-prediction accuracy but also detecting uncertain data is significant for s... | [] | [
{
"authors": [
"Roberto Cipolla"
],
"title": "Bayesian segnet: Model uncertainty in deep convolutional encoder-decoder architectures for scene understanding",
"venue": "Proceedings of the British Machine Vision Conference (BMVC),",
"year": 2017
},
{
"authors": [
"M. Christoph... | [
{
"heading": "1 INTRODUCTION",
"text": "Uncertainty evaluation is a core technique in practical applications of deep neural networks (DNNs). As an example, let us consider the Cyber-Physical Systems (CPS) such as the automated driving system. In the past decade, machine learning methods are widely utilized ... | 2,020 | null |
SP:72d1283f3602edc22896934271fcec5b03f25d9e | [
"This paper studies the differential private synthetic dataset generation. Unlike previous DP based GAN models, this paper aims to boost the sample quality of after the training stage. In particular, the final synthetic dataset is sampled from the sequence of generators obtained during GAN training. The distributio... | Differentially private GANs have proven to be a promising approach for generating realistic synthetic data without compromising the privacy of individuals. Due to the privacy-protective noise introduced in the training, the convergence of GANs becomes even more elusive, which often leads to poor utility in the output g... | [
{
"affiliations": [],
"name": "POST-GAN BOOSTING"
},
{
"affiliations": [],
"name": "Marcel Neunhoeffer"
},
{
"affiliations": [],
"name": "Zhiwei Steven Wu"
}
] | [
{
"authors": [
"Martin Abadi",
"Andy Chu",
"Ian Goodfellow",
"H. Brendan McMahan",
"Ilya Mironov",
"Kunal Talwar",
"Li Zhang"
],
"title": "Deep learning with differential privacy",
"venue": "In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Co... | [
{
"heading": "1 INTRODUCTION",
"text": "The vast collection of detailed personal data, including everything from medical history to voting records, to GPS traces, to online behavior, promises to enable researchers from many disciplines to conduct insightful data analyses. However, many of these datasets con... | 2,021 | null |
SP:a6280b6605e621403de6ac4c3fc80fa71184ab6d | [
"In this paper, the authors propose a post-processing method for removing bias from a trained model. The bias is defined as conditional statistical parity — for a given partitioning of the data, the predicted label should be conditionally uncorrelated with the sensitive (bias inducing) attribute for each partition.... | We present an efficient and scalable algorithm for debiasing trained models, including deep neural networks (DNNs), which we prove to be near-optimal by bounding its excess Bayes risk. Unlike previous black-box reduction methods to cost-sensitive classification rules, the proposed algorithm operates on models that have... | [] | [
{
"authors": [
"M.A. Bruckner"
],
"title": "The promise and perils of algorithmic lenders’ use of big data,",
"venue": "Chi.-Kent L. Rev.,",
"year": 2018
},
{
"authors": [
"R.C. Deo"
],
"title": "Machine learning in medicine,",
"venue": "Circulation,",
"year":... | [
{
"heading": "1 INTRODUCTION",
"text": "Machine learning is increasingly applied to critical decisions which can have a lasting impact on individual lives, such as for credit lending (Bruckner, 2018), medical applications (Deo, 2015), and criminal justice (Brennan et al., 2009). Consequently, it is imperati... | 2,020 | A NEAR-OPTIMAL ALGORITHM FOR DEBIASING TRAINED MACHINE LEARNING MODELS |
SP:90ffef024018f59b3bde23aa2e2a4677602d41e8 | [
"This paper presents a variant of Transformer where low-dimension matrix multiplications and single-head attention are used. Stacked group-linear-transformation (GLT) are applied on input of each layer to perform dimension growth and then reduction. The paper is well-written and easy to follow. Experiments demonstr... | We introduce a deep and light-weight transformer, DeLighT, that delivers similar or better performance than standard transformer-based models with significantly fewer parameters. DeLighT more efficiently allocates parameters both (1) within each Transformer block using the DeLighT transformation, a deep and lightweight... | [
{
"affiliations": [],
"name": "LIGHT-WEIGHT TRANSFORMER"
},
{
"affiliations": [],
"name": "Sachin Mehta"
},
{
"affiliations": [],
"name": "Marjan Ghazvininejad"
},
{
"affiliations": [],
"name": "Srinivasan Iyer"
},
{
"affiliations": [],
"name": "Luke Zettlemoy... | [
{
"authors": [
"Ashish Vaswani",
"Noam Shazeer",
"Niki Parmar",
"Jakob Uszkoreit",
"Llion Jones",
"Aidan N Gomez",
"Łukasz Kaiser",
"Illia Polosukhin"
],
"title": "Attention is all you need",
"venue": "In Advances in neural information processing syste... | [
{
"heading": "1 INTRODUCTION",
"text": "Attention-based transformer networks (Vaswani et al., 2017) are widely used for sequence modeling tasks, including language modeling and machine translation. To improve performance, models are often scaled to be either wider, by increasing the dimension of hidden laye... | 2,021 | null |
SP:c83ecc74eb885df5f29e5a7080a8c60d1ee0a3b0 | [
"This paper shows a relationship between the project rule weights of a Hopfield network (HN) and the interaction weights in a corresponding restricted Boltzmann machine (RBM). The mapping from HN to RBM is facilitated by realising that the partition function of BN can be seen as the partition function of a binary-c... | Hopfield networks (HNs) and Restricted Boltzmann Machines (RBMs) are two important models at the interface of statistical physics, machine learning, and neuroscience. Recently, there has been interest in the relationship between HNs and RBMs, due to their similarity under the statistical mechanics formalism. An exact m... | [
{
"affiliations": [],
"name": "BOLTZMANN MACHINES"
},
{
"affiliations": [],
"name": "Matthew Smart"
},
{
"affiliations": [],
"name": "Anton Zilman"
}
] | [
{
"authors": [
"David H Ackley",
"Geoffrey E Hinton",
"Terrence J. Sejnowski"
],
"title": "A learning algorithm for boltzmann machines",
"venue": "Cognitive Science,",
"year": 1985
},
{
"authors": [
"Elena Agliari",
"Adriano Barra",
"Andrea De Antoni",... | [
{
"heading": "1 INTRODUCTION",
"text": "Hopfield networks (HNs) (Hopfield, 1982; Amit, 1989) are a classical neural network architecture that can store prescribed patterns as fixed-point attractors of a dynamical system. In their standard formulation with binary valued units, HNs can be regarded as spin gla... | 2,021 | null |
SP:3d705a1b70254d2b9d05277efff8ac08b0539086 | [
"The authors present a way to learn the action of an arbitrary orthogonal matrix on a vector via a map from $\\mathbb{R}^{n\\times n}$ onto $\\operatorname{O}(n)$. They show that the map is surjective, and give conditions under which they can invert this action. They then compare against previous proposed schemes i... | Orthogonal weight matrices are used in many areas of deep learning. Much previous work attempt to alleviate the additional computational resources it requires to constrain weight matrices to be orthogonal. One popular approach utilizes many Householder reflections. The only practical drawback is that many reflections c... | [] | [
{
"authors": [
"Martin Arjovsky",
"Amar Shah",
"Yoshua Bengio"
],
"title": "Unitary Evolution Recurrent Neural Networks",
"venue": "In ICML,",
"year": 2016
},
{
"authors": [
"Nitin Bansal",
"Xiaohan Chen",
"Zhangyang Wang"
],
"title": "Can We G... | [
{
"heading": null,
"text": "Orthogonal weight matrices are used in many areas of deep learning. Much previous work attempt to alleviate the additional computational resources it requires to constrain weight matrices to be orthogonal. One popular approach utilizes many Householder reflections. The only pract... | 2,020 | null |
SP:0cb862cf3806c4f04d2d30f200c25841a1cb52a8 | [
"This paper proposes to learn patient-specific representation using patient physiological signals. The authors design a PCP representation for each patient, which is learned to agree with signals from the same patients and disagrees with the remaining patients. In the supervised part, the classifier is generated f... | Many clinical deep learning algorithms are population-based and difficult to interpret. Such properties limit their clinical utility as population-based findings may not generalize to individual patients and physicians are reluctant to incorporate opaque models into their clinical workflow. To overcome these obstacles,... | [] | [
{
"authors": [
"A K Akobeng"
],
"title": "Understanding randomised controlled trials",
"venue": "Archives of Disease in Childhood,",
"year": 2005
},
{
"authors": [
"Erick A Perez Alday",
"Annie Gu",
"Amit Shah",
"Chad Robichaux",
"An-Kwok Ian Wong",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Modern medical research is arguably anchored around the “gold standard” of evidence provided by randomized control trials (RCTs) (Cartwright, 2007). However, RCT-derived conclusions are population-based and fail to capture nuances at the individual patient level (Akob... | null | PCPS: PATIENT CARDIAC PROTOTYPES |
SP:b7a45906d972644e9d0e757a83ff50fd3ad7cde3 | [
"Either putting the uncertainty on the weights (e.g., Bayes by BP) or on the activation (e.g., fast dropout or variants of natural-parameter networks [2,3] or Bayesian dark knowledge [4]) or both [1] have been investigated before. The idea of moving the uncertainty from the weight to the activation function is not ... | Current approaches for uncertainty estimation in deep learning often produce too confident results. Bayesian Neural Networks (BNNs) model uncertainty in the space of weights, which is usually high-dimensional and limits the quality of variational approximations. The more recent functional BNNs (fBNNs) address this only... | [
{
"affiliations": [],
"name": "Pablo Morales-Álvarez"
},
{
"affiliations": [],
"name": "Daniel Hernández-Lobato"
}
] | [
{
"authors": [
"F. Agostinelli",
"M. Hoffman",
"P. Sadowski",
"P. Baldi"
],
"title": "Learning activation functions to improve deep neural networks",
"venue": "arXiv preprint arXiv:1412.6830,",
"year": 2014
},
{
"authors": [
"P. Baldi",
"P. Sadowski",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep Neural Networks (DNNs) have achieved state-of-the-art performance in many different tasks, such as speech recognition (Hinton et al., 2012), natural language processing (Mikolov et al., 2013) or computer vision (Krizhevsky et al., 2012). In spite of their predict... | 2,021 | ACTIVATION-LEVEL UNCERTAINTY IN DEEP NEURAL NETWORKS |
SP:4d94ef57fdaf5f1100b6b09331d5cff5264fcdf6 | [
"In this paper, the authors argue that the mini-batch method and local SGD method suffers generalization performance degradation for large local mini-batch size. An asynchronous method is proposed to improve the generalization performance. A sublinear convergence rate is provided for the non-convex objective. As th... | Distributed variants of stochastic gradient descent (SGD) are central to training deep neural networks on massive datasets. Several scalable versions of data-parallel SGD have been developed, leveraging asynchrony, communicationcompression, and local gradient steps. Current research seeks a balance between distributed ... | [
{
"affiliations": [],
"name": "MEETS ASYNCHRONY"
}
] | [
{
"authors": [
"Dan Alistarh",
"Demjan Grubic",
"Jerry Li",
"Ryota Tomioka",
"M. Vojnovic"
],
"title": "Qsgd: Communicationefficient sgd via gradient quantization and encoding",
"venue": null,
"year": 2017
},
{
"authors": [
"Nils Berglund"
],
"... | [
{
"heading": "1 INTRODUCTION",
"text": "In this paper, we consider the classic problem of minimizing an empirical risk, defined simply as min x∈Rd ∑ i∈[I] fi(x), (1) where d is the dimension, x ∈ Rd denotes the set of model parameters, [I] is the training set, and fi(x) : Rd → R is the loss on the training ... | 2,020 | null |
SP:3dffd0add054e13be141cfe939e367f6f6785eb8 | [
"This paper deals with the problem of natural language generation for a dialogue system involved in complex communication tasks such as negotiation or persuasion. The proposed architecture consists of two encoders: one for the utterance and the other for dialogue acts and negotiation strategies. The decoder is an R... | To successfully negotiate a deal, it is not enough to communicate fluently: pragmatic planning of persuasive negotiation strategies is essential. While modern dialogue agents excel at generating fluent sentences, they still lack pragmatic grounding and cannot reason strategically. We present DIALOGRAPH, a negotiation s... | [
{
"affiliations": [],
"name": "NEGOTIATION DIALOGUES"
},
{
"affiliations": [],
"name": "Rishabh Joshi"
},
{
"affiliations": [],
"name": "Vidhisha Balachandran"
},
{
"affiliations": [],
"name": "Shikhar Vashishth"
},
{
"affiliations": [],
"name": "Alan W Black"... | [
{
"authors": [
"Nicholas Asher",
"Julie Hunter",
"Mathieu Morey",
"Benamara Farah",
"Stergos Afantenos"
],
"title": "Discourse structure and dialogue acts in multiparty dialogue: the STAC corpus",
"venue": "In Proceedings of the Tenth International Conference on Languag... | [
{
"heading": "1 INTRODUCTION",
"text": "Negotiation is ubiquitous in human interaction, from e-commerce to the multi-billion dollar sales of companies. Learning how to negotiate effectively involves deep pragmatic understanding and planning the dialogue strategically (Thompson; Bazerman et al., 2000b; Pruit... | 2,021 | null |
SP:3b3e7833784c53527eb32d5f6ac8d720f9d764bd | [
"The paper studies a problem of learning step-size policy for L-BFGS algorithm. This paper falls into a general category of meta-learning algorithms that try to derive a data-driven approach to learn one of the parameters of the learning algorithm. In this case, it is the learning rate of L-BFGS. The paper is very ... | We consider the problem of how to learn a step-size policy for the LimitedMemory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. This is a limited computational memory quasi-Newton method widely used for deterministic unconstrained optimization but currently avoided in large-scale problems for requiring step sizes... | [] | [
{
"authors": [
"A. Agrawal",
"B. Amos",
"S. Barratt",
"S. Boyd",
"S. Diamond",
"Z. Kolter"
],
"title": "Differentiable convex optimization layers",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"... | [
{
"heading": "1 INTRODUCTION",
"text": "Consider the unconstrained optimization problem\nminimize x f(x) (1)\nwhere f : Rn → R is an objective function that is differentiable for all x ∈ Rn, with n being the number of decision variables forming x. Let ∇xf(x0) be the gradient of f(x) evaluated at some x0 ∈ R... | 2,020 | null |
SP:7a92beaba926a93a627208abebe4a455ae3e0400 | [
"This paper proposes a new calibration error measurement named UCE (Uncertainty Calibration Error) for deep classification models. It consists in doing a calibration in order to achieve \"perfect calibration\" (i.e., the uncertainty provided is equivalent to the classification error at all levels in [0, 1]), relyin... | Various metrics have recently been proposed to measure uncertainty calibration of deep models for classification. However, these metrics either fail to capture miscalibration correctly or lack interpretability. We propose to use the normalized entropy as a measure of uncertainty and derive the Uncertainty Calibration E... | [] | [
{
"authors": [
"Arsenii Ashukha",
"Alexander Lyzhov",
"Dmitry Molchanov",
"Dmitry Vetrov"
],
"title": "Pitfalls of in-domain uncertainty estimation and ensembling in deep learning",
"venue": "In ICLR,",
"year": 2020
},
{
"authors": [
"Christopher M Bishop"
... | [
{
"heading": "1 INTRODUCTION",
"text": "Advances in deep learning have led to superior accuracy in classification tasks, making deep learning classifiers an attractive choice for safety-critical applications like autonomous driving (Chen et al., 2015) or computer-aided diagnosis (Esteva et al., 2017). Howev... | 2,020 | null |
SP:92d112388a1eac20c2208f0596cdfcdcca685c8f | [
"This paper presents a model and a corresponding training approach for multi-scale invertible models. The presented model is defined on multiple scales with information on finer scales being conditioned on coarser scales. Data generation is hence done sequentially from a coarser to finer scale. The authors argue th... | High-dimensional Bayesian inference problems cast a long-standing challenge in generating samples, especially when the posterior has multiple modes. For a wide class of Bayesian inference problems equipped with the multiscale structure that low-dimensional (coarse-scale) surrogate can approximate the original highdimen... | [] | [
{
"authors": [
"Lynton Ardizzone",
"Jakob Kruse",
"Sebastian Wirkert",
"Daniel Rahner",
"Eric W Pellegrini",
"Ralf S Klessen",
"Lena Maier-Hein",
"Carsten Rother",
"Ullrich Köthe"
],
"title": "Analyzing inverse problems with invertible neural network... | [
{
"heading": "1 INTRODUCTION",
"text": "Bayesian inference provides a powerful framework to blend prior knowledge, data generation process and (possibly small) data for statistical inference. With some prior knowledge ⇢ (distribution) for the quantity of interest x 2 Rd, and some (noisy) measurement y 2 Rdy... | 2,020 | null |
SP:077926a214f87b9fdcd5a5f9d818d6313437cd90 | [
"This study is presented clearly, and the core idea is interesting. However, the presented novelty is limited to a globally (for all tasks) and locally (task-specific) learning paradigm using a framework inspired by (Badirli et al., 2020). The authors have presented experimental results for both regression and cla... | Meta-optimization is an effective approach that learns a shared set of parameters across tasks for parameter initialization in meta-learning. A key challenge for metaoptimization based approaches is to determine whether an initialization condition can be generalized to tasks with diverse distributions to accelerate lea... | [] | [
{
"authors": [
"Ferran Alet",
"Tomás Lozano-Pérez",
"Leslie P Kaelbling"
],
"title": "Modular meta-learning",
"venue": "arXiv preprint arXiv:1806.10166,",
"year": 2018
},
{
"authors": [
"Jacob Andreas",
"Marcus Rohrbach",
"Trevor Darrell",
"Dan K... | [
{
"heading": "1 INTRODUCTION",
"text": "While humans can learn quickly with a few samples with prior knowledge and experiences, artificial intelligent algorithms face challenges in dealing with such situations. Learning to learn (or metalearning) (Vilalta & Drissi, 2002) emerges as the common practice to ad... | 2,020 | null |
SP:2969ff98eb93abe37242a962df458541311090ff | [
"The paper explores adversarial robustness in a new setting of test-time adaptation. It shows this new problem of “test-time-adapted adversarial robustness” is strictly weaker than the “traditional adversarial robustness” when assuming the training data is available for the “test-time-adapted adversarial robustness... | This paper studies test-time adaptation in the context of adversarial robustness. We formulate an adversarial threat model for test-time adaptation, where the defender may have a unique advantage as the adversarial game becomes a maximin game, instead of a minimax game as in the classic adversarial robustness threat mo... | [] | [
{
"authors": [
"Anish Athalye",
"Nicholas Carlini",
"David A. Wagner"
],
"title": "Obfuscated gradients give a false sense of security: Circumventing defenses to adversarial examples",
"venue": "Proceedings of the 35th International Conference on Machine Learning,",
"year": 201... | [
{
"heading": "1 INTRODUCTION",
"text": "There is a surge of interest to study test-time adaptation to help generalization to unseen domains (e.g., recent work by Sun et al. (2020); Wang et al. (2020); Nado et al. (2020)). At the high level, a generic test-time adaptation can be modeled as an algorithm Γ whi... | 2,020 | null |
SP:b7532fd6e281d88fff5a0a89c73ae3e6651f8827 | [
"This paper presents an approach to deep subspace clustering based on minimizing the correntropy induced metric (CIM), with the goal of establishing when training should be stopped and generalizing to unseen data. The main contribution over the existing S2ConfSCN method is a change from squared error loss to CIM wh... | Deep subspace clustering (SC) algorithms recently gained attention due to their ability to successfully handle nonlinearities in data. However, the insufficient capability of existing SC methods to deal with data corruption of unknown (arbitrary) origin hinders their generalization ability and capability to address rea... | [] | [
{
"authors": [
"Vincent Vanhoucke",
"Vijay Vasudevan",
"Fernanda Viégas",
"Oriol Vinyals",
"Pete Warden",
"Martin Wattenberg",
"Martin Wicke",
"Yuan Yu",
"Xiaoqiang Zheng"
],
"title": "TensorFlow: Large-scale machine learning on heterogeneous systems... | [
{
"heading": "1 INTRODUCTION",
"text": "Subspace clustering approaches have achieved encouraging performance when compared with the clustering algorithms that rely on proximity measures between data points. The main idea behind the subspace model is that the data can be drawn from low-dimensional subspaces ... | 2,020 | null |
SP:f0e0d909df518f25eb9243837939225d7db1196e | [
"The authors present a new Algorithm for performing unsupervised anomaly detection in diverse applications such as visual, audio and text data. They propose a two-step method in which first they utilise contrastive learning in order to find a semantically dense map of the data onto the unit-hypersphere. Then, they ... | In this paper we present SemSAD, a simple and generic framework for detecting examples that lie out-of-distribution (OOD) for a given training set. The approach is based on learning a semantic similarity measure to find for a given test example the semantically closest example in the training set and then using a discr... | [] | [
{
"authors": [
"Faruk Ahmed",
"Aaron C. Courville"
],
"title": "Detecting semantic anomalies",
"venue": "In AAAI,",
"year": 2020
},
{
"authors": [
"Samaneh Azadi",
"Catherine Olsson",
"Trevor Darrell",
"Ian J. Goodfellow",
"Augustus Odena"
],... | [
{
"heading": "1 INTRODUCTION",
"text": "Anomaly detection or novelty detection aims at identifying patterns in data that are significantly different to what is expected. This problem is inherently a binary classification problem that classifies examples either as in-distribution or out-of-distribution, give... | 2,020 | null |
SP:7c44bf5a4a8d5e5ee1e86ee4582c42186e2df72c | [
"The paper proposes a generative method for 3D objects (voxels representation). Given an initial voxels configuration (e.g. partial shape, or even a single voxel), the method learns a local transition kernel for a Markov chain to decide how to evolve the configuration; sampling iteratively from these probabilities ... | We present a probabilistic 3D generative model, named Generative Cellular Automata, which is able to produce diverse and high quality shapes. We formulate the shape generation process as sampling from the transition kernel of a Markov chain, where the sampling chain eventually evolves to the full shape of the learned d... | [
{
"affiliations": [],
"name": "Dongsu Zhang"
},
{
"affiliations": [],
"name": "Changwoon Choi"
},
{
"affiliations": [],
"name": "Jeonghwan Kim"
},
{
"affiliations": [],
"name": "Young Min Kim"
}
] | [
{
"authors": [
"Panos Achlioptas",
"Olga Diamanti",
"Ioannis Mitliagkas",
"Leonidas Guibas"
],
"title": "Learning representations and generative models for 3D point clouds",
"venue": "Proceedings of Machine Learning Research,",
"year": 2018
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Probabilistic 3D shape generation aims to learn and sample from the distribution of diverse 3D shapes and has applications including 3D contents generation or robot interaction. Specifically, learning the distribution of shapes or scenes can automate the process of ge... | 2,021 | GENERATIVE CELLULAR AUTOMATA |
SP:9326f169cc5e8d2f4268dcf39af31590ee004d98 | [
"This paper extends the results for actor-critic with stochastic policies of [Zhang, ICML 2018] to deterministic policies and offers the proof of convergence under some specific assumptions. The authors consider both the on-policy setting and the off-policy setting and offers some convincing derivation. It provides... | [Zhang, ICML 2018] provided the first decentralized actor-critic algorithm for 1 multi-agent reinforcement learning (MARL) that offers convergence guarantees. In 2 that work, policies are stochastic and are defined on finite action spaces. We extend 3 those results to offer a provably-convergent decentralized actor-cri... | [] | [
{
"authors": [
"Zhang"
],
"title": "The desired result holds since Step 1 and Step 2 of the proof of Theorem",
"venue": null,
"year": 2018
},
{
"authors": [
"Benveniste"
],
"title": "492 proof is now similar to the proof of Lemma 2 on page",
"venue": null,
"ye... | [
{
"heading": null,
"text": "[Zhang, ICML 2018] provided the first decentralized actor-critic algorithm for1 multi-agent reinforcement learning (MARL) that offers convergence guarantees. In2 that work, policies are stochastic and are defined on finite action spaces. We extend3 those results to offer a provab... | 2,020 | Decentralized Deterministic Multi-Agent Reinforcement Learning |
SP:cc282126b689c7311c3a28f0d173a004ed24382f | [
"The paper proposes a new training objective for fine-tuning pre-trained models: a weighted sum of the classical cross-entropy (CE) and a new supervised contrastive learning term (SCP). The latter uses the (negated) softmax over the embedding distances (i.e. dot products) between a training instance and all other i... | State-of-the-art natural language understanding classification models follow twostages: pre-training a large language model on an auxiliary task, and then finetuning the model on a task-specific labeled dataset using cross-entropy loss. However, the cross-entropy loss has several shortcomings that can lead to sub-optim... | [
{
"affiliations": [],
"name": "Beliz Gunel"
},
{
"affiliations": [],
"name": "Jingfei Du"
},
{
"affiliations": [],
"name": "Alexis Conneau"
},
{
"affiliations": [],
"name": "Ves Stoyanov"
}
] | [
{
"authors": [
"Armen Aghajanyan",
"Akshat Shrivastava",
"Anchit Gupta",
"Naman Goyal",
"Luke Zettlemoyer",
"Sonal Gupta"
],
"title": "Better fine-tuning by reducing representational collapse",
"venue": null,
"year": 2008
},
{
"authors": [
"Phili... | [
{
"heading": "1 INTRODUCTION",
"text": "State-of-the-art for most existing natural language processing (NLP) classification tasks is achieved by models that are first pre-trained on auxiliary language modeling tasks and then fine-tuned on the task of interest with cross-entropy loss (Radford et al., 2019; H... | 2,021 | PRE-TRAINED LANGUAGE MODEL FINE-TUNING |
SP:7eb0d8278168465270570233e4af64ebb3f2f154 | [
"Paper proposes to attack the challenging problem of RL with sparse feedback by leveraging a few demonstrations and learnable reward redistribution. The redistributed reward is computed by aligning the key events (a set of clustered symbols) to the demonstrations via PSSM-based seq matching. Experiments on two arti... | Reinforcement Learning algorithms require a large number of samples to solve complex tasks with sparse and delayed rewards. Complex tasks are often hierarchically composed of sub-tasks. Solving a sub-task increases the return expectation and leads to a step in the Q-function. RUDDER identifies these steps and then redi... | [
{
"affiliations": [],
"name": "STRATIONS BY"
},
{
"affiliations": [],
"name": "REWARD REDISTRIBUTION"
}
] | [
{
"authors": [
"P. Abbeel",
"A.Y. Ng"
],
"title": "Apprenticeship learning via inverse reinforcement learning",
"venue": "In Proceedings of the Twenty-First International Conference on Machine Learning, pp",
"year": 2004
},
{
"authors": [
"S.F. Altschul",
"W. Gish... | [
{
"heading": null,
"text": "Reinforcement Learning algorithms require a large number of samples to solve complex tasks with sparse and delayed rewards. Complex tasks are often hierarchically composed of sub-tasks. Solving a sub-task increases the return expectation and leads to a step in the Q-function. RUD... | 2,021 | ALIGN-RUDDER: LEARNING FROM FEW DEMON- |
SP:233335a3dc327cf153bd2e8d35a9e4594cf5bc67 | [
"This paper proposes a novel approach to modeling uncertainty, as an layer added-on to an otherwise black-box system. The ChePAN uses a neural network to estimate per-quantile roots of a chebyshev polynomial, then uses a quantile regression loss to fit these coefficients using backpropagation. Importantly, the Cheb... | Most predictive systems currently in use do not report any useful information for auditing their associated uncertainty and evaluating the corresponding risk. Taking it for granted that their replacement may not be advisable in the short term, in this paper we propose a novel approach to modelling confidence in such sy... | [] | [
{
"authors": [
"M. Abadi",
"P. Barham",
"J. Chen",
"Z. Chen",
"A. Davis",
"J. Dean",
"M. Devin",
"S. Ghemawat",
"G. Irving",
"M. Isard"
],
"title": "Tensorflow: A system for large-scale machine learning",
"venue": "In 12th tUSENIXu Symposiu... | [
{
"heading": "1 INTRODUCTION",
"text": "The present paper proposes a novel method for adding aleatoric uncertainty estimation to any pointwise predictive system currently in use. Considering the system as a black box, i.e. avoiding any hypothesis about the internal structure of the system, the method offers... | 2,020 | null |
SP:eff774eddcc60e943c0a41207c21a1c9d6d5d950 | [
"This paper proposes an approach to improve (supervised and unsupervised) representation learning for text using constrastive learning. The proposed approach augments standard contrastive learning with: (1) Spectral-norm regularization of the critic to estimate the Wasserstein distance instead of the KL (as in the ... | There has been growing interest in representation learning for text data, based on theoretical arguments and empirical evidence. One important direction involves leveraging contrastive learning to improve learned representations. We propose an application of contrastive learning for intermediate textual feature pairs, ... | [] | [
{
"authors": [
"Martin Arjovsky",
"Soumith Chintala",
"Léon Bottou"
],
"title": "Wasserstein generative adversarial networks",
"venue": "In ICML,",
"year": 2017
},
{
"authors": [
"Sanjeev Arora",
"Hrishikesh Khandeparkar",
"Mikhail Khodak",
"Ores... | [
{
"heading": "1 INTRODUCTION",
"text": "Representation learning is one of the pivotal topics in natural language processing (NLP), in both supervised and unsupervised settings. It has been widely recognized that some forms of “general representation” exist beyond specific applications (Oord et al., 2018). T... | 2,020 | null |
SP:a8bb14b514e474691be63b51582544a9befa7125 | [
"The paper finds that at extreme sparsities (>95%), existing approaches to pruning neural networks at initialization devolve to worse than random pruning. The paper posits that this degenerate behavior is due to the fact that weights are pruned in groups, though the saliency metrics only capture pointwise changes. ... | Recent studies have shown that skeletonization (pruning parameters) of networks at initialization provides all the practical benefits of sparsity both at inference and training time, while only marginally degrading their performance. However, we observe that beyond a certain level of sparsity (approx 95%), these approa... | [
{
"affiliations": [],
"name": "Pau de Jorge"
},
{
"affiliations": [],
"name": "Amartya Sanyal"
},
{
"affiliations": [],
"name": "Harkirat S. Behl"
},
{
"affiliations": [],
"name": "Puneet K. Dokania"
}
] | [
{
"authors": [
"Guillaume Bellec",
"David Kappel",
"Wolfgang Maass",
"Robert Legenstein"
],
"title": "Deep rewiring: Training very sparse deep networks",
"venue": "In International Conference on Learning Representations,",
"year": 2018
},
{
"authors": [
"Mig... | [
{
"heading": "1 INTRODUCTION",
"text": "The majority of pruning algorithms for Deep Neural Networks require training dense models and often fine-tuning sparse sub-networks in order to obtain their pruned counterparts. In Frankle & Carbin (2019), the authors provide empirical evidence to support the hypothes... | 2,021 | PROGRESSIVE SKELETONIZATION: TRIMMING MORE FAT FROM A NETWORK AT INITIALIZATION |
SP:ee89d3273df8b3b082c0e72a8768dff7cd3b7f56 | [
"Paper proposed to generate the communication message in MARL with the predicted trajectories of all the agents (include the agent itself). An extra self-attention model is also stacked over the trajectories to trade off the length of prediction and the possible explaining away issue. The whole model is trained vi... | Communication is one of the core components for learning coordinated behavior in multi-agent systems. In this paper, we propose a new communication scheme named Intention Sharing (IS) for multi-agent reinforcement learning in order to enhance the coordination among agents. In the proposed IS scheme, each agent generate... | [
{
"affiliations": [],
"name": "INTENTION SHARING"
},
{
"affiliations": [],
"name": "Woojun Kim"
},
{
"affiliations": [],
"name": "Jongeui Park"
},
{
"affiliations": [],
"name": "Youngchul Sung"
}
] | [
{
"authors": [
"Abhishek Das",
"Théophile Gervet",
"Joshua Romoff",
"Dhruv Batra",
"Devi Parikh",
"Mike Rabbat",
"Joelle Pineau"
],
"title": "Tarmac: Targeted multi-agent communication",
"venue": "In International Conference on Machine Learning,",
"year"... | [
{
"heading": "1 INTRODUCTION",
"text": "Reinforcement learning (RL) has achieved remarkable success in various complex control problems such as robotics and games (Gu et al. (2017); Mnih et al. (2013); Silver et al. (2017)). Multi-agent reinforcement learning (MARL) extends RL to multi-agent systems, which ... | 2,021 | null |
SP:b24e79d30d19c99f1093779bdba8bd8b2aed9ec0 | [
"In this paper, the authors focus on keystroke inference attacks in which an attacker leverages machine learning approaches, In particular, a new framework is proposed for low-resource video domain adaptation using supervised disentangled learning, and another method to assess the threat of keystroke inference att... | Keystroke inference attacks are a form of side-channels attacks in which an attacker leverages various techniques to recover a user’s keystrokes as she inputs information into some display (for example, while sending a text message or entering her pin). Typically, these attacks leverage machine learning approaches, but... | [] | [
{
"authors": [
"M. Backes",
"M. Dürmuth",
"D. Unruh"
],
"title": "Compromising reflections-or-how to read lcd monitors around the corner",
"venue": "IEEE Symposium on Security and Privacy (sp",
"year": 2008
},
{
"authors": [
"M. Backes",
"T. Chen",
"M.... | [
{
"heading": "1 INTRODUCTION",
"text": "We are exceedingly reliant on our mobile devices in our everyday lives. Numerous activities, such as banking, communications, and information retrieval, have gone from having separate channels to collapsing into one: through our mobile phones. While this has made many... | 2,020 | DISENTANGLING STYLE AND CONTENT FOR LOW RESOURCE VIDEO DOMAIN ADAPTATION: A CASE STUDY ON KEYSTROKE INFERENCE ATTACKS |
SP:181ce6eaacf4be8ede3fbdd82c63200278f63cc4 | [
"The paper considers the problem of approximating Sinkhorn divergence and corresponding transportation plan by combining low-rank and sparse approximation for the Sinkhorn kernel and using Nystrom iterations as a substitute for Sinkhorn's iterations. The corresponding approach is amenable to differentiation and can... | Optimal transport (OT) is a cornerstone of many machine learning tasks. The current best practice for computing OT is via entropy regularization and Sinkhorn iterations. This algorithm runs in quadratic time and requires calculating the full pairwise cost matrix, which is prohibitively expensive for large sets of objec... | [] | [
{
"authors": [
"Pierre Ablin",
"Gabriel Peyré",
"Thomas Moreau"
],
"title": "Super-efficiency of automatic differentiation for functions defined as a minimum",
"venue": "In ICML,",
"year": 2020
},
{
"authors": [
"Mokhtar Z. Alaya",
"Maxime Berar",
"Gil... | [
{
"heading": "1 INTRODUCTION",
"text": "Measuring the distance between two distributions or sets of objects is a central problem in machine learning. One common method of solving this is optimal transport (OT). OT is concerned with the problem of finding the transport plan for moving a source distribution (... | 2,020 | null |
SP:06414ad3c4b2438227a6d0749755106ee30f1564 | [
"The submission presents three contributions. First, the authors show the inconsistencies in the existing annealed Langevin sampling used in score-matching generative models and propose to correct it with the newly proposed Consistent Annealed Sampling (CAS) algorithm. The second contribution claimed is in providin... | Denoising Score Matching with Annealed Langevin Sampling (DSM-ALS) has recently found success in generative modeling. The approach works by first training a neural network to estimate the score of a distribution, and then using Langevin dynamics to sample from the data distribution assumed by the score network. Despite... | [] | [
{
"authors": [
"Yang Song",
"Stefano Ermon"
],
"title": "Generative modeling by estimating gradients of the data distribution",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"Aapo Hyvärinen"
],
"title": "Estimat... | [
{
"heading": "1 INTRODUCTION",
"text": "Song and Ermon (2019) recently proposed a novel method of generating samples from a target distribution through a combination of Denoising Score Matching (DSM) (Hyvärinen, 2005; Vincent, 2011; Raphan and Simoncelli, 2011) and Annealed Langevin Sampling (ALS) (Welling ... | null | ADVERSARIAL SCORE MATCHING AND IMPROVED SAMPLING FOR IMAGE GENERATION |
SP:f61e427d087e7f8b176a518af6088bde2ab75167 | [
"This paper proposes an approach based on Fourier transforms to predict ratings in collaborative filtering problems. The paper’s scope (“smooth reconstruction functions”) gets immediately narrowed down to Fourier transforms--it would be nice to provide some motivation for this choice over alternative smooth functio... | The problem of predicting the rating of a set of users to a set of items in a recommender system based on partial knowledge of the ratings is widely known as collaborative filtering. In this paper, we consider a mapping of the items into a vector space and study the prediction problem by assuming an underlying smooth p... | [] | [
{
"authors": [
"Rianne van den Berg",
"Thomas N Kipf",
"Max Welling"
],
"title": "Graph convolutional matrix completion",
"venue": "arXiv preprint arXiv:1706.02263,",
"year": 2017
},
{
"authors": [
"James Davidson",
"Benjamin Liebald",
"Junning Liu",
... | [
{
"heading": null,
"text": "The problem of predicting the rating of a set of users to a set of items in a recommender system based on partial knowledge of the ratings is widely known as collaborative filtering. In this paper, we consider a mapping of the items into a vector space and study the prediction pr... | 2,020 | null |
SP:97471b69a8e0ce6d2bbb202cc3f9cd786e77ddea | [
"The theoretical analysis is clearly stated in an well-organized way and the derived sparsity bound is reasonable. With FFNN and CNN, a theorem is given to show that the model is trainable only when the initialization on Edge of Chaos (EOC) and also provided a rescaling method to make the pruned NN into EOC regime.... | Overparameterized Neural Networks (NN) display state-of-the-art performance. However, there is a growing need for smaller, energy-efficient, neural networks to be able to use machine learning applications on devices with limited computational resources. A popular approach consists of using pruning techniques. While the... | [
{
"affiliations": [],
"name": "ROBUST PRUNING"
},
{
"affiliations": [],
"name": "AT INITIALIZATION"
},
{
"affiliations": [],
"name": "Soufiane Hayou"
},
{
"affiliations": [],
"name": "Jean-Francois Ton"
},
{
"affiliations": [],
"name": "Arnaud Doucet"
}
] | [
{
"authors": [
"J.M. Alvarez",
"M. Salzmann"
],
"title": "Compression-aware training of deep networks",
"venue": "In 31st Conference in Neural Information Processing Systems,",
"year": 2017
},
{
"authors": [
"S. Arora",
"S. Du",
"W. Hu",
"Z. Li",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Overparameterized deep NNs have achieved state of the art (SOTA) performance in many tasks (Nguyen and Hein, 2018; Du et al., 2019; Zhang et al., 2016; Neyshabur et al., 2019). However, it is impractical to implement such models on small devices such as mobile phones.... | 2,021 | null |
SP:934bf46c7ff0d3a3b1f0b75e48235dd0c902558c | [
"This paper study the fundamental relationship between adversarial transferability and knowledge transferability. Theoretical analysis is conducted, revealing that adversarial transferability can indicate knowledge transferability. In this procedure, two quantities are formally defined to measure adversarial transf... | Despite the immense success that deep neural networks (DNNs) have achieved, adversarial examples, which are perturbed inputs that aim to mislead DNNs to make mistakes, have recently led to great concerns. On the other hand, adversarial examples exhibit interesting phenomena, such as adversarial transferability. DNNs al... | [] | [
{
"authors": [
"Alessandro Achille",
"Michael Lam",
"Rahul Tewari",
"Avinash Ravichandran",
"Subhransu Maji",
"Charless C Fowlkes",
"Stefano Soatto",
"Pietro Perona"
],
"title": "Task2vec: Task embedding for meta-learning",
"venue": "In Proceedings of ... | [
{
"heading": null,
"text": "Despite the immense success that deep neural networks (DNNs) have achieved, adversarial examples, which are perturbed inputs that aim to mislead DNNs to make mistakes, have recently led to great concerns. On the other hand, adversarial examples exhibit interesting phenomena, such... | 2,020 | DOES ADVERSARIAL TRANSFERABILITY INDICATE KNOWLEDGE TRANSFERABILITY? |
SP:4a6f5bb1d0f72df5782a09a1ffc5e19504010e36 | [
"This work proposes an effective modification of language model token-level distribution during the training which prevents some forms of degeneration such as repetitions and dullness. The approach is based on the idea of encouraging the model to use tokens which were not observed in the previous context so far. In... | Advanced large-scale neural language models have led to significant success in many natural language generation tasks. However, the most commonly used training objective, Maximum Likelihood Estimation (MLE), has been shown to be problematic, where the trained model prefers using dull and repetitive phrases. In this wor... | [] | [
{
"authors": [
"Ilya Sutskever",
"Dario Amodei"
],
"title": "Language models are few-shot",
"venue": "learners. arXiv,",
"year": 2020
},
{
"authors": [
"Jacob Devlin",
"Ming-Wei Chang",
"Kenton Lee",
"Kristina Toutanova"
],
"title": "BERT: Pre-... | [
{
"heading": "1 INTRODUCTION",
"text": "Text generation has been one of the most important research problems in natural language processing (NLP) (Reiter & Dale, 2000). Thanks to the advances in neural architectures, models are now capable of generating texts that are of better quality than before (Brown et... | 2,020 | null |
SP:2062ab9c65e0d10e5d6d0112aaeaca208f131afd | [
"In this paper, the authors augment the instance-level self-supervised learning with cluster-aware learning mechanism during the training procedure. Specifically, for each training batch, the authors project the instances into a clustering space and then utilize a cluster-aware contrastive loss to push the augmente... | Learning visual representations using large-scale unlabelled images is a holy grail for most of computer vision tasks. Recent contrastive learning methods have focused on encouraging the learned visual representations to be linearly separable among the individual items regardless of their semantic similarity; however, ... | [] | [
{
"authors": [
"YM Asano",
"C Rupprecht",
"A Vedaldi"
],
"title": "Self-labelling via simultaneous clustering and representation learning",
"venue": "In International Conference on Learning Representations,",
"year": 2019
},
{
"authors": [
"Hyojin Bahng",
"S... | [
{
"heading": "1 INTRODUCTION",
"text": "Learning to extract generalized representations from a high-dimensional image is essential in solving various down-stream tasks in computer vision. Though a supervised learning framework has shown to be useful in learning discriminative representations for pre-trainin... | 2,020 | null |
SP:b47032cd0c8bf0189504e1c6562b058ba8f0e8ae | [
"The paper studies generalization under distribution shift, and tries to answer the question: why do ERM-based classifiers learn to rely on \"spurious\" features? They present a class of distributions called \"easy-to-learn\" that rules out several explanations given in recent work and isolates the spurious correla... | Empirical studies suggest that machine learning models often rely on features, such as the background, that may be spuriously correlated with the label only during training time, resulting in poor accuracy during test-time. In this work, we identify the fundamental factors that give rise to this behavior, by explaining... | [
{
"affiliations": [],
"name": "Vaishnavh Nagarajan"
},
{
"affiliations": [],
"name": "Anders Andreassen"
},
{
"affiliations": [],
"name": "Behnam Neyshabur"
}
] | [
{
"authors": [
"Isabela Albuquerque",
"João Monteiro",
"Mohammad Darvishi",
"Tiago H. Falk",
"Ioannis Mitliagkas"
],
"title": "Generalizing to unseen domains via distribution matching, 2020",
"venue": null,
"year": 2020
},
{
"authors": [
"Martı́n Arjov... | [
{
"heading": "1 INTRODUCTION",
"text": "A machine learning model in the wild (e.g., a self-driving car) must be prepared to make sense of its surroundings in rare conditions that may not have been well-represented in its training set. This could range from conditions such as mild glitches in the camera to s... | 2,021 | UNDERSTANDING THE FAILURE MODES OF OUT-OF- DISTRIBUTION GENERALIZATION |
SP:698104525f6955ba58aee1331a9487f77a542f13 | [
"This paper proposes a dataset of tasks to help evaluate learned optimizers. The learned optimizers are evaluated by the loss that they achieve on held-out tasks after 10k steps. Using this dataset, the main strategy considered is to use search spaces that parametrize optimizers and learn a list of hyperparameter c... | We present TaskSet, a dataset of tasks for use in training and evaluating optimizers. TaskSet is unique in its size and diversity, containing over a thousand tasks ranging from image classification with fully connected or convolutional neural networks, to variational autoencoders, to non-volume preserving flows on a va... | [] | [
{
"authors": [
"Martín Abadi",
"Paul Barham",
"Jianmin Chen",
"Zhifeng Chen",
"Andy Davis",
"Jeffrey Dean",
"Matthieu Devin",
"Sanjay Ghemawat",
"Geoffrey Irving",
"Michael Isard"
],
"title": "Tensorflow: A system for large-scale machine learni... | [
{
"heading": null,
"text": "We present TaskSet, a dataset of tasks for use in training and evaluating optimizers. TaskSet is unique in its size and diversity, containing over a thousand tasks ranging from image classification with fully connected or convolutional neural networks, to variational autoencoders... | 2,020 | TASKSET: A DATASET OF OPTIMIZATION TASKS |
SP:4bda50ce81c790cf9b19a24d81db4c07ec3729c1 | [
"The purpose of the paper seems clear: it proposes an attack to the recently proposed algorithm called Instahide (ICML 2020) which is a probabilistic algorithm for generating synthetic private data in the distributed setting. The attack proposed in this paper is considered for the case where the private data is i.i... | In this work, we examine the security of InstaHide, a scheme recently proposed by Huang et al. (2020b) for preserving the security of private datasets in the context of distributed learning. To generate a synthetic training example to be shared among the distributed learners, InstaHide takes a convex combination of pri... | [
{
"affiliations": [],
"name": "SPARSE MA"
},
{
"affiliations": [],
"name": "TRIX FACTORIZATION"
},
{
"affiliations": [],
"name": "Sitan Chen"
},
{
"affiliations": [],
"name": "Xiaoxiao Li"
},
{
"affiliations": [],
"name": "Danyang Zhuo"
}
] | [
{
"authors": [
"Sébastien Bubeck",
"Yin Tat Lee",
"Eric Price",
"Ilya Razenshteyn"
],
"title": "Adversarial examples from computational constraints",
"venue": "In International Conference on Machine Learning (ICML),",
"year": 2019
},
{
"authors": [
"T Tony C... | [
{
"heading": null,
"text": "In this work, we examine the security of InstaHide, a scheme recently proposed by Huang et al. (2020b) for preserving the security of private datasets in the context of distributed learning. To generate a synthetic training example to be shared among the distributed learners, Ins... | 2,021 | null |
SP:a1c54d5c42097b8ba971ac20470de864ae87dd4e | [
"In this work the authors propose a framework to perform object detection when there is noise present in class labels as well as bounding box annotations. The authors propose a two-step process, where in the first step the bounding boxes are corrected in class-agnostic way, and in the second step knowledge distilla... | Training deep object detectors requires large amounts of human-annotated images with accurate object labels and bounding box coordinates, which are extremely expensive to acquire. Noisy annotations are much more easily accessible, but they could be detrimental for learning. We address the challenging problem of trainin... | [] | [
{
"authors": [
"Eric Arazo",
"Diego Ortego",
"Paul Albert",
"Noel E. O’Connor",
"Kevin McGuinness"
],
"title": "Unsupervised label noise modeling and loss correction",
"venue": "In ICML,",
"year": 2019
},
{
"authors": [
"Devansh Arpit",
"Stanisla... | [
{
"heading": "1 INTRODUCTION",
"text": "The remarkable success of modern object detectors largely relies on large-scale datasets with extensive bounding box annotations. However, it is extremely expensive and time-consuming to acquire high-quality human annotations. For example, annotating each bounding box... | 2,020 | null |
SP:4fde35c9931ca15ab6cd53b171323e1abf0224db | [
"This paper proposes an approach to self-supervised learning from videos. The approach takes advantage of compressed videos, using the encoded residuals and motion vectors within the video codec. Using encoded videos has been shown to reduce computation time required by decoding videos. Previous works have explored... | Self-supervised learning of video representations has received great attention. Existing methods typically require frames to be decoded before being processed, which increases compute and storage requirements and ultimately hinders largescale training. In this work, we propose an efficient self-supervised approach to l... | [
{
"affiliations": [],
"name": "Youngjae Yu"
},
{
"affiliations": [],
"name": "Sangho Lee"
},
{
"affiliations": [],
"name": "Gunhee Kim"
}
] | [
{
"authors": [
"Relja Arandjelovic",
"Andrew Zisserman"
],
"title": "Look, listen and learn",
"venue": "In ICCV,",
"year": 2017
},
{
"authors": [
"Andrew Brock",
"Jeff Donahue",
"Karen Simonyan"
],
"title": "Large scale GAN training for high fidelity... | [
{
"heading": null,
"text": "Self-supervised learning of video representations has received great attention. Existing methods typically require frames to be decoded before being processed, which increases compute and storage requirements and ultimately hinders largescale training. In this work, we propose an... | 2,021 | SELF-SUPERVISED LEARNING OF COMPRESSED VIDEO REPRESENTATIONS |
SP:2d804ce6cd9917277ac5c4d6c72cceeb14bf0641 | [
"The paper presents two algorithms - one for the deterministic and one for stochastic bilevel optimization. The paper claims the methods are lower cost in computational complexity for various terms and easy to implement. A finite-time convergence proof is provided for the algorithms. Empirical results are presente... | Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensi... | [
{
"affiliations": [],
"name": "BILEVEL OPTI"
}
] | [
{
"authors": [
"Luca Bertinetto",
"Joao F Henriques",
"Philip Torr",
"Andrea Vedaldi"
],
"title": "Meta-learning with differentiable closed-form solvers",
"venue": "In International Conference on Learning Representations (ICLR),",
"year": 2018
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Bilevel optimization has received significant attention recently and become an influential framework in various machine learning applications including meta-learning (Franceschi et al., 2018; Bertinetto et al., 2018; Rajeswaran et al., 2019; Ji et al., 2020a), hyperpa... | 2,020 | null |
SP:2c5537aa2c173582e193c903eb85dd63aabc7366 | [
"In this paper, the authors propose a novel manifold learning method, via adding a locally isometric smoothness constraint, which preserves topological and geometric properties of data manifold. Empirical results demonstrate the efficacy of their approach. The authors also show that the reliability of tangent space... | It is widely believed that a dimension reduction (DR) process drops information inevitably in most practical scenarios. Thus, most methods try to preserve some essential information of data after DR, as well as manifold based DR methods. However, they usually fail to yield satisfying results, especially in high-dimensi... | [] | [
{
"authors": [
"Jens Behrmann",
"Will Grathwohl",
"Ricky T.Q. Chen",
"David Duvenaud",
"Jörn-Henrik Jacobsen"
],
"title": "Invertible residual networks",
"venue": "In Proceedings of the 36th International Conference on Machine Learning (ICML), Proceedings of Machine Lea... | [
{
"heading": "1 INTRODUCTION",
"text": "In real-world scenarios, it is widely believed that the loss of data information is inevitable after dimension reduction (DR), though the goal of DR is to preserve as much information as possible in the low-dimensional space. In the case of linear DR, compressed sensi... | 2,020 | null |
SP:26c214e61671b012baa8824a39772738a861e44b | [
"This paper introduces a Transformer-based image recognition model that is fully built on the Transformer layers (multi-head self-attention + point-wise MLP) without any standard convolution layers. Basically, it splits an image into patches and takes as input the set of linear embeddings of the patches and their p... | While the Transformer architecture has become the de-facto standard for natural language processing tasks, its applications to computer vision remain limited. In vision, attention is either applied in conjunction with convolutional networks, or used to replace certain components of convolutional networks while keeping ... | [
{
"affiliations": [],
"name": "Alexey Dosovitskiy"
},
{
"affiliations": [],
"name": "Lucas Beyer"
},
{
"affiliations": [],
"name": "Alexander Kolesnikov"
},
{
"affiliations": [],
"name": "Dirk Weissenborn"
},
{
"affiliations": [],
"name": "Xiaohua Zhai"
},
... | [
{
"authors": [
"Samira Abnar",
"Willem Zuidema"
],
"title": "Quantifying attention flow in transformers",
"venue": "In ACL,",
"year": 2020
},
{
"authors": [
"Alexei Baevski",
"Michael Auli"
],
"title": "Adaptive input representations for neural language mo... | [
{
"heading": "1 INTRODUCTION",
"text": "Self-attention-based architectures, in particular Transformers (Vaswani et al., 2017), have become the model of choice in natural language processing (NLP). The dominant approach is to pre-train on a large text corpus and then fine-tune on a smaller task-specific data... | 2,021 | null |
SP:87507439ef121d5d243502d2cb45eafec175f2bc | [
"Temporal smoothness is a recurring feature of real-world data that has been unaccounted for when training neural networks. Much of the random sampling in training neural networks is done to remove the temporal correlations originally present when the data is collected. This work aims to propose a method to train o... | Events in the real world are correlated across nearby points in time, and we must learn from this temporally “smooth” data. However, when neural networks are trained to categorize or reconstruct single items, the common practice is to randomize the order of training items. What are the effects of temporally smooth trai... | [] | [
{
"authors": [
"Christopher Baldassano",
"Uri Hasson",
"Kenneth A Norman"
],
"title": "Representation of real-world event schemas during narrative perception",
"venue": "Journal of Neuroscience,",
"year": 2018
},
{
"authors": [
"Yoshua Bengio",
"Jérôme Loura... | [
{
"heading": "1 INTRODUCTION",
"text": "Events in the world are correlated in time: the information that we receive at one moment is usually similar to the information that we receive at the next. For example, when having a conversation with someone, we see multiple samples of the same face from different a... | 2,020 | null |
SP:d460957c05007cafe286b0590ffed111c806dd48 | [
"The authors study the problem of global non-convex optimization with access only to function valuations. Specifically, they propose an approach to automatically control the hyper-parameters of Directional Gaussian Smoothing (DGS) a recently proposed solution for the problem. Their proposed solution trade-offs some... | The local gradient points to the direction of the steepest slope in an infinitesimal neighborhood. An optimizer guided by the local gradient is often trapped in local optima when the loss landscape is multi-modal. A directional Gaussian smoothing (DGS) approach was recently proposed in (Zhang et al., 2020) and used to ... | [
{
"affiliations": [],
"name": "SMOOTHING GRADIENT"
}
] | [
{
"authors": [
"Youhei Akimoto",
"Nikolaus Hansen"
],
"title": "Projection-based restricted covariance matrix adaptation for high dimension",
"venue": "In Proceedings of the Genetic and Evolutionary Computation Conference 2016,",
"year": 2016
},
{
"authors": [
"Larry Ar... | [
{
"heading": "1 INTRODUCTION",
"text": "We consider the problem of black-box optimization, where we search for the optima of a loss function F : Rd → R given access to only its function queries. This type of optimization finds applications in many machine learning areas where the loss function’s gradient is... | 2,020 | ADADGS: AN ADAPTIVE BLACK-BOX OPTIMIZATION METHOD WITH A NONLOCAL DIRECTIONAL GAUSSIAN |
SP:253566b5271d22d4d6492ef9def2e67fb99c5d57 | [
"The paper is addressing an important and challenging problem of end-to-end training of deep nets in fixed-point, in this case, with 8-bit precision. A good solution to this problem can have a major impact on the deployability of deep nets on embedded hardware. The basic idea is to introduce an additional term (the... | Quantization of neural network parameters and activations has emerged as a successful approach to reducing model size and inference time on hardware that supports native low-precision arithmetic. Fully quantized training would facilitate further computational speed-ups as well as enable model training on embedded devic... | [] | [
{
"authors": [
"Pulkit Bhuwalka",
"Alan Chiao",
"Suharsh Sivakumar",
"Raziel Alvarez",
"Feng Liu",
"Lawrence Chan",
"Skirmantas Kligys",
"Yunlu Li",
"Khanh LeViet",
"Billy Lambert",
"Mark Daoust",
"Tim Davis",
"Sarah Sirajuddin",
... | [
{
"heading": "1 INTRODUCTION",
"text": "As state-of-the-art deep learning models for vision, language understanding and speech grow increasingly large and computationally burdensome (He et al., 2017; Devlin et al., 2018; Karita et al., 2019), there is increasing antithetical demand, motivated by latency, se... | 2,020 | null |
SP:d9155553fae947cc53d87a221fdd1d57b44f5ec6 | [
"I read this paper with great interest. The authors propose an easy-to-understand, easy-to-implement baseline method for detecting when inputs to a ML model is out of distribution. The method involves augmenting the training dataset with an out of distribution dataset and adding an additional class in the classifi... | Refraining from confidently predicting when faced with categories of inputs different from those seen during training is an important requirement for the safe deployment of deep learning systems. While simple to state, this has been a particularly challenging problem in deep learning, where models often end up making o... | [] | [
{
"authors": [
"Loïc Barrault",
"Fethi Bougares",
"Lucia Specia",
"Chiraag Lala",
"Desmond Elliott",
"Stella Frank"
],
"title": "Findings of the third shared task on multimodal machine translation",
"venue": "In Proceedings of the Third Conference on Machine Trans... | [
{
"heading": "1 INTRODUCTION AND RELATED WORK",
"text": "Most of supervised machine learning has been developed with the assumption that the distribution of classes seen at train and test time are the same. However, the real-world is unpredictable and open-ended, and making machine learning systems robust t... | 2,020 | null |
SP:70b8c75426f18a3dc4a359c8a8cd7dd2076953a0 | [
"The authors propose to address the robustness over outliers for optimal transport (OT). They propose a new formulation based on penalizing the contaminated probability measures by a signed measure (which shares a close relation with unbalanced OT). The authors further derive an equivalent formulation by adjusting ... | Optimal transport (OT) provides a way of measuring distances between distributions that depends on the geometry of the sample space. In light of recent advances in solving the OT problem, OT distances are widely used as loss functions in minimum distance estimation. Despite its prevalence and advantages, however, OT is... | [] | [
{
"authors": [
"David Alvarez-Melis",
"Tommi S Jaakkola"
],
"title": "Gromov-Wasserstein alignment of word embedding spaces",
"venue": null,
"year": 2018
},
{
"authors": [
"Yogesh Balaji",
"Rama Chellappa",
"Soheil Feizi"
],
"title": "Robust optimal ... | [
{
"heading": "1 INTRODUCTION",
"text": "Optimal transport is a fundamental problem in applied mathematics. In its original form (Monge, 1781), the problem entails finding the minimum cost way to transport mass from a prescribed probability distribution µ on X to another prescribed distribution ν on X . Kant... | 2,020 | OUTLIER-ROBUST OPTIMAL TRANSPORT |
SP:198d7f650c930a1423f7f30688cd2f73d2719920 | [
"This paper aims to extend the continuous optimization approach to causal discovery to handle interventional data as well as observational data. It describes a method for learning the causal structure over a set of categorical variables and reports strong empirical performance. However, no theoretical guarantee or ... | Promising results have driven a recent surge of interest in continuous optimization methods for Bayesian network structure learning from observational data. However, there are theoretical limitations on the identifiability of underlying structures obtained from observational data alone. Interventional data provides muc... | [] | [
{
"authors": [
"Bruce Abramson",
"John Brown",
"Ward Edwards",
"Allan Murphy",
"Robert L Winkler"
],
"title": "Hailfinder: A bayesian system for forecasting severe weather",
"venue": "International Journal of Forecasting,",
"year": 1996
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Structure learning concerns itself with the recovery of the graph structure of Bayesian networks (BNs) from data samples. A natural application of Bayesian networks is to describe cause-effect relationships between variables. In that context, one may speak of causal s... | 2,020 | null |
SP:d8c4980cf2187b549f2f2a4fbb2fba4101337459 | [
".** Autoregressive models have demonstrate their potential utility for modeling images and other types of complex data with high flexibility (particularly in density estimation). However, its sampling ability is not that good as explained in the paper. Authors show that one of the main weaknesses of autoregressive... | While autoregressive models excel at image compression, their sample quality is often lacking. Although not realistic, generated images often have high likelihood according to the model, resembling the case of adversarial examples. Inspired by a successful adversarial defense method, we incorporate randomized smoothing... | [
{
"affiliations": [],
"name": "Chenlin Meng"
},
{
"affiliations": [],
"name": "Jiaming Song"
},
{
"affiliations": [],
"name": "Yang Song"
},
{
"affiliations": [],
"name": "Shengjia Zhao"
},
{
"affiliations": [],
"name": "Stefano Ermon"
}
] | [
{
"authors": [
"Guillaume Alain",
"Yoshua Bengio"
],
"title": "What regularized auto-encoders learn from the data-generating distribution",
"venue": "The Journal of Machine Learning Research,",
"year": 2014
},
{
"authors": [
"Jens Behrmann",
"Will Grathwohl",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Autoregressive models have exhibited promising results in a variety of downstream tasks. For instance, they have shown success in compressing images (Minnen et al., 2018), synthesizing speech (Oord et al., 2016a) and modeling complex decision rules in games (Vinyals e... | 2,021 | IMPROVED AUTOREGRESSIVE MODELING WITH DISTRIBUTION SMOOTHING |
SP:5918a2c105a901f8de4bba248dc283a476d9beac | [
"This work considers an important problem of generating adversarial examples to attack a black-box model. The paper proposes a new approach to consider an adversarial example as a result of a sequence of pixel changes from a benign instance. Therefore, the adversarial generation problem can be considered as a bandi... | We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching over the structured space can be approx... | [] | [
{
"authors": [
"Abdullah Al-Dujaili",
"Una-May O’Reilly"
],
"title": "Sign bits are all you need for black-box attacks",
"venue": "In International Conference on Learning Representations,",
"year": 2020
},
{
"authors": [
"Abdullah Al-Dujaili",
"Una-May O’Reilly"
... | [
{
"heading": null,
"text": "We present a new method for score-based adversarial attack, where the attacker queries the loss-oracle of the target model. Our method employs a parameterized search space with a structure that captures the relationship of the gradient of the loss function. We show that searching... | 2,020 | CORRATTACK: BLACK-BOX ADVERSARIAL ATTACK |
SP:9403fa2679f18af78aed2e81b75eb39abeb722eb | [
"The paper develops a density diffusion theory to reveal how minima selection quantitatively depends on the minima sharpness and the hyperparameters. It shows theoretically and empirically that SGD favors flat minima exponentially more than sharp minima. In particular, the paper analyzed the dependence of mean esca... | Stochastic Gradient Descent (SGD) and its variants are mainstream methods for training deep networks in practice. SGD is known to find a flat minimum that often generalizes well. However, it is mathematically unclear how deep learning can select a flat minimum among so many minima. To answer the question quantitatively... | [
{
"affiliations": [],
"name": "DESCENT EXPONEN"
},
{
"affiliations": [],
"name": "TIALLY FAVORS"
},
{
"affiliations": [],
"name": "FLAT MINIMA"
},
{
"affiliations": [],
"name": "Zeke Xie"
},
{
"affiliations": [],
"name": "Issei Sato"
},
{
"affiliations... | [
{
"authors": [
"Alessandro Achille",
"Stefano Soatto"
],
"title": "Where is the information in a deep neural network",
"venue": "arXiv preprint arXiv:1905.12213,",
"year": 2019
},
{
"authors": [
"George B Arfken",
"Hans J Weber"
],
"title": "Mathematical m... | [
{
"heading": "1 INTRODUCTION",
"text": "In recent years, deep learning (LeCun et al., 2015) has achieved great empirical success in various application areas. Due to the over-parametrization and the highly complex loss landscape of deep networks, optimizing deep networks is a difficult task. Stochastic Grad... | 2,021 | A DIFFUSION THEORY FOR DEEP LEARNING DYNAM- |
SP:d92fe94e29672783f906710a2ecb7a02aa4bd67d | [
"The value of the optimal objective as a function of the cost vector $c$ can be written as $z^*(c) = c^T u^*(c)$ where the optimal solution $u^*$ also depends on $c$. The function $u^*(c)$ is piecewise constant -- there are finitely (resp. countably) many feasible solutions; candidates for $u^*$ -- and so the funct... | Combinatorial problems with linear objective function play a central role in many computer science applications, and efficient algorithms for solving them are well known. However, the solutions to these problems are not differentiable with respect to the parameters specifying the problem instance – for example, shortes... | [] | [
{
"authors": [
"Akshay Agrawal",
"Brandon Amos",
"Shane Barratt",
"Stephen Boyd",
"Steven Diamond",
"J Zico Kolter"
],
"title": "Differentiable convex optimization layers",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{... | [
{
"heading": "1 INTRODUCTION",
"text": "Combinatorial optimization problems, such as shortest path in a weighted directed graph, minimum spanning tree in a weighted undirected graph, or optimal assignment of tasks to workers, play a central role in many computer science applications. We have highly refined,... | 2,020 | null |
SP:16d9ab54eb8e4f24314ceca6e0f86f4ca586d7f1 | [
"This paper provides the interesting method that leverages GPU memory resources more efficiently for supernet (meta-graph) of differentiable NAS. For this, this paper proposes binary neural architecture search and consecutive model parallel (CMP). CMP parallelizes one supernet with multiple GPUs, which allows NAS m... | Neural architecture search (NAS) automatically designs effective network architectures. Differentiable NAS with supernets that encompass all potential architectures in a large graph cuts down search overhead to few GPU days or less. However, these algorithms consume massive GPU memory, which will restrain NAS from larg... | [] | [
{
"authors": [
"Han Cai",
"Ligeng Zhu",
"Song Han"
],
"title": "ProxylessNAS: Direct neural architecture search on target task and hardware",
"venue": "In International Conference on Learning Representations,",
"year": 2019
},
{
"authors": [
"Tianqi Chen",
"... | [
{
"heading": "1 INTRODUCTION",
"text": "Neural architecture search (NAS) has revolutionized architecture designs of deep learning from manually to automatically in various applications, such as image classification (Zoph & Le, 2016) and semantic segmentation (Liu et al., 2019a). Reinforcement learning (Zoph... | 2,020 | null |
SP:d10957cc11891e1aad6ecac21a73d589bfac341d | [
"This paper proposes a method called Temporal Abstract Latent Dynamics (TALD). TALD is built up on RSSM (Hafner et al. 2019) but with hierarchical dynamics. The experiments are conducted on moving MNIST, GQN 3D Mazes, and KTH. Results are qualitatively better than other methods in term of maintaining long-term cons... | Deep learning has shown promise for accurately predicting high-dimensional video sequences. Existing video prediction models succeeded in generating sharp but often short video sequences. Toward improving long-term video prediction, we study hierarchical latent variable models with levels that process at different time... | [] | [
{
"authors": [
"Mohammad Babaeizadeh",
"Chelsea Finn",
"Dumitru Erhan",
"Roy H. Campbell",
"Sergey Levine"
],
"title": "Stochastic variational video",
"venue": "prediction. CoRR,",
"year": 2017
},
{
"authors": [
"Lars Buesing",
"Theophane Weber",... | [
{
"heading": null,
"text": "1 INTRODUCTION\nDeep learning has enabled predicting video sequences from large datasets (Chiappa et al., 2017; Oh et al., 2015; Vondrick et al., 2016). For high-dimensional inputs such as video, there likely exists a more compact representation of the scene that facilitates long... | 2,020 | null |
SP:6082a5b51b24315dfdbfe147de1aef2c53cd113d | [
"This paper extends the Wasserstein autoencoder for learning disentangled representations from sequential data. The latent variable model considered contains separate latent variables capturing global and local information respectively, each of which is regularized by a divergence measuring the marginal posterior $... | Learning disentangled representations leads to interpretable models and facilitates data generation with style transfer, which has been extensively studied on static data such as images in an unsupervised learning framework. However, only a few works have explored unsupervised disentangled sequential representation lea... | [
{
"affiliations": [],
"name": "DISENTANGLED RECURRENT"
},
{
"affiliations": [],
"name": "WASSERSTEIN AUTOEN"
},
{
"affiliations": [],
"name": "Jun Han"
},
{
"affiliations": [],
"name": "Martin Renqiang Min"
},
{
"affiliations": [],
"name": "Ligong Han"
},
... | [
{
"authors": [
"Niki Aifanti",
"Christos Papachristou",
"Anastasios Delopoulos"
],
"title": "The mug facial expression database",
"venue": "In 11th International Workshop on Image Analysis for Multimedia Interactive Services WIAMIS",
"year": 2010
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Unsupervised representation learning is an important research topic in machine learning. It embeds high-dimensional sensory data such as images and videos into a low-dimensional latent space in an unsupervised learning framework, aiming at extracting essential data va... | 2,021 | null |
SP:60894f74f40addd7a2a35a003dcdce6cf70ffef4 | [
"The paper extends prior work on equivalence between predictive coding and backprop in layered neural networks to arbitrary computation graphs. This is empirically tested first on a simple nonlinear scalar function, and then on a few commonly used architectures (CNNs, RNNs, LSTMs), confirming the theoretical result... | Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. Recently it has been shown that backprop in multilayer-perceptrons (MLPs) can be approximated using predictive coding, a biologically-plausible process theory of cortical computati... | [
{
"affiliations": [],
"name": "APPROXIMATES BACKPROP"
}
] | [
{
"authors": [
"Mohamed Akrout",
"Collin Wilson",
"Peter Humphreys",
"Timothy Lillicrap",
"Douglas B Tweed"
],
"title": "Deep learning without weight transport",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep learning has seen stunning successes in the last decade in computer vision (Krizhevsky et al., 2012; Szegedy et al., 2015), natural language processing and translation (Vaswani et al., 2017; Radford et al., 2019; Kaplan et al., 2020), and computer game playing (M... | 2,020 | null |
SP:9e6b5b7d9e7459c015130f4b80f7bc75424de050 | [
"This paper proposes a simple scheme for training with multiple augmentations of training data in one iteration and reweighting the instances by their relative loss. As authors note in their related works, the idea of reweighting examples based on their relative loss has been widely studied in a variety of machine ... | Data augmentation is an effective technique to improve the generalization of deep neural networks. However, previous data augmentation methods usually treat the augmented samples equally without considering their individual impacts on the model. To address this, for the augmented samples from the same training example,... | [
{
"affiliations": [],
"name": "ING THE"
},
{
"affiliations": [],
"name": "MAXIMAL EXPECTED LOSS"
},
{
"affiliations": [],
"name": "Mingyang Yi"
},
{
"affiliations": [],
"name": "Lu Hou"
},
{
"affiliations": [],
"name": "Lifeng Shang"
},
{
"affiliations... | [
{
"authors": [
"S. Behpour",
"K. Kitani",
"B. Ziebart"
],
"title": "Ada: Adversarial data augmentation for object detection",
"venue": "In IEEE Winter Conference on Applications of Computer Vision,",
"year": 2019
},
{
"authors": [
"Y. Bengio",
"J. Louradour"... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep neural networks have achieved state-of-the-art results in various tasks in natural language processing (NLP) tasks (Sutskever et al., 2014; Vaswani et al., 2017; Devlin et al., 2019) and computer vision (CV) tasks (He et al., 2016; Goodfellow et al., 2016). One a... | 2,021 | REWEIGHTING AUGMENTED SAMPLES BY MINIMIZ- |
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