q > r, and Dm denoting the matrix of the vectors in Bm. As we show in more detail in the appendix, our constraint from equation 4 requires eigenvectors of MTmMm to be w i m, with Vm containing r orthogonal vectors (v1m,v 2 m, ...,v r m) from Dm and (sinm − r) vectors from the null space of M .\nWe are especially interested in how Mm changes w.r.t. input in terms of Dm, i.e., we express LRR in terms of Dm. By construction, each input dim can be represented as a linear combination via a vector of coefficients cim that multiplies Dm so that d i m =Dmc i m. Since Mmdm = UmΣmV T mdm, the loss LRR of layer m can be rewritten as\nLRRm = (MTmMmdm − dm)2 = (VmΣTmΣmV Tmdm − dm)2\n= (VmΣ T mΣmV T mDmcm −Dmcm)2,\n(6)\nwhere we can assume that the coefficient vector cm is accumulated over the training data set size t via cm = ∑t i=1 c i m, since eventually every single datum inDm will contribute to LRRm . The central component of equation 6 is V TmDm. For a successful minimization, Vm needs to retain those w i m with the largest cm coefficients. As Vm is typically severely limited in terms of its representational capabilities by the number of adjustable weights in a network, it needs to focus on the most important eigenvectors in terms of cm in order to establish a small distance to Dmcm. Thus, features that appear multiple times in the input data with a corresponding factor in cm will more strongly contribute to minimizing LRRm . To summarize, Vm is driven towards containing r orthogonal vectors wim that represent the most frequent features of the input data, i.e., the dominant features. Additionally, due to the column vectors of Vm being mutually orthogonal, Mm is encouraged to extract different features from the input. By the sake of being distinct and representative for the data set, these features have the potential to be useful for new inference tasks. The feature vectors embedded inMm can be extracted from the network weights in practical settings, as we will demonstrate below.\nRealization in Neural Networks Calculating MTmMm is usually very expensive due to the dimensionality of Mm. Instead of building it explicitly, we constrain intermediate results to realize equation 3 when training. A regular training typically starts with a chosen network structure and\nUnder review as a conference paper at ICLR 2021\ntrains the model weights for a given task via a suitable loss function. Our approach fully retains this setup and adds a second pass that reverses the initial structure while reusing all weights and biases. E.g., for a typical fully connected layer in the forward pass with dm+1 = Mmdm +bm, the reverse pass operation is given by d ′\nm = M T m(dm+1 − bm), where d\n′ m denotes the reconstructed input.\na m\no un\nt o f s\nh a\nre d\nin fo\nrm a\ntio n\nb et\nw e\ne n\nla ye\nr a\nn d\namount of shared information between layer ℒ and 𝑋\n𝐿𝑎𝑦𝑒𝑟 1 2 3 𝑚 𝑛 𝐿𝑎𝑦𝑒𝑟 𝑚 𝒅𝒎 𝒅𝒎 𝟏\n𝒅𝒎 𝟏𝒅𝒎\n𝒅𝟏\n𝒅𝟏\n|\n𝑓\n𝐵𝑁 𝑐𝑜𝑛𝑣 𝑓 (𝑀 , 𝒃𝒎)\n𝑑𝑒𝑐𝑜𝑛𝑣\n(𝑀 , −𝒃𝒎) 𝐵𝑁\n𝑤𝑒𝑖𝑔ℎ𝑡 𝑠ℎ𝑎𝑟𝑖𝑛𝑔\n𝒗 𝒗 𝒗 0.427 0.400 0.489\n0.420 0.394 0.445 0.213 0.216 0.153 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n𝒗\n𝒗\n𝒅\n…\n…\n(50 images)\n(50 images)\n(1,0)\n(0,1)\n𝒅 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n𝑎 𝑀𝑁𝐼𝑆𝑇 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\n𝑏 𝑃𝑒𝑎𝑘 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝐵𝑁 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈 𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛\n(0,0,0,0,0,0,0,1,0,0) (0,0,1,0,0,0,0,0,0,0)\n0.499\n0.502\n0.499\n0.491\n0.233\n0.202\n𝒅 𝒅\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\nmore information about both 𝑋&𝑌 more information about 𝑌 only\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\nless shared information between layer ℒ and 𝑋\nle s s\nsh ar\ned in\nfo rm\nat io n b et w ee n la ye r ℒ a n d\n𝑌\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\n0.9\n0.92\n0.94\n0.96\n0.98\n1\n1.02\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝒟 𝒟 𝒟 𝒟 𝒟 𝒟\n𝑆𝑡𝑑\n𝐼( 𝒟\n;𝑌 )\nmore information about 𝑋 only\n(a) Mutual Information Plane, How to Read (b) Mutual Information for Task A (c) After fine-tuning for A (d) After fine-tuning for B\nno information\n…\n…\n…\n…\n𝒅𝟐\n𝒅𝟐\n|\n𝒅𝟑\n𝒅𝟑\n|\n𝒅𝒏\n𝒅𝒏\n| 𝒐𝒖𝒕𝒑𝒖𝒕\nLayers of 𝑅𝑅 models exhibit strong MI with in- & output\n𝒗 𝒗 0.43 0.49\n0.42 0.45 0.21 0.22 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n𝒗\n𝒗\n𝒅\n(1,0)\n(0,1)\n𝒅\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\n𝑏 𝑊𝑖𝑡ℎ 𝐵𝑁 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈\n7 2\n0.50\n0.50\n0.50\n0.49\n0.23\n0.20\n𝒅 𝒅\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\n……(50 imgs)\n……(50 imgs)\nOur goal with the reverse pass is to transpose all operations of the forward pass to obtain identical intermediate activations between the layers with matching dimensionality. We can then constrain the intermediate results of each layer of the forward pass to match the results of the backward pass, as illustrated in figure 2. Unlike greedy layer-wise autoencoder pretraining, which trains each layer separately and only constrains d1 and d ′\n1, we jointly train all layers and constrain all intermediate results. Due to the symmetric structure of the two passes, we can use a simple L2 difference to drive the network towards aligning the results:\nLRR = n∑\nm=1\nλm\n∥∥∥dm − d′m∥∥∥2\nF . (7)\nHere dm denotes the input of layer m in the forward pass and d ′\nm the output of layer m for the reverse pass. λm denotes a scaling factor for the loss of layer m, which, however, is typically constant in our tests across all layers. Note that with our notation, d1 and d ′\n1 refer to the input data, and the reconstructed input, respectively.\nNext, we show how this setup realizes the regularization from equation 3. For clarity, we use a fully connected layer with bias. In a neural network with n hidden layers, the forward process for a layer m is given by dm+1 = Mmdm +bm,, with d1 and dn+1 denoting in- and output, respectively. For our pretraining, we build a reverse pass network with transposed operations starting with the final output where dn+1 = d ′ n+1, and the intermediate results d ′ m+1:\nd ′\nm = M T m(d\n′ m+1 − bm), (8) which yields ∥∥∥dm − d′m∥∥∥2 F = ∥∥MTmMmdm − dm∥∥2F . When this difference is minimized via equation 7, we obtain activated intermediate content during the reverse pass that reconstructs the values computed in the forward pass, i.e. d ′\nm+1 = dm+1 holds. As in equation 10 the reverse pass activation d ′\nm depends on dm+1 ′, this formulation yields a full reverse pass from output to input, which\nwe use for most training runs below. In this case\nd ′\nm = M T m(d\n′ m+1 − bm) = MTm(dm+1 − bm) = MTmMmdm , (9)\nwhich is consistent with equation 3, and satisfies the original constraint MTmMmdm − dm = 0. This version is preferable if a unique path from output to input exists. For architectures where the path is not unique, e.g., in the presence of additive residual connections, we use a local formulation\nd ′\nm = M T m(dm+1 − bm), (10)\nwhich employs dm+1 for jointly constraining all intermediate activations in the reverse pass.\nUp to now, the discussion focused on simplified neural networks without activation functions or extensions such as batch normalization (BN). While we leave incorporating such extensions for future work, our experiments consistently show that the inherent properties of our pretraining remain valid: even with activations and BN, our approach successfully extracts dominant structures and yields improved generalization. In the appendix, we give details on how to ensure that the latent space content for forward and reverse pass is aligned such that differences can be minimized.\nTo summarize, we realize the loss formulation of equation 7 to minimize ∑n\nm=1((M T mMm−I)dm)2\nwithout explicitly having to construct MTmMm. Following the notation above, we will refer to networks trained with the added reverse structure and the additional loss terms as RR variants. We consider two variants for the reverse pass: a local pretraining equation 10 using the datum dm+1 of a given layer, and a full version via equation 8 which uses d ′\nm+1 incoming from the next layer during the reverse pass.\nEmbedding Singular Values Below, Std denotes a regular training run (in orange color in graphs below), while RR denotes our models (in green). Pre and Ort will denote regular autoencoder pretraining and orthogonality, respectively, while a subscript will denote the task variant the model was trained for, e.g., StdT for task T. While we typically use all layers of a network in the constraints, a reduced variant that we compare to below only applies the constraint for the input data, i.e., m=1. A network trained with this variant, denoted by RR1A, is effectively trained to only reconstruct the input. It contains no constraints for the inner activations and layers of the network. For the Ort models, we use the Spectral Restricted Isometry Property algorithm (Bansal et al., 2018).\nWe verify that the column vectors of Vm of models from RR training contain the dominant features of the input with the help of a classification test, employing a single fully connected layer, i.e. d2 = M1d1, with batch normalization and activation. To quantify this similarity, we compute an LPIPS distance (Zhang et al., 2018b) between vim and the training data (lower values being better).\n𝒗\n𝒗 0.43\n0.49\n0.42\n0.45 0.21\n0.22\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\n7\n2\n𝒅 𝒅\n𝐋𝐏𝐈𝐏𝐒𝐏𝐫𝐞\n0.14\n0.31\n𝒗\n𝒗\n1,0 𝑎𝑛𝑑 (0,1)\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.50\n0.50\n0.50\n0.49\n0.23\n0.20\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\n…… ……\n𝑃𝑟𝑒 𝐋𝐏𝐈𝐏𝐒𝐏𝐫𝐞\n0.24\n0.40\n+\nWe employ a training data set constructed from two dominant classes (a peak in the top left, and bottom right quadrant, respectively), augmented with noise in the form of random scribbles. Based on the analysis above, we expect the RR training to extract the two dominant peaks during training. The LPIPS measurements confirm our SVD argumentation above, with average scores of 0.217±0.022 for RR, 0.319±0.114 for Pre, 0.495± 0.006 for Ort, and 0.500± 0.002 for Std. I.e., the RR model fares significantly better than the others. At the same time, the peaks are clearly visible for RR models, an example is shown in figure 3(b), while the other models fail to extract structures that resemble the input. Thus, by training with the full network and the original training objective, our pretraining yields structures that are interpretable and be inspected by humans.\nThe results above experimentally confirm our formulation of the RR loss and its ability to extract dominant and generalizing structures from the training data. Next, we will focus on quantified metrics and turn to measurements in terms of mutual information to illustrate the behavior of our pretraining for deeper networks."},{"heading":"3 EVALUATION IN TERMS OF MUTUAL INFORMATION","text":"As our approach hinges on the introduction of the reverse pass, we will show that it succeeds in terms of establishing mutual information (MI) between the input and the constrained intermediates inside a network. More formally, MI I(X;Y ) of random variablesX and Y measures how different the joint distribution ofX and Y is w.r.t. the product of their marginal distributions, i.e., the Kullback-Leibler divergence I(X;Y ) = DKL[P(X,Y )||PXPY ]. (Tishby & Zaslavsky, 2015) proposed MI plane to analyze trained models, which show the MI between the input X and activations of a layer Dm, i.e., I(X;Dm) and I(Dm;Y ), i.e., MI of layer Dm with output Y . These two quantities indicate how much information about the in- and output distributions are retained at each layer, and we use them to show to which extent our pretraining succeeds at incorporating information about the inputs throughout training.\nThe following tests employ networks with six fully connected layers with the objective to learn the mapping from 12 binary inputs to 2 binary output digits (Shwartz-Ziv & Tishby, 2017), with results accumulated over five runs. We compare the versions StdA, PreA, OrtA, RRA, and a variant of the latter: RR1A, i.e. a version where only the input d1 is constrained to be reconstructed. While figure 4a) visually summarizes the content of the MI planes, the graph in (b) highlights that training with the RR loss correlates input and output distributions across all layers: the cluster of green points in the center of the graph shows that all layers contain balanced MI between in- as well as output and the activations of each layer. RR1A fares slightly worse, while StdA and OrtA almost exclusively focus on the output with I(Dm;Y ) being close to one. PreA instead only focuses on reconstructing inputs. Thus, the early layers cluster in the right-top corner, while the last layer I(D7;Y ) fails to\nalign with the outputs. Once we continue fine-tuning these models without regularization, the MI naturally shifts towards the output, as shown in figure 4 (c). Here, RRAA outperforms the other models in terms of final performance. Likewise, RRAB performs best for a transfer task B with switched output digits, as shown in graph (d). The final performance for both tasks across all runs is summarized in figure 5. These graphs visualize that the proposed pretraining succeeds in robustly establishing mutual information between inputs and targets across a full network, in addition to extracting reusable features.\nMI has received attention recently as a learning objective, e.g., in the form of the InfoGAN approach (Chen et al., 2016) for learning disentangled and interpretable latent representations. While MI is typically challenging to assess and estimate (WaltersWilliams & Li, 2009), the results above show that our approach provides a straightforward and robust way for including it as a learning objective. In this way, we can, e.g., reproduce the disentangling results from (Chen et al., 2016), which are shown in figure 1(c). A generative model with our pretraining extracts intuitive latent dimensions for the different digits, line thickness, and orientation without any additional modifications of the loss function. The joint training of the full network with the proposed reverse structure, including non-linearities and normalization, yields a natural and intuitive decomposition."},{"heading":"4 EXPERIMENTAL RESULTS","text":"We now turn to a broad range of network structures, i.e., CNNs, Autoencoders, and GANs, with a variety of data sets and tasks to show our approach succeeds in improving inference accuracy and generality for modern day applications and architectures.\nTransfer-learning Benchmarks We first evaluate our approach with two state-of-the-art benchmarks for transfer learning. The first one uses the texture-shape data set from (Geirhos et al., 2018), which contains challenging images of various shapes combined with patterns and textures to be classified. The results below are given for 10 runs each. For the stylized data shown in figure 6 (a), the accuracy of PreTS is low with 20.8%. This result is in line with observations in previous work and confirms the detrimental effect of classical pretraining. StdTS yields a performance of 44.2%, and OrtTS improves the performance to 47.0%, while RRTS yields a performance of 54.7% (see figure 6b). Thus, the accuracy of RRTS is 162.98% higher than PreTS, 23.76% higher than StdTS, and 16.38% higher than OrtTS. To assess generality, we also apply the models to new data without\n0.208\n0.085\n0.159\n0.442\n0.158\n0.331\n0.470\n0.203\n0.372\n0.547\n0.237\n0.408\nS T Y L I Z E D D A T A E D G E D A T A F I L L E D D A T A\nEdge data setStylized data set Filled data set\nAccuracy Comparisons of 𝑂𝑟𝑡 , 𝑆𝑡𝑑 and 𝑅𝑅\n𝑆𝑡𝑑 𝑅𝑅\n𝑂𝑟𝑡\nEdge data\nFilled data\nStylized data\n(𝑎) (𝑏) (𝑐) (𝑑)\n𝑃𝑟𝑒\nFigure 6: (a) Examples from texture-shape data set. (b, c, d) Texture-shape test accuracy comparisons of PreTS, OrtTS, StdTS and RRTS for different data sets.\n𝑂𝑢𝑡𝑝𝑢𝑡\n𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑆𝑡𝑑 𝑅𝑅\n𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑒𝑑 𝐼𝑛𝑝𝑢𝑡 (𝒅 )\n𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑆𝑡𝑑 𝑅𝑅\nre-training, i.e. an edge and a filled data set, also shown in figure 6 (a). For the edge data set, RRTS outperforms PreTS, StdTS and OrtTS by 178.82%, 50% and 16.75%, respectively,\nIt is worth pointing out that the additional constraints of our training approach lead to moderately increased requirements for memory and computations, e.g., 41.86% more time per epoch than regular training for the texture-shape test. On the other hand, it allows us to train smaller models: we can reduce the weight count by 32% for the texture-shape case while still being on-par with OrtTS in terms of classification performance. By comparison, regular layer-wise pretraining requires a significant overhead and fundamental changes to the training process. Our pretraining fully integrates with existing training methodologies and can easily be deactivated via λm = 0.\nAs a second test case, we use a CIFAR-based task transfer (Recht et al., 2019) that measures how well models trained on the original CIFAR 10, generalize to a new data set (CIFAR 10.1) collected according to the same principles as the original one. Here we use a Resnet110 with 110 layers and 1.7 million parameters, Due to the consistently low performance of the Pre models (Alberti et al., 2017), we focus on Std, Ort and RR for this test case. In terms of accuracy across 5 runs, OrtC10 outperforms StdC10 by 0.39%, while RRC10 outperforms OrtC10 by another 0.28% in terms of absolute test accuracy (figure 7). This increase for RR training matches the gains reported for orthogonality in previous work (Bansal et al., 2018), thus showing that our approach yields substantial practical improvements over the latter. It is especially interesting how well performance for CIFAR 10 translates into transfer performance for CIFAR 10.1. Here, RRC10 still outperforms OrtC10 and StdC10 by 0.22% and 0.95%, respectively. Hence, the models from our pretraining very successfully translate gains in performance from the original task to the new one, indicating that the models have successfully learned a set of more general features. To summarize, both benchmark cases confirm that the proposed pretraining benefits generalization.\nGenerative Adversarial Models In this section, we employ our pretraining in the context of generative models for transferring from synthetic to real-world data from the ScalarFlow data set (Eckert et al., 2019). As super-resolution taskA, we first use a fully-convolutional generator network, adversarially trained with a discriminator network on the synthetic flow data. While regular pretraining is\nmore amenable to generative tasks than orthogonal regularization, it can not be directly combined with adversarial training. Hence, we pretrain a model Pre for a reconstruction task at high-resolution without discriminator instead. Figure 8 a) demonstrates that our method works well in conjunction with the GAN training: As shown in the bottom row, the trained generator succeeds in recovering the input via the reverse pass without modifications. A regular model StdA, only yields a black image in this case. For PreA, the layer-wise nature of the pretraining severely limits its capabilities to learn the correct data distribution (Zhou et al., 2014), leading to a low performance.\nWe now mirror the generator model from the previous task to evaluate an autoencoder structure that we apply to two different data sets: the synthetic smoke data used for the GAN training (task B1), and a real-world RGB data set of smoke clouds (task B2). Thus both variants represent transfer tasks, the second one being more difficult due to the changed data distribution. The resulting losses, summarized in figure 8 b), show that RR training performs best for both autoencoder tasks: the L2 loss of RRAB1 is 68.88% lower than StdAB1 , while it is 13.3% lower for task B2. The proposed pretraining also clearly outperforms the Pre variants. Within this series of tests, the RR performance for task B2 is especially encouraging, as this task represents a synthetic to real transfer.\nWeather Forecasting Pretraining is particularly attractive in situations where the amount of data for training is severely limited. Weather forecasting is such a case, as systematic and accurate data for many relevant quantities are only available for approximately 50 years. We target three-day forecasts of pressure, ground temperature, and mid-level atmospheric temperature based on a public benchmark dataset (Rasp et al., 2020). This dataset contains worldwide observations from ERA5 (Hersbach et al., 2020) in six-hour intervals with a 5.625◦ resolution. For the joint inference of atmospheric pressure (500 hPa geopotential, Z500), ground temperature (T2M), and atmospheric temperature (at 850 hPa, T850), we use a convolutional ResNet architecture with 19 residual blocks. As regular pretraining is not compatible with residual connections, we omit it here.\nWe train a model regular model (about 6.36M trainable parameters) with data from 1979 to 2015, and compare its inference accuracy across all datapoints from years 2017 and 2018 to a similar model that employs our pretraining. While the regular model was trained for 25 epochs, the RR model was pretrained for 10 epochs and fine-tuned for another 15 epochs. Across all three physical quantities, the RR model clearly outperforms the regular model, as summarized in figure 1 (d) and figure 9 (details are given in the appendix). Especially for the latitude-weighted RMSE of Z500, it yields improvements of 5.5%. These improvements point to an improved generalization of the RR model via the pretraining and highlight its importance for domains where data is scarce."},{"heading":"5 CONCLUSIONS","text":"We have proposed a novel pretraining approach inspired by classic methods for unsupervised autoencoder pretraining and orthogonality constraints. In contrast to the classical methods, we employ a constrained reverse pass for the full non-linear network structure and include the original learning objective. We have shown for a wide range of scenarios, from mutual information, over transfer learning benchmarks to weather forecasting, that the proposed pretraining yields networks with better generalizing capabilities. Our training approach is general, easy to integrate, and imposes no requirements regarding network structure or training methods. Most importantly, our results show that unsupervised pretraining has not lost its relevance in today’s deep learning environment.\nAs future work, we believe it will be exciting to evaluate our approach in additional contexts, e.g., for temporal predictions (Hochreiter & Schmidhuber, 1997; Cho et al., 2014), and for training explainable and interpretable models (Zeiler & Fergus, 2014; Chen et al., 2016; Du et al., 2018)."},{"heading":"A APPENDIX","text":"To ensure reproducibility, source code and data for all tests will be published. Runtimes were measured on a machine with Nvidia GeForce GTX 1080 Ti GPUs and an Intel Core i7-6850K CPU."},{"heading":"A.1 DISCUSSION OF RELATED WORK","text":"Greedy layer-wise pretraining was first proposed by (Bengio et al., 2007), and influenced a large number of follow up works, providing a crucial method for enabling stable training runs of deeper networks. A detailed evaluation was performed by (Erhan et al., 2010), also highlighting cases were it can be detrimental. These problems were later on also detailed in other works, e.g., by (Alberti et al., 2017). The transferability of learned features was likewise a topic of interest for transfer learning applications (Yosinski et al., 2014). Sharing similarities with our approach, (Rasmus et al., 2015) combined supervised and unsupervised learning objectives, but focused on denoising autoencoders and a layer-wise approach without weight sharing. We demonstrate the importance of leveraging state-of-the-art methods for training deep networks, i.e. without decomposing or modifying the network structure. This not only improves performance, but also very significantly simplifies the adoption of the pretraining pass in new application settings.\nExtending the classic viewpoint of unsupervised autoencoder pretraining, several prior methods employed ”hard orthogonal constraints” to improve weight orthogonality via singular value decomposition (SVD) at training time (Huang et al., 2018; Jia et al., 2017; Ozay & Okatani, 2016). Bansal et al. (Bansal et al., 2018) additionally investigated efficient formulations of the orthogonality constraints. In practice, these constraints are difficult to satisfy, and correspondingly only weakly imposed. In addition, these methods focus on improving performance for a known, given task. This means the training process only extracts features that the network considers useful for improving the performance of the current task, not necessarily improving generalization or transfer performance (Torrey & Shavlik, 2010). While our approach shares similarities with SVD-based constraints, it can be realized with a very efficient L2-based formulation, and takes the full input distribution into account.\nRecovering all input information from hidden representations of a network is generally very difficult (Dinh et al., 2016; Mahendran & Vedaldi, 2016), due to the loss of information throughout the layer transformations. In this context, (Tishby & Zaslavsky, 2015) proposed the information bottleneck principle, which states that for an optimal representation, information unrelated to the current task is omitted. This highlights the common specialization of conventional training approaches.\nReversed network architectures were proposed in previous work (Ardizzone et al., 2018; Jacobsen et al., 2018; Gomez et al., 2017), but mainly focus on how to make a network fully invertible via augmenting the network with special structures. As a consequence, the path from input to output is different from the reverse path that translates output to input. Besides, the augmented structures of these approaches can be challenging to apply to general network architectures. In contrast, our approach fully preserves an existing architecture for the backward path, and does not require any operations that were not part of the source network. As such, it can easily be applied in new settings, e.g., adversarial training (Goodfellow et al., 2014). While methods using reverse connections were previously proposed (Zhang et al., 2018a; Teng & Choromanska, 2019), these modules primarily focus on transferring information between layers for a given task, and on auto-encoder structures for domain adaptation, respectively."},{"heading":"A.2 PRETRAINING AND SINGULAR VALUE DECOMPOSITION","text":"In this section we give a more detailed derivation of our loss formulation, extending Section 3 of the main paper. As explained there, our loss formulation aims for minimizing\nLRR = n∑\nm=1\n(MTmMmd i m − dim)2, (11)\nwhere Mm ∈ Rs out m×s in m denotes the weight matrix of layer m, and data from the input data set Dm is denoted by dim ⊂ Rs in m , i = 1, 2, ..., t. Here t denotes the number of samples in the input data set.\nMinimizing equation 11 is mathematically equivalent to\nMTmMmd i m − dim = 0 (12)\nfor all dim. Hence, perfectly fulfilling equation 11 would require all d i m to be eigenvectors of MTmMm with corresponding eigenvalues being 1. As in Sec. 3 of the main paper, we make use of an auxiliary orthonormal basis Bm : 〈 w1m, ...,w q m 〉 , for which q (with q ≤ t) denotes the number of linearly independent entries inDm. While Bm never has to be explicitly constructed for our method, it can, e.g., be obtained via Gram-Schmidt. The matrix consisting of the vectors in Bm is denoted by Dm.\nSince the whm(h = 1, 2, ...q) necessarily can be expressed as linear combinations of d i m, equation 11 similarly requires whm to be eigenvectors of M T mMm with corresponding eigenvalues being 1, i.e.:\nMTmMmw h m −whm = 0 (13)\nWe denote the vector of coefficients to express dim via Dm with c i m, i.e. d i m = Dmc i m. Then equation 12 can be rewritten as:\nMTmMmDmc i m −Dmcim = 0 (14)\nVia an SVD of the matrix Mm in equation 14 we obtain\nMTmMmDmcm −Dmcm\n= q∑ h=1 MTmMmw h mcmh −whmcmh\n= q∑ h=1 VmΣ T mΣmV T mw h mcmh −whmcmh\n(15)\nwhere the coefficient vector cm is accumulated over the training data set size t via cm = ∑t i=1 c i m. Here we assume that over the course of a typical training run eventually every single datum in Dm will contribute to LRRm . This form of the loss highlights that minimizing LRR requires an alignment of VmΣTmΣmV T mw h mcmh and w h mcmh .\nBy construction, Σm contains the square roots of the eigenvalues of MTmMm as its diagonal entries. The matrix has rank r = rank(MTmMm), and since all eigenvalues are required to be 1 by equation 13, the multiplication with Σm in equation 15 effectively performs a selection of r column vectors from Vm. Hence, we can focus on the interaction between the basis vectors wm and the r active column vectors of Vm:\nVmΣ T mΣmV T mw h mcmh −whmcmh\n= cmh(VmΣ T mΣmV T mw h m −whm)\n= cmh( r∑ f=1 (vfm) Twhmv f m −whm).\n(16)\nAs Vm is obtained via an SVD it contains r orthogonal eigenvectors of MTmMm. equation 13 requires w1m, ...,w q m to be eigenvectors of M T mMm, but since typically the dimension of the input data set is much larger than the dimension of the weight matrix, i.e. r ≤ q, in practice only r vectors from Bm can fulfill equation 13. This means the vectors v1m, ...,vrm in Vm are a subset of the orthonormal basis vectors Bm : 〈 w1m, ...,w q m 〉 with (whm)\n2 = 1. Then for any whm we have{ (vfm) Twhm = 1, if v f m = w h m\n(vfm) Twhm = 0, otherwise.\n(17)\nThus if Vm contains whm, we have r∑\nf=1\n(vfm) Twhmv f m = w h m, (18)\nand we trivially fulfill the constraint\ncmh( r∑ f=1 (vfm) Twhmv f m −whm) = 0. (19)\nHowever, due to r being smaller than q in practice, Vm typically can not include all vectors from Bm. Thus, if Vm does not contain whm, we have (vfm)Twhm = 0 for every vector vfm in Vm, which means\nr∑ f=1 (vfm) Twhmv f m = 0. (20)\nAs a consequence, the constraint equation 12 is only partially fulfilled:\ncmh( r∑ f=1 (vfm) Twhmv f m −whm) = −cmhwhm . (21)\nAs the whm have unit length, the factors cm determine the contribution of a datum to the overall loss. A feature whm that appears multiple times in the input data will have a correspondingly larger factor in cm and hence will more strongly contribute to LRR. The L2 formulation of equation 11 leads to the largest contributors being minimized most strongly, and hence the repeating features of the data, i.e., dominant features, need to be represented in Vm to minimize the loss. Interestingly, this argumentation holds when additional loss terms are present, e.g., a loss term for classification. In such a case, the factors cm will be skewed towards those components that fulfill the additional loss terms, i.e. favor basis vectors whm that contain information for about the loss terms. This, e.g., leads to clear digit structures being embedded in the weight matrices for the MNIST example below.\nIn summary, to minimize LRR, Vm is driven towards containing r orthogonal vectors whm which represent the most frequent features of the input data, i.e. the dominant features. It is worth emphasizing that above Bm is only an auxiliary basis, i.e., the derivation does not depend on any particular choice of Bm."},{"heading":"A.3 EXAMPLES OF NETWORK ARCHITECTURES WITH PRETRAINING","text":"While the proposed pretraining is significantly more easy to integrate into training pipelines than classic autoencoder pretraining, there are subtleties w.r.t. the order of the operations in the reverse pass that we clarify with examples in the following sections. To specify NN architectures, we use the following notation: C(k, l, q), and D(k, l, q) denote convolutional and deconvolutional operations, respectively, while fully connected layers are denoted with F (l), where k, l, q denote kernel size, output channels and stride size, respectively. The bias of a CNN layer is denoted with b. I/O(z) denote input/output, their dimensionality is given by z. Ir denotes the input of the reverse pass network. tanh, relu, lrelu denote hyperbolic tangent, ReLU, and leaky ReLU activation functions (AF), where we typically use a leaky tangent of 0.2 for the negative half-space. UP , MP and BN denote 2× nearest-neighbor up-sampling, max pooling with 2 × 2 filters and stride 2, and batch normalization, respectively.\nBelow we provide additional examples how to realize the pretraining loss Lrr in a neural network architecture. As explained in the main document, the constraint equation 11 is formulated via\nLrr = n∑\nm=1\nλm ∥∥∥dm − d′m∥∥∥2 F , (22)\nwith dm, and λm denoting the vector of activated intermediate data in layer m from the forward pass, and a scaling factor, respectively. d ′\nm denotes the activations of layer m from the reverse pass. E.g., let Lm() denote the operations of a layer m in the foward pass, and L′m() the corresponding operations for the reverse pass. Then dm+1 = Lm(dm), and d ′ m = L ′ m(d ′ m+1).\nWhen equation 22 is minimized, we obtain activated intermediate content during the reverse pass that reconstructs the values computed in the forward pass, i.e. d ′ m+1 = dm+1 holds. Then d ′\nm can be reconstructed from the incoming activations from the reverse pass, i.e., d ′\nm+1, or from the output of layer m, i.e., dm+1. Using d ′ m+1 results in a global coupling of input and output throughout\na m\no un\nt o f s\nh a\nre d\nin fo\nrm a\ntio n\nb et\nw e\ne n\nla ye\nr a\nn d\namount of shared information between layer ℒ and 𝑋\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\nmore information about both 𝑋&𝑌 more information about 𝑌 only\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\nless shared information between layer ℒ and 𝑋\nle s s\nsh ar\ned in\nfo rm\nat io n b et w ee n la ye r ℒ a n d\n𝑌\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\n0.9\n0.92\n0.94\n0.96\n0.98\n1\n1.02\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝐼( 𝒟\n;𝑌 )\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1\n1.1\n0 2 4 6 8 10 12\n𝑂𝑟𝑡\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\n𝐼(𝑋; 𝒟 )\n𝒟 𝒟 𝒟 𝒟 𝒟 𝒟\n𝑆𝑡𝑑\n𝐼( 𝒟\n;𝑌 )\nmore information about 𝑋 only\n(a) Mutual Information Plane, How to Read (b) Mutual Information for Task A (c) After fine-tuning for A (d) After fine-tuning for B\nno information\nLayers of 𝑅𝑅 models exhibit strong MI with in- & output\nall layers, i.e., the full loss variant. On the other hand, dm+1 yields a variant that ensures local reversibility of each layer, and yields a very similar performance, as we will demonstrate below. We employ this local loss for networks without a unique, i.e., bijective, connection between two layers. Intuitively, whe inputs cannot be reliably reconstructed from outputs.\nFull Network Pretraining: An illustration of a CNN structure with AF and BN and a full loss is shown in figure 10. While the construction of the reverse pass is straight-forward for all standard operations, i.e., fully connected layers, convolutions, pooling, etc., slight adjustments are necessary for AF and BN. It is crucial for our formulation that dm and d ′\nm contain the same latent space content in terms of range and dimensionality, such that they can be compared in the loss. Hence, we use the BN parameters and the AF of layer m − from the forward pass for layer m in the reverse pass. An example is shown in figure 14.\nTo illustrate this setup, we consider an example network employing convolutions with mixed AFs, BN, and MP. Let the network receives a field of 322 scalar values as input. From this input, 20, 40, and 60 feature maps are extracted in the first three layers. Besides, the kernel sizes are decreased from 5× 5 to 3× 3. To clarify the structure, we use ReLU activation for the first convolution, while the second one uses a hyperbolic tangent, and the third one a sigmoid function. With the notation outlined above, the first three layers of the network are\nI(32, 32, 1) = d1 → C1(5, 20, 1) + b1 → BN1 → relu → d2 →MP → C2(4, 40, 1) + b2 → BN2 → tanh → d3 →MP → C3(3, 60, 1) + b3 → BN3 → sigm → d4 → ...\n(23)\nThe reverse pass for evaluating the loss re-uses all weights of the forward pass and ensures that all intermediate vectors of activations, dm and d ′\nm, have the same size and content in terms of normalization and non-linearity. We always consider states after activation for Lrr. Thus, dm denotes activations before pooling in the forward pass and d ′\nm contains data after up-sampling in the reverse pass, in order to ensure matching dimensionality. Thus, the last three layers of the reverse network for computing Lrr take the form:\n...→ d ′\n4 → −b3 → D3(3, 40, 1)→ BN2 → tanh→ UP\n→ d ′\n3 → −b2 → D2(4, 20, 1)→ BN1 → relu→ UP\n→ d ′\n2 → −b1 → D1(5, 3, 1)\n→ d ′\n1 = O(32, 32, 1).\n(24)\nHere, the de-convolutions Dx in the reverse network share weights with Cx in the forward network. I.e., the 4× 4× 20× 40 weight matrix of C2 is reused in its transposed form as a 4× 4× 40× 20 matrix in D2. Additionally, it becomes apparent that AF and BN of layer 3 from the forward pass do not appear in the listing of the three last layers of the reverse pass. This is caused by the fact that both are required to establish the latent space of the fourth layer. Instead, d3 in our example represents the activations after the second layer (with BN2 and tanh), and hence the reverse pass for d ′ 3 reuses both functions. This ensures that dm and d ′\nm contain the same latent space content in terms of range and dimensionality, and can be compared in equation 22.\nFor the reverse pass, we additionally found it beneficial to employ an AF for the very last layer if the output space has suitable content. E.g., for inputs in the form of RGB data we employ an additional activation with a ReLU function for the output to ensure the network generates only positive values.\nLocalized Pretraining: In the example above, we use a full pretraining with d ′\nm+1 to reconstruct the activations d ′\nm. The full structure establishes a slightly stronger relationship among the loss terms of different layers, and allows earlier layers to decrease the accumulated loss of later layers. However, if the architecture of the original network makes use of operations between layers that are not bijective, we instead use the local loss. E.g., this happens for residual connections with an addition or non-invertible pooling operations such as max-pooling. In the former, we cannot uniquely determine the b, c in a = b + c given only a. And unless special care is taken (Bruna et al., 2013), the source neuron of an output is not known for regular max-pooling operations. Note that our loss formulation has no problems with irreversible operations within a layer, e.g., most convolutional or fully-connected layers typically are not fully invertible. In all these cases the loss will drive the network towards a state that is as-invertible-as-possible for the given input data set. However, this requires a reliable vector of target activations in order to apply the constraints. If the connection betweeen layers is not bijective, we cannot reconstruct this target for the constraints, as in the examples given above.\nIn such cases, we regard every layer as an individual unit to which we apply the constraints by building a localized reverse pass. For example, given a simple convolutional architecture with\nd1 → C1(5, 20, 1) + b1 = d2 (25) in the forward pass, we calculate d ′\n1 with\n(d2 − b1)→ D1(5, 3, 1) = d ′\n1, (26) We, e.g., use this local loss in the Resnet110 network below. It is important to note that despite being closer to regular autoencoder pretraining, this formulation still incorporates all non-linearities of the original network structure, and jointly trains full networks while taking into account the original learning objective.\nA.4 MNIST AND PEAK TESTS\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\n0.312\n0.392\n0.298\n0.334\n0.231\n0.114\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n(a) One hidden layer, without BN, without AF\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\n0.499\n0.502\n0.499\n0.491\n0.239\n0.400\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n(b) One hidden layer, with BN, with ReLU AF\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.356\n0.437\n0.405\n0.433\n0.196\n0.267\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n… … …\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.200\n0.110\n𝑅𝑅\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.233\n0.202\n𝑅𝑅\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × ,\n(c) Two hidden layers, with BN, with ReLU AF\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.175\n0.181\n𝑅𝑅\n…\n𝒗\n0.502\n0.491\n0.400\n𝒗\n0.440\n0.395\n0.524\n𝒗\n0.401\n0.377\n0.417\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\n0.395\n0.528\n0.420\n0.465\n0.225\n0.441\n(4)\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\n0.582\n0.447\n0.389\n0.411\n0.214\n0.370\n(5)\n0.3190.500 0.495\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.377 0.426 0.408\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3360.407 0.412\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3330.462 0.443\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.2920.514 0.400\nSTDEV LPIPS 0.002 0.006 0.114 0.020 0.017 0.209 STDEV LPIPS 0.009 0.050 0.115 STDEV LPIPS\n0.094 0.032 0.153 STDEV LPIPS\n0.095 0.016 0.110 STDEV LPIPS\n0.202\n0.217\n𝑅𝑅\n0.022\n0.244\n𝑅𝑅 0.243\n0.001\n0.219\n𝑅𝑅 0.183\n0.051\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.142\n0.232\n𝑅𝑅 0.187\n0.064\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.306\n0.240\n𝑅𝑅\n0.273\n0.047\nBelow we give details for the peak tests from Sec. 3 of the main paper and show additional tests with the MNIST data set.\nPeak Test: For the Peak test we generated a data set of 110 images shown in figure 12. 55 images contain a peak located in the upper left corner of the image. The other 55 contain a peak located in the bottom right corner. We added random scribbles in the images to complicate the task. All 110 images were labeled with a one-hot encoding of the two possible positions of the peak. We use 100 images as training data set, and the remaining 10 for testing. All peak models are trained for 5000 epochs with a learning rate of 0.0001, with λ = 1e − 6 for RRA. To draw reliable conclusions, we show results for five repeated runs here. The neural network in this case contains one fully connected layer, with BN and ReLU activation. The results are shown in figure 13, with both peak modes being consistently embedded into the weight matrix of RRA, while regular and orthogonal training show primarily random singular vectors.\nWe also use different network architectures in figure 14 to verify that the dominant features are successfully extracted when using more complex\n16\ntraining data test data\nFigure 12: Data set used for the peak tests.\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.312\n0.392\n0.298\n0.334\n0.231\n0.114\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n(a) One hidden layer, without BN, without AF\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.499\n0.502\n0.499\n0.491\n0.239\n0.400\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n(b) One hidden layer, with BN, with ReLU AF\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.356\n0.437\n0.405\n0.433\n0.196\n0.267\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × (c) Two hidden layers, with BN, with ReLU AF\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n… … …\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.200\n0.110\n𝑅𝑅\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.233\n0.202\n𝑅𝑅\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × ,\n(c) Two hidden layers, with BN, with ReLU AF\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.175\n0.181\n𝑅𝑅\n… 𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.499\n0.502\n0.499\n0.491\n0.239\n0.400\n𝑅𝑒𝑝𝑒𝑎𝑡𝑒𝑑 𝑃𝑒𝑎𝑘 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝐵𝑁, 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈 𝐴𝐹 (1)\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.412\n0.440\n0.420\n0.395\n0.229\n0.524\n(2)\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.414\n0.401\n0.448\n0.377\n0.255\n0.417\n(3)\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.395\n0.528\n0.420\n0.465\n0.225\n0.441\n(4)\n𝒗\n𝒗\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.582\n0.447\n0.389\n0.411\n0.214\n0.370\n(5)\n0.3190.500 0.495\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.377 0.426 0.408\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3360.407 0.412\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3330.462 0.443\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.2920.514 0.400\nSTDEV LPIPS 0.002 0.006 0.114 0.020 0.017 0.209 STDEV LPIPS 0.009 0.050 0.115 STDEV LPIPS\n0.094 0.032 0.153 STDEV LPIPS\n0.095 0.016 0.110 STDEV LPIPS\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.233\n0.202\n0.217\n𝑅𝑅\n0.022\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.243\n0.244\n𝑅𝑅 0.243\n0.001\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.147\n0.219\n𝑅𝑅 0.183\n0.051\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.142\n0.232\n𝑅𝑅 0.187\n0.064\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\n0.306\n0.240\n𝑅𝑅\n0.273\n0.047\nFigure 13: Five repeated tests with the peak data shown in Sec. 3 of the main paper. RRA robustly extracts dominant features from the data set. The two singular vectors strongly resemble the two peak modes of the training data. This is confirmed by the LPIPS measurements.\nnetwork structures. Even for two layers with BN and ReLU activations, our pretraining clearly extracts the two modes of the training data. The visual resemblance is slightly reduced in this case, as the network has the freedom to embed the features in both layers. Across all three cases (for which we performed 5 runs each), our pretraining clearly outperforms regular training and the orthogonality constraint in terms of extracting and embedding the dominant structures of the training data set in the weight matrix.\nMNIST Test: We additionally verify that the column vectors of Vm of models from RR training contain the dominant features of the input with MNIST tests, which employ a single fully connected layer, i.e. d2 = M1d1. In the first MNIST test, the training data consists only of 2 different images. All MNIST models are trained for 1000 epochs with a learning rate of 0.0001, and λ = 1e − 5 for RRA. After training, we compute the SVD for M1. SVDs of the weight matrices of trained models can be seen in figure 11. The LPIPS scores show that features embedded in the weights of RR are consistently closer to the training data set than all other methods, i.e., regular training Std, classic autoencoder pretraining Pre, and regularization via orthogonalization Ort. While the vectors of Std and Ort contain no recognizable structures.\nOverall, our experiments confirm the motivation of our pretraining formulation. They additionally show that employing an SVD of the network weights after our pretraining yields a simple and convenient method to give humans intuition about the features learned by a network."},{"heading":"B MUTUAL INFORMATION","text":"This section gives details of the mutual information and disentangled representation tests from Sec. 4 of the main paper."},{"heading":"B.1 MUTUAL INFORMATION TEST","text":"Mutual information (MI) measures the dependence of two random variables, i.e., higher MI means that there is more shared information between two parameters. More formally, the mutual information I(X;Y ) of random variables X and Y measures how different the joint distribution of X and Y is w.r.t. the product of their marginal distributions, i.e., the Kullback-Leibler divergence I(X;Y ) = KL[P(X,Y )||PXPY ], where KL denotes the Kullback-Leibler divergence. Let I(X;Dm) denote the mutual information between the activations of a layer Dm and input X. Similarly I(Dm;Y ) denotes the MI between layer m and the output Y . We use MI planes in the main paper, which show I(X;Dm) and I(Dm;Y ) in a 2D graph for the activations of each layer Dm of a network after training. This visualizes how much information about input and output distribution is retained at each layer, and how these relationships change within the network. For regular training, the information bottleneck principle (Tishby & Zaslavsky, 2015) states that early layers contain more information about the input, i.e., show high values for I(X;Dm) and I(Dm;Y ). Hence in the MI plane visualizations, these layers are often visible at the top-right corner. Later layers typically share a large amount of information with the output after training, i.e. show large I(Dm;Y ) values, and correlate less with the input (low I(X;Dm)). Thus, they typically show up in the top-left corner of the MI plane graphs.\nTraining Details: We use the same numerical studies as in (Shwartz-Ziv & Tishby, 2017) as task A, i.e. a regular feed-forward neural network with 6 fully-connected layers. The input variable X contains 12 binary digits that represent 12 uniformly distributed points on a 2D sphere. The learning objective is to discover binary decision rules which are invariant under O(3) rotations of the sphere. X has 4096 different patterns, which are divided into 64 disjoint orbits of the rotation group, forming a minimal sufficient partition for spherically symmetric rules (Kazhdan et al., 2003). To generate the input-output distribution P (X,Y ), We apply the stochastic rule p(y = 1|x) = Ψ(f(x) − θ), (x ∈ X, y ∈ Y ), where Ψ is a standard sigmoidal function Ψ(u) = 1/(1 + exp(−γu)), following (Shwartz-Ziv & Tishby, 2017). We then use a spherically symmetric real valued function of the pattern f(x), evaluated through its spherical harmonics power spectrum (Kazhdan et al., 2003), and compare with a threshold θ, which was selected to make p(y = 1) = ∑ x p(y = 1|x)p(x) ≈ 0.5, with uniform p(x). γ is high enough to keep the mutual information I(X;Y ) ≈ 0.99 bits.\nFor the transfer learning taskB, we reverse output labels to check whether the model learned specific or generalizing features. E.g., if the output is [0,1] in the original data set, we swap the entries to [1,0]. 80% of the data (3277 data pairs) are used for training and rests (819 data pairs) are used for testing.\nFor the MI comparison in Fig. 4 of the main paper, we discuss models before and after fine-tuning separately, in order to illustrate the effects of regularization. We include a model with greedy layerwise pretraining Pre, a regular model StdA, one with orthogonality constraints OrtA, and our regular model RRA, all before fine-tuning. For the model RRA all layers are constrained to be recovered in the backward pass. We additionally include the version RR1A, i.e. a model trained with only one loss term λ1|d1 − d ′\n1|2, which means that only the input is constrained to be recovered. Thus, RR1A represents a simplified version of our approach which receives no constraints that intermediate results of the forward and backward pass should match. For OrtA, we used the Spectral Restricted Isometry Property (SRIP) regularization (Bansal et al., 2018),\nLSRIP = βσ(WTW − I), (27)\nwhere W is the kernel, I denotes an identity matrix, and β represents the regularization coefficient. σ(W ) = supz∈Rn,z 6=0 ‖Wz‖ ‖z‖ denotes the spectral norm of W .\nAs explained in the main text, all layers of the first stage, i.e. from RRA, RR 1 A, OrtA , PreA and StdA are reused for training the fine-tuned models without regularization, i.e. RRAA, RR 1 AA, OrtAA , PreAA and StdAA. Likewise, all layers of the transfer task models RRAB, RR 1 AB, OrtAB , PreAB and StdAB are initialized from the models of the first training stage.\nAnalysis of Results: We first compare the version only constraining input and output reconstruction (RR1A) and the full loss version RRA. Fig. 4(b) of the main paper shows that all points of RRA are located in a central region of the MI place, which means that all layers successfully encode information about the inputs as well as the outputs. This also indicates that every layer contains a similar amount of information about X and Y , and that the path from input to output is similar to the path from output to input. The points of RR1A, on the other hand, form a diagonal line. I.e., this network has different amounts of mutual information across its layers, and potentially a very different path for each direction. This difference in behavior is caused by the difference of the constraints in these two versions: RR1A is only constrained to be able to regenerate its input, while the full loss for RRA ensures that the network learns features which are beneficial for both directions. This test highlights the importance of the constraints throughout the depth of a network in our formulation. In contrast, the I(X;D) values of later layers for StdA and OrtA exhibit small values (points near the left side), while I(D;Y ) is high throughout. This indicates that the outputs were successfully encoded and that increasing amounts of information about the inputs are discarded. Hence, more specific features about the given output data-set are learned by StdA and OrtA. This shows that both models are highly specialized for the given task, and potentially perform worse when applied to new tasks. PreA only focuses on decreasing the reconstruction loss, which results in high I(X;D) values for early layers, and low I(D;Y ) values for later layers. During the fine-tuning phase for task A (i.e. regularizers being disabled), all models focus on the output and maximize I(D;Y ). There are differences in the distributions of the points along the y-axis, i.e., how much MI with the output is retained, as shown in Fig. 4(c) of the main paper. For model RRAA, the I(D;Y ) value is higher than for StdAA, OrtAA, PreAA and RR1AA, which means outputs of RRAA are more closely related to the outputs, i.e., the ground truth labels for task A. Thus, RRAA outperforms the other variants for the original task.\nIn the fine-tuning phase for task B, StdAB stands out with very low accuracy in Fig. 5 of the main paper. This model from a regular training run has large difficulties to adapt to the new task. PreA aims at extracting features from inputs and reconstructed them. PreAB outperforms StdAB, which means features helpful for task B are extracted by PreA, however, it’s hard to guide the feature extracting process. Model OrtAB also performs worse than StdB. RRAB shows the best performance in this setting, demonstrating that our loss formulation yielded more generic features, improving the performance for related tasks such as the inverted outputs for B.\nWe also analyze the two variants of our pretraining: the local variant lRRA and the full version RRA in terms of mutual information. figure 15 shows the MI planes for these two models, also\nshowing RR1A for comparison. Despite the local nature of lRRA it manages to establish MI for the majority of the layers, as indicated by the cluster of layers in the center of the MI plane. Only the first layer moves towards the top right corner, and the second layer is affected slightly. I.e., these layers exhibit a stronger relationship with the distribution of the outputs. Despite this, the overall performance when fine-tuning or for the task transfer remains largely unaffected, e.g., the lRRA still clearly outperforms RR1A. This confirms our choice to use the full pretraining when network connectivity permits, and employ the local version in all other cases."},{"heading":"B.2 DISENTANGLED REPRESENTATIONS","text":"The InfoGAN approach (Chen et al., 2016) demonstrated the possibility to control the output of generative models via maximizing mutual information between outputs and structured latent variables. However, mutual information is very hard to estimate in practice (Walters-Williams & Li, 2009). The previous section and Fig. 4(b) of the main paper demonstrated that models from our pretraining (both RR1A and RRA) can increase the mutual information between network inputs and outputs. Intuitively, the pretraining explicitly constrains the model to recover an input given an output, which directly translates into an increase of mutual information between input and output distributions compared to regular training runs. For highlighting how our pretraining can yield disentangled representations (as discussed in the later paragraphs of Sec. 4 of the main text), we follow the experimental setup of InfoGAN (Chen et al., 2016): the input dimension of our network is 74, containing 1 ten-dimensional category code c1, 2 continuous latent codes c2, c3 ∼ U(−1, 1) and 62 noise variables. Here, U denotes a uniform distribution.\nTraining Details: As InfoGAN focuses on structuring latent variables and thus only increases the mutual information between latent variables and the output, we also focus the pretraining on the corresponding latent variables. I.e., the goal is to maximize their mutual information with the output of the generative model. Hence, we train a model RR1 for which only latent dimensions c1, c2, c3 of the input layer are involved in the loss. We still employ a full reverse pass structure in the neural network architecture. c1 is a ten-dimensional category code, which is used for controlling the output digit category, while c2 and c3 are continuous latent codes, to represent (previously unknown) key properties of the digits, such as orientation or thickness. Building relationship between c1 and outputs is more difficult than for c2 or c3, since the 10 different digit outputs need to be encoded in a sinlge continuous variable c1. Thus, for the corresponding loss term for c1 we use a slightly larger λ factor (by 33%) than for c2 and c3. Details of our results are shown in figure 16. Models are trained using a GAN loss (Goodfellow et al., 2014) as the loss function for the outputs.\nAnalysis of Results: In figure 16 we show additional results for the disentangling test case. It is visible that our pretraining of the RR1 model yields distinct and meaningful latent space dimensions for c1,2,3. While c1 controls the digit, c2,3 control the style and orientation of the digits. For comparison, a regular training run with model Std does result in meaningful or visible changes when adjusting the latent space dimensions. This illustrates how strongly the pretraining can shape the\nUnder review as a conference paper at ICLR 2021\nVarying 𝑐\nV arying in put no ise\nV arying in put no ise\nVarying 𝑐\nV arying in put no ise\nVarying 𝑐\nV arying in put no ise\nVarying 𝑐\nlatent space, and in addition to an intuitive embedding of dominant features, yield a disentangled representation."},{"heading":"C DETAILS OF EXPERIMENTAL RESULTS","text":"C.1 TEXTURE-SHAPE BENCHMARK\n0.208\n0.085\n0.159\n0.442\n0.158\n0.331\n0.470\n0.203\n0.372\n0.547\n0.237\n0.408\nS T Y L I Z E D D A T A E D G E D A T A F I L L E D D A T A\nEdge data setStylized data set Filled data set\nAccuracy Comparisons of 𝑂𝑟𝑡 , 𝑆𝑡𝑑 and 𝑅𝑅\n𝑆𝑡𝑑 𝑅𝑅\n𝑂𝑟𝑡\nEdge data\nFilled data\nStylized data\n(𝑎) (𝑏) (𝑐) (𝑑)\n𝑃𝑟𝑒\nTraining Details: All training data of the texture-shape tests were obtained from (Geirhos et al., 2018). The stylized data set contains 1280 images, 1120 images are used as training data, and 160 as test data. Both edge and filled data sets contain 160 images each, all of which are used for testing only. All three sets (stylized, edge, and filled) contain data for 16 different classes.\nAnalysis of Results: For a detailed comparison, we list per-class accuracy of stylized data training runs for OrtTS, StdTS, PreTS and RRTS in figure 17. RRTS out-\nperforms the other three models for most of the classes. RRTS requires an additional 41.86% for training compared to StdTS, but yields a 23.76% higher performance. (Training times for these models are given in the supplementary document.) All models saturated, i.e. training StdTS or OrtTS longer does not increase classification accuracy any further. We also investigated how much we can reduce model size when using our pretraining in comparison to the baselines. A reduced model only uses 67.94% of the parameters, while still outperforming OrtTS."},{"heading":"C.2 GENERATIVE ADVERSARIAL MODELS","text":"Training Details: The data set of smoke simulation was generated with a Navier-Stokes solver from an open-source library (Thuerey & Pfaff, 2018). We generated 20 randomized simulations with 120 frames each, with 10% of the data being used for training. The low-resolution data were down-sampled from the high-resolution data by a factor of 4. Data augmentation, such as flipping and rotation was used in addition. As outlined in the main text, we consider building an autoencoder model for the synthetic data as task B1, and a generating samples from a real-world smoke data set as task B2. The smoke capture data set for B2 contains 2500 smoke images from the ScalarFlow data set (Eckert et al., 2019), and we again used 10% of these images as training data set.\nTask A: We use a fully convolutional CNN-based architecture for generator and discriminator networks. Note that the inputs of the discriminator contain high resolution data (64, 64, 1), as well as low resolution (16, 16, 1), which is up-sampled to (64, 64, 1) and concatenated with the high resolution data. In line with previous work (Xie et al., 2018), RRA and StdA are trained with a\nnon-saturating GAN loss, feature space loss and L2 loss as base loss function. All generator layers are involved in the pretraining loss. As greedy layer-wise autoencoder pretraining is not compatible with adversarial training, we pretrain PreA for reconstructing the high resolution data instead.\nTask B1: All encoder layers are initialized from RRA and StdA when training RRAB1 and StdAB1 . It is worth noting that the reverse pass of the generator is also constrained when training PreA and RRA. So both encoder and decoder are initialized with parameters from PreA and RRA when training PreAB1 and RRAB1 , respectively. This is not possible for a regular network like StdAB1 , as the weights obtained with a normal training run are not suitable to be transposed. Hence, the deconvolutions of StdAB1 are initialized randomly.\nTask B2: As the data set for the task B2 is substantially different and contains RBG images (instead of single channel gray-scale images), we choose the following setups for the RRA, PreA and StdA models: parameters from all six layers of StdA and RRA are reused for initializing decoder part of StdAB2 and RRAB2 , parameters from all six layers of PreA are reused for initializing the encoder part of PreAB2 . Specially, when initializing the last layer of PreAB2 , StdAB2 and RRAB2 , we copy and stack the parameters from the last layer of PreA, StdA and RRA, respectively, into three channels to match the dimenions of the outputs for taskB2. Here, the encoder part of RRAB2 and the decoder of PreAB2 are not initialized with RRA and PreA, due to the significant gap between training data sets of taskB1 and taskB2. Our experiments show that only initializing the decoder part of RRAB2 (avg. loss:1.56e7, std. dev.:3.81e5) outperforms initializing both encoder and decoder (avg. loss:1.82e7± 2.07e6), and only initializing the encoder part of PreAB2 (avg. loss:4.41e7 ± 6.36e6) outperforms initializing both encoder and decoder (avg. loss:9.42e7 ± 6.11e7). We believe the reason is that initializing both encoder and decoder part makes it more difficult to adjust the parameters for new data set that is very different from the data set of the source task.\nAnalysis of Results: Example outputs of PreAB1 , StdAB1 and RRAB1 are shown in figure 18. It is clearly visible that RRAB1 gives the best performance among these models. We similarly illustrate the behavior of the transfer learning task B2 for images of real-world fluids. This example\n387 1057 2070 1990 2005 2189\nlikewise uses an autoencoder structure. Visual comparisons are provided in figure 19, where RRAB2 generates results that are closer to the reference. Overall, these results demonstrate the benefits of our pretraining for GANs, and indicate its potential to obtain more generic features from synthetic data sets that can be used for tasks involving real-world data."},{"heading":"C.3 WEATHER FORECASTING","text":"Training Details: The weather forecasting scenario discussed in the main text follows the methodology of the WeatherBench benchmark (Rasp et al., 2020). This benchmark contains 40 years of data from the ERA5 reanalysis project Hersbach et al. (2020) which was re-sampled to a 5.625◦ resolution, yielding 32× 64 grid points in ca. two-hour intervals. Data from the year of 1979 to 2015 (i.e., 162114 samples) are used for training, the year of 2016 for validation. The last two years (2017 and 2018) are used as test data. All RMSE measurements are latitude-weighted to account for area distortions from the spherical projection.\nThe neural networks for the forecasting tasks employ a ResNet architecture with 19 layers, all of which contain 128 features with 3 × 3 kernels (apart from 7 × 7 in the first layer). All layers use batch normalization, leaky ReLU activation (tangent 0.3), and dropout with strength 0.1. As inputs, the model receives feature-wise concatenated data from the WeatherBench data for 3 consecutive time steps, i.e., t, t − 6h, and t − 12h, yielding 117 channels in total. The last convolution jointly generates all three output fields, i.e., pressure at 500 hPa (Z500), temperature at 850 hPa (T850), and the 2-meter temperature (T2M).\nAnalysis of Results: In addition to the quantitative results for both years of test data given in the main text, figure 20 and 21 contain additional example visualizations from the test data set. A visualization of the spatial error distribution w.r.t. ground truth result in also shown in figure 21. It becomes apparent that our pretraining achieves reduced errors across the whole range of samples. Both temperature targets contain a larger number of smaller scale features than the pressure fields. While the gains from our pretraining approach are not huge (on the order of 3% in both cases), they represent important steps forward. The learning objective is highly non-trivial, and the improvements were achieved with the same limited set of training data. Being very easy to integrate into existing training pipelines, these results indicate that the proposed pretraining methodology has the potential to yield improved learning results for a wide range of problem settings."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"While approaches such as greedy layer-wise autoencoder pretraining (Bengio et al., 2007; Vincent et al., 2010; Erhan et al., 2010) arguably paved the way for many fundamental concepts of today’s methodologies in deep learning, the pressing need for pretraining neural networks has been diminished in recent years. This was primarily caused by numerous advances in terms of regularization (Srivastava et al., 2014; Hanson & Pratt, 1989; Weigend et al., 1991), network architectures (Ronneberger et al., 2015; He et al., 2016; Vaswani et al., 2017), and improved optimization algorithms (Kingma & Ba, 2014; Loshchilov & Hutter, 2017; Reddi et al., 2019). Despite these advances, training deep neural networks that generalize well to a wide range of previously unseen tasks remains a fundamental challenge (Neyshabur et al., 2017; Kawaguchi et al., 2017; Frankle & Carbin, 2018).\\nInspired by techniques for orthogonalization (Ozay & Okatani, 2016; Jia et al., 2017; Bansal et al., 2018), we re-visit the classic idea of unsupervised autoencoder pretraining in the context of reversible network architectures. Hence, we propose a modified variant that relies on a full reverse\\npass trained in conjunction with a given training task. A key insight is that there is no need for ”greediness”, i.e., layer-wise decompositions of the network structure, and it is additionally beneficial to take into account a specific problem domain at the time of pretraining. We establish links between singular value decomposition (SVD) and pretraining, and show how our approach yields an embedding of problem-aware dominant features in the weight matrices. An SVD can then be leveraged to conveniently gain insights about learned structures. Most importantly, we demonstrate that the proposed pretraining yields an improved performance for a variety of learning and transfer tasks. Our formulation incurs only a very moderate computational cost, is very easy to integrate, and widely applicable.\\nThe structure of our networks is influenced by invertible network architectures that have received significant attention in recent years (Gomez et al., 2017; Jacobsen et al., 2018; Zhang et al., 2018a). However, instead of aiming for a bijective mapping that reproduces inputs, we strive for learning a general representation by constraining the network to represent an as-reversible-as-possible process for all intermediate layer activations. Thus, even for cases where a classifier can, e.g., rely on color for inference of an object type, the model is encouraged to learn a representation that can recover the input. Hence, not only the color of the input should be retrieved, but also, e.g., its shape. In contrast to most structures for invertible networks, our approach does not impose architectural restrictions. We demonstrate the benefits of our pretraining for a variety of architectures, from fully connected layers to convolutional neural networks (CNNs), over networks with and without batch normalization, to GAN architectures. We discuss other existing approaches and relate them to the proposed method in the appendix.\\nBelow, we will first give an overview of our formulation and its connection to singular values, before evaluating our model in the context of transfer learning. For a regular, i.e., a non-transfer task, the goal usually is to train a network that gives optimal performance for one specific goal. During a regular training run, the network naturally exploits any observed correlations between input and output distribution. An inherent difficulty in this setting is that typically no knowledge about the specifics of the new data and task domains is available when training the source model. Hence, it is common practice to target broad and difficult tasks hoping that this will result in features that are applicable in new domains (Zamir et al., 2018; Gopalakrishnan et al., 2017; Ding et al., 2017). Motivated by autoencoder pretraining, we instead leverage a pretraining approach that takes into account the data distribution of the inputs. We demonstrate the gains in accuracy for original and new tasks below for a wide range of applications, from image classification to data-driven weather forecasting.\"\n },\n {\n \"heading\": \"2 METHOD\",\n \"text\": \"With state-of-the-art methods, there is no need for breaking down the training process into single layers. Hence, we consider approaches that target whole networks, and especially orthogonalization regularizers as a starting point (Huang et al., 2018). Orthogonality constraints were shown to yield improved training performance in various settings (Bansal et al., 2018), and can be formulated as:\\nLort = n∑\\nm=1 ∥∥MTmMm − I∥∥2F , (1) i.e., enforcing the transpose of the weight matrix Mm ∈ Rs out m×s in m for all layers m to yield its inverse when being multiplied with the original matrix. I denotes the identity matrix with I = (e1m, ...e sinm m ), ejm denoting the jth column unit vector. Minimizing equation 1, i.e. M T mMm− I = 0 is mathematically equivalent to: MTmMme j m − ejm = 0, j = 1, 2, ..., sinm, (2)\\nwith rank(MTmMm) = s in m, and e j m as eigenvectors of M T mMm with eigenvalues of 1. This formulation highlights that equation 2 does not depend on the training data, and instead only targets the content of Mm. Inspired by the classical unsupervised pretraining, we re-formulate the orthogonality constraint in a data-driven manner to take into account the set of inputs Dm for the current layer (either activation from a previous layer or the training data D1), and instead minimize\\nLRR = n∑\\nm=1\\n(MTmMmd i m − dim)2 = n∑ m=1 ((MTmMm − I)dim)2, (3)\\nwhere dim ∈ Dm ⊂ Rs in m . Due to its reversible nature, we will denote our approach with an RR subscript in the following. In contrast to classical autoencoder pretraining, we are minimizing this loss jointly for all layers of a network, and while orthogonality only focuses onMm, our formulation allows for minimizing the loss by extracting the dominant features of the input data.\\nLet q denote the number of linearly independent entries in Dm, i.e. its dimension, and t the size of the training data, i.e. |Dm| = t, usually with q < t. For every single datum dim, i = 1, 2, ..., t, equation 3 results in\\nMTmMmd i m − dim = 0, (4)\\nand hence dim are eigenvectors of M T mMm with corresponding eigenvalues being 1. Thus, instead of the generic constraint MTmMm = I that is completely agnostic to the data at hand, the proposed formulation of equation 4 is aware of the training data, which improves the generality of the learned representation, as we will demonstrate in detail below.\\nAs by construction, rank(Mm) = r 6 min(sinm, s out m ), the SVD of Mm yields:\\nMm = UmΣmV T m , with\\n{ Um = (u 1 m,u 2 m, ...,u r m,u r+1 m , ...,u soutm m ) ∈ Rs out m×s out m ,\\nVm = (v 1 m,v 2 m, ...,v r m,v r+1 m , ...,v sinm m ) ∈ Rs in m×s in m ,\\n(5)\\nwith left and right singular vectors in Um and Vm, respectively, and Σm having square roots of the r eigenvalues of MTmMm on its diagonal. u k m and v k m(k = 1, ..., r) are the eigenvectors of MmM T m and MTmMm, respectively (Wall et al., 2003). Here, especially the right singular vectors in V T m are important, as they determine which structures of the input are processed by the transformation Mm. The original orthogonality constraint with equation 2 yields r unit vectors ejm as the eigenvectors of MTmMm. Hence, the influence of equation 2 on Vm is completely independent of training data and learning objectives.\\nNext, we show that LRR facilitates learning dominant features from a given data set. For this, we consider an arbitrary basis for spanning the space of inputsDm for layerm. Let Bm : 〈 w1m, ...,w q m 〉 denote a set of q orthonormal basis vectors obtained via a Gram-Schmidt process, with t> q > r, and Dm denoting the matrix of the vectors in Bm. As we show in more detail in the appendix, our constraint from equation 4 requires eigenvectors of MTmMm to be w i m, with Vm containing r orthogonal vectors (v1m,v 2 m, ...,v r m) from Dm and (sinm − r) vectors from the null space of M .\\nWe are especially interested in how Mm changes w.r.t. input in terms of Dm, i.e., we express LRR in terms of Dm. By construction, each input dim can be represented as a linear combination via a vector of coefficients cim that multiplies Dm so that d i m =Dmc i m. Since Mmdm = UmΣmV T mdm, the loss LRR of layer m can be rewritten as\\nLRRm = (MTmMmdm − dm)2 = (VmΣTmΣmV Tmdm − dm)2\\n= (VmΣ T mΣmV T mDmcm −Dmcm)2,\\n(6)\\nwhere we can assume that the coefficient vector cm is accumulated over the training data set size t via cm = ∑t i=1 c i m, since eventually every single datum inDm will contribute to LRRm . The central component of equation 6 is V TmDm. For a successful minimization, Vm needs to retain those w i m with the largest cm coefficients. As Vm is typically severely limited in terms of its representational capabilities by the number of adjustable weights in a network, it needs to focus on the most important eigenvectors in terms of cm in order to establish a small distance to Dmcm. Thus, features that appear multiple times in the input data with a corresponding factor in cm will more strongly contribute to minimizing LRRm . To summarize, Vm is driven towards containing r orthogonal vectors wim that represent the most frequent features of the input data, i.e., the dominant features. Additionally, due to the column vectors of Vm being mutually orthogonal, Mm is encouraged to extract different features from the input. By the sake of being distinct and representative for the data set, these features have the potential to be useful for new inference tasks. The feature vectors embedded inMm can be extracted from the network weights in practical settings, as we will demonstrate below.\\nRealization in Neural Networks Calculating MTmMm is usually very expensive due to the dimensionality of Mm. Instead of building it explicitly, we constrain intermediate results to realize equation 3 when training. A regular training typically starts with a chosen network structure and\\nUnder review as a conference paper at ICLR 2021\\ntrains the model weights for a given task via a suitable loss function. Our approach fully retains this setup and adds a second pass that reverses the initial structure while reusing all weights and biases. E.g., for a typical fully connected layer in the forward pass with dm+1 = Mmdm +bm, the reverse pass operation is given by d ′\\nm = M T m(dm+1 − bm), where d\\n′ m denotes the reconstructed input.\\na m\\no un\\nt o f s\\nh a\\nre d\\nin fo\\nrm a\\ntio n\\nb et\\nw e\\ne n\\nla ye\\nr a\\nn d\\namount of shared information between layer ℒ and 𝑋\\n𝐿𝑎𝑦𝑒𝑟 1 2 3 𝑚 𝑛 𝐿𝑎𝑦𝑒𝑟 𝑚 𝒅𝒎 𝒅𝒎 𝟏\\n𝒅𝒎 𝟏𝒅𝒎\\n𝒅𝟏\\n𝒅𝟏\\n|\\n𝑓\\n𝐵𝑁 𝑐𝑜𝑛𝑣 𝑓 (𝑀 , 𝒃𝒎)\\n𝑑𝑒𝑐𝑜𝑛𝑣\\n(𝑀 , −𝒃𝒎) 𝐵𝑁\\n𝑤𝑒𝑖𝑔ℎ𝑡 𝑠ℎ𝑎𝑟𝑖𝑛𝑔\\n𝒗 𝒗 𝒗 0.427 0.400 0.489\\n0.420 0.394 0.445 0.213 0.216 0.153 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n𝒗\\n𝒗\\n𝒅\\n…\\n…\\n(50 images)\\n(50 images)\\n(1,0)\\n(0,1)\\n𝒅 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n𝑎 𝑀𝑁𝐼𝑆𝑇 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\\n𝑏 𝑃𝑒𝑎𝑘 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝐵𝑁 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈 𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛\\n(0,0,0,0,0,0,0,1,0,0) (0,0,1,0,0,0,0,0,0,0)\\n0.499\\n0.502\\n0.499\\n0.491\\n0.233\\n0.202\\n𝒅 𝒅\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\nmore information about both 𝑋&𝑌 more information about 𝑌 only\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\nless shared information between layer ℒ and 𝑋\\nle s s\\nsh ar\\ned in\\nfo rm\\nat io n b et w ee n la ye r ℒ a n d\\n𝑌\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\n0.9\\n0.92\\n0.94\\n0.96\\n0.98\\n1\\n1.02\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝒟 𝒟 𝒟 𝒟 𝒟 𝒟\\n𝑆𝑡𝑑\\n𝐼( 𝒟\\n;𝑌 )\\nmore information about 𝑋 only\\n(a) Mutual Information Plane, How to Read (b) Mutual Information for Task A (c) After fine-tuning for A (d) After fine-tuning for B\\nno information\\n…\\n…\\n…\\n…\\n𝒅𝟐\\n𝒅𝟐\\n|\\n𝒅𝟑\\n𝒅𝟑\\n|\\n𝒅𝒏\\n𝒅𝒏\\n| 𝒐𝒖𝒕𝒑𝒖𝒕\\nLayers of 𝑅𝑅 models exhibit strong MI with in- & output\\n𝒗 𝒗 0.43 0.49\\n0.42 0.45 0.21 0.22 𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n𝒗\\n𝒗\\n𝒅\\n(1,0)\\n(0,1)\\n𝒅\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\\n𝑏 𝑊𝑖𝑡ℎ 𝐵𝑁 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈\\n7 2\\n0.50\\n0.50\\n0.50\\n0.49\\n0.23\\n0.20\\n𝒅 𝒅\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\\n……(50 imgs)\\n……(50 imgs)\\nOur goal with the reverse pass is to transpose all operations of the forward pass to obtain identical intermediate activations between the layers with matching dimensionality. We can then constrain the intermediate results of each layer of the forward pass to match the results of the backward pass, as illustrated in figure 2. Unlike greedy layer-wise autoencoder pretraining, which trains each layer separately and only constrains d1 and d ′\\n1, we jointly train all layers and constrain all intermediate results. Due to the symmetric structure of the two passes, we can use a simple L2 difference to drive the network towards aligning the results:\\nLRR = n∑\\nm=1\\nλm\\n∥∥∥dm − d′m∥∥∥2\\nF . (7)\\nHere dm denotes the input of layer m in the forward pass and d ′\\nm the output of layer m for the reverse pass. λm denotes a scaling factor for the loss of layer m, which, however, is typically constant in our tests across all layers. Note that with our notation, d1 and d ′\\n1 refer to the input data, and the reconstructed input, respectively.\\nNext, we show how this setup realizes the regularization from equation 3. For clarity, we use a fully connected layer with bias. In a neural network with n hidden layers, the forward process for a layer m is given by dm+1 = Mmdm +bm,, with d1 and dn+1 denoting in- and output, respectively. For our pretraining, we build a reverse pass network with transposed operations starting with the final output where dn+1 = d ′ n+1, and the intermediate results d ′ m+1:\\nd ′\\nm = M T m(d\\n′ m+1 − bm), (8) which yields ∥∥∥dm − d′m∥∥∥2 F = ∥∥MTmMmdm − dm∥∥2F . When this difference is minimized via equation 7, we obtain activated intermediate content during the reverse pass that reconstructs the values computed in the forward pass, i.e. d ′\\nm+1 = dm+1 holds. As in equation 10 the reverse pass activation d ′\\nm depends on dm+1 ′, this formulation yields a full reverse pass from output to input, which\\nwe use for most training runs below. In this case\\nd ′\\nm = M T m(d\\n′ m+1 − bm) = MTm(dm+1 − bm) = MTmMmdm , (9)\\nwhich is consistent with equation 3, and satisfies the original constraint MTmMmdm − dm = 0. This version is preferable if a unique path from output to input exists. For architectures where the path is not unique, e.g., in the presence of additive residual connections, we use a local formulation\\nd ′\\nm = M T m(dm+1 − bm), (10)\\nwhich employs dm+1 for jointly constraining all intermediate activations in the reverse pass.\\nUp to now, the discussion focused on simplified neural networks without activation functions or extensions such as batch normalization (BN). While we leave incorporating such extensions for future work, our experiments consistently show that the inherent properties of our pretraining remain valid: even with activations and BN, our approach successfully extracts dominant structures and yields improved generalization. In the appendix, we give details on how to ensure that the latent space content for forward and reverse pass is aligned such that differences can be minimized.\\nTo summarize, we realize the loss formulation of equation 7 to minimize ∑n\\nm=1((M T mMm−I)dm)2\\nwithout explicitly having to construct MTmMm. Following the notation above, we will refer to networks trained with the added reverse structure and the additional loss terms as RR variants. We consider two variants for the reverse pass: a local pretraining equation 10 using the datum dm+1 of a given layer, and a full version via equation 8 which uses d ′\\nm+1 incoming from the next layer during the reverse pass.\\nEmbedding Singular Values Below, Std denotes a regular training run (in orange color in graphs below), while RR denotes our models (in green). Pre and Ort will denote regular autoencoder pretraining and orthogonality, respectively, while a subscript will denote the task variant the model was trained for, e.g., StdT for task T. While we typically use all layers of a network in the constraints, a reduced variant that we compare to below only applies the constraint for the input data, i.e., m=1. A network trained with this variant, denoted by RR1A, is effectively trained to only reconstruct the input. It contains no constraints for the inner activations and layers of the network. For the Ort models, we use the Spectral Restricted Isometry Property algorithm (Bansal et al., 2018).\\nWe verify that the column vectors of Vm of models from RR training contain the dominant features of the input with the help of a classification test, employing a single fully connected layer, i.e. d2 = M1d1, with batch normalization and activation. To quantify this similarity, we compute an LPIPS distance (Zhang et al., 2018b) between vim and the training data (lower values being better).\\n𝒗\\n𝒗 0.43\\n0.49\\n0.42\\n0.45 0.21\\n0.22\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n𝑎 𝐿𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑑𝑒𝑙\\n7\\n2\\n𝒅 𝒅\\n𝐋𝐏𝐈𝐏𝐒𝐏𝐫𝐞\\n0.14\\n0.31\\n𝒗\\n𝒗\\n1,0 𝑎𝑛𝑑 (0,1)\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.50\\n0.50\\n0.50\\n0.49\\n0.23\\n0.20\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑅𝑅\\n…… ……\\n𝑃𝑟𝑒 𝐋𝐏𝐈𝐏𝐒𝐏𝐫𝐞\\n0.24\\n0.40\\n+\\nWe employ a training data set constructed from two dominant classes (a peak in the top left, and bottom right quadrant, respectively), augmented with noise in the form of random scribbles. Based on the analysis above, we expect the RR training to extract the two dominant peaks during training. The LPIPS measurements confirm our SVD argumentation above, with average scores of 0.217±0.022 for RR, 0.319±0.114 for Pre, 0.495± 0.006 for Ort, and 0.500± 0.002 for Std. I.e., the RR model fares significantly better than the others. At the same time, the peaks are clearly visible for RR models, an example is shown in figure 3(b), while the other models fail to extract structures that resemble the input. Thus, by training with the full network and the original training objective, our pretraining yields structures that are interpretable and be inspected by humans.\\nThe results above experimentally confirm our formulation of the RR loss and its ability to extract dominant and generalizing structures from the training data. Next, we will focus on quantified metrics and turn to measurements in terms of mutual information to illustrate the behavior of our pretraining for deeper networks.\"\n },\n {\n \"heading\": \"3 EVALUATION IN TERMS OF MUTUAL INFORMATION\",\n \"text\": \"As our approach hinges on the introduction of the reverse pass, we will show that it succeeds in terms of establishing mutual information (MI) between the input and the constrained intermediates inside a network. More formally, MI I(X;Y ) of random variablesX and Y measures how different the joint distribution ofX and Y is w.r.t. the product of their marginal distributions, i.e., the Kullback-Leibler divergence I(X;Y ) = DKL[P(X,Y )||PXPY ]. (Tishby & Zaslavsky, 2015) proposed MI plane to analyze trained models, which show the MI between the input X and activations of a layer Dm, i.e., I(X;Dm) and I(Dm;Y ), i.e., MI of layer Dm with output Y . These two quantities indicate how much information about the in- and output distributions are retained at each layer, and we use them to show to which extent our pretraining succeeds at incorporating information about the inputs throughout training.\\nThe following tests employ networks with six fully connected layers with the objective to learn the mapping from 12 binary inputs to 2 binary output digits (Shwartz-Ziv & Tishby, 2017), with results accumulated over five runs. We compare the versions StdA, PreA, OrtA, RRA, and a variant of the latter: RR1A, i.e. a version where only the input d1 is constrained to be reconstructed. While figure 4a) visually summarizes the content of the MI planes, the graph in (b) highlights that training with the RR loss correlates input and output distributions across all layers: the cluster of green points in the center of the graph shows that all layers contain balanced MI between in- as well as output and the activations of each layer. RR1A fares slightly worse, while StdA and OrtA almost exclusively focus on the output with I(Dm;Y ) being close to one. PreA instead only focuses on reconstructing inputs. Thus, the early layers cluster in the right-top corner, while the last layer I(D7;Y ) fails to\\nalign with the outputs. Once we continue fine-tuning these models without regularization, the MI naturally shifts towards the output, as shown in figure 4 (c). Here, RRAA outperforms the other models in terms of final performance. Likewise, RRAB performs best for a transfer task B with switched output digits, as shown in graph (d). The final performance for both tasks across all runs is summarized in figure 5. These graphs visualize that the proposed pretraining succeeds in robustly establishing mutual information between inputs and targets across a full network, in addition to extracting reusable features.\\nMI has received attention recently as a learning objective, e.g., in the form of the InfoGAN approach (Chen et al., 2016) for learning disentangled and interpretable latent representations. While MI is typically challenging to assess and estimate (WaltersWilliams & Li, 2009), the results above show that our approach provides a straightforward and robust way for including it as a learning objective. In this way, we can, e.g., reproduce the disentangling results from (Chen et al., 2016), which are shown in figure 1(c). A generative model with our pretraining extracts intuitive latent dimensions for the different digits, line thickness, and orientation without any additional modifications of the loss function. The joint training of the full network with the proposed reverse structure, including non-linearities and normalization, yields a natural and intuitive decomposition.\"\n },\n {\n \"heading\": \"4 EXPERIMENTAL RESULTS\",\n \"text\": \"We now turn to a broad range of network structures, i.e., CNNs, Autoencoders, and GANs, with a variety of data sets and tasks to show our approach succeeds in improving inference accuracy and generality for modern day applications and architectures.\\nTransfer-learning Benchmarks We first evaluate our approach with two state-of-the-art benchmarks for transfer learning. The first one uses the texture-shape data set from (Geirhos et al., 2018), which contains challenging images of various shapes combined with patterns and textures to be classified. The results below are given for 10 runs each. For the stylized data shown in figure 6 (a), the accuracy of PreTS is low with 20.8%. This result is in line with observations in previous work and confirms the detrimental effect of classical pretraining. StdTS yields a performance of 44.2%, and OrtTS improves the performance to 47.0%, while RRTS yields a performance of 54.7% (see figure 6b). Thus, the accuracy of RRTS is 162.98% higher than PreTS, 23.76% higher than StdTS, and 16.38% higher than OrtTS. To assess generality, we also apply the models to new data without\\n0.208\\n0.085\\n0.159\\n0.442\\n0.158\\n0.331\\n0.470\\n0.203\\n0.372\\n0.547\\n0.237\\n0.408\\nS T Y L I Z E D D A T A E D G E D A T A F I L L E D D A T A\\nEdge data setStylized data set Filled data set\\nAccuracy Comparisons of 𝑂𝑟𝑡 , 𝑆𝑡𝑑 and 𝑅𝑅\\n𝑆𝑡𝑑 𝑅𝑅\\n𝑂𝑟𝑡\\nEdge data\\nFilled data\\nStylized data\\n(𝑎) (𝑏) (𝑐) (𝑑)\\n𝑃𝑟𝑒\\nFigure 6: (a) Examples from texture-shape data set. (b, c, d) Texture-shape test accuracy comparisons of PreTS, OrtTS, StdTS and RRTS for different data sets.\\n𝑂𝑢𝑡𝑝𝑢𝑡\\n𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑆𝑡𝑑 𝑅𝑅\\n𝑅𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑒𝑑 𝐼𝑛𝑝𝑢𝑡 (𝒅 )\\n𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑆𝑡𝑑 𝑅𝑅\\nre-training, i.e. an edge and a filled data set, also shown in figure 6 (a). For the edge data set, RRTS outperforms PreTS, StdTS and OrtTS by 178.82%, 50% and 16.75%, respectively,\\nIt is worth pointing out that the additional constraints of our training approach lead to moderately increased requirements for memory and computations, e.g., 41.86% more time per epoch than regular training for the texture-shape test. On the other hand, it allows us to train smaller models: we can reduce the weight count by 32% for the texture-shape case while still being on-par with OrtTS in terms of classification performance. By comparison, regular layer-wise pretraining requires a significant overhead and fundamental changes to the training process. Our pretraining fully integrates with existing training methodologies and can easily be deactivated via λm = 0.\\nAs a second test case, we use a CIFAR-based task transfer (Recht et al., 2019) that measures how well models trained on the original CIFAR 10, generalize to a new data set (CIFAR 10.1) collected according to the same principles as the original one. Here we use a Resnet110 with 110 layers and 1.7 million parameters, Due to the consistently low performance of the Pre models (Alberti et al., 2017), we focus on Std, Ort and RR for this test case. In terms of accuracy across 5 runs, OrtC10 outperforms StdC10 by 0.39%, while RRC10 outperforms OrtC10 by another 0.28% in terms of absolute test accuracy (figure 7). This increase for RR training matches the gains reported for orthogonality in previous work (Bansal et al., 2018), thus showing that our approach yields substantial practical improvements over the latter. It is especially interesting how well performance for CIFAR 10 translates into transfer performance for CIFAR 10.1. Here, RRC10 still outperforms OrtC10 and StdC10 by 0.22% and 0.95%, respectively. Hence, the models from our pretraining very successfully translate gains in performance from the original task to the new one, indicating that the models have successfully learned a set of more general features. To summarize, both benchmark cases confirm that the proposed pretraining benefits generalization.\\nGenerative Adversarial Models In this section, we employ our pretraining in the context of generative models for transferring from synthetic to real-world data from the ScalarFlow data set (Eckert et al., 2019). As super-resolution taskA, we first use a fully-convolutional generator network, adversarially trained with a discriminator network on the synthetic flow data. While regular pretraining is\\nmore amenable to generative tasks than orthogonal regularization, it can not be directly combined with adversarial training. Hence, we pretrain a model Pre for a reconstruction task at high-resolution without discriminator instead. Figure 8 a) demonstrates that our method works well in conjunction with the GAN training: As shown in the bottom row, the trained generator succeeds in recovering the input via the reverse pass without modifications. A regular model StdA, only yields a black image in this case. For PreA, the layer-wise nature of the pretraining severely limits its capabilities to learn the correct data distribution (Zhou et al., 2014), leading to a low performance.\\nWe now mirror the generator model from the previous task to evaluate an autoencoder structure that we apply to two different data sets: the synthetic smoke data used for the GAN training (task B1), and a real-world RGB data set of smoke clouds (task B2). Thus both variants represent transfer tasks, the second one being more difficult due to the changed data distribution. The resulting losses, summarized in figure 8 b), show that RR training performs best for both autoencoder tasks: the L2 loss of RRAB1 is 68.88% lower than StdAB1 , while it is 13.3% lower for task B2. The proposed pretraining also clearly outperforms the Pre variants. Within this series of tests, the RR performance for task B2 is especially encouraging, as this task represents a synthetic to real transfer.\\nWeather Forecasting Pretraining is particularly attractive in situations where the amount of data for training is severely limited. Weather forecasting is such a case, as systematic and accurate data for many relevant quantities are only available for approximately 50 years. We target three-day forecasts of pressure, ground temperature, and mid-level atmospheric temperature based on a public benchmark dataset (Rasp et al., 2020). This dataset contains worldwide observations from ERA5 (Hersbach et al., 2020) in six-hour intervals with a 5.625◦ resolution. For the joint inference of atmospheric pressure (500 hPa geopotential, Z500), ground temperature (T2M), and atmospheric temperature (at 850 hPa, T850), we use a convolutional ResNet architecture with 19 residual blocks. As regular pretraining is not compatible with residual connections, we omit it here.\\nWe train a model regular model (about 6.36M trainable parameters) with data from 1979 to 2015, and compare its inference accuracy across all datapoints from years 2017 and 2018 to a similar model that employs our pretraining. While the regular model was trained for 25 epochs, the RR model was pretrained for 10 epochs and fine-tuned for another 15 epochs. Across all three physical quantities, the RR model clearly outperforms the regular model, as summarized in figure 1 (d) and figure 9 (details are given in the appendix). Especially for the latitude-weighted RMSE of Z500, it yields improvements of 5.5%. These improvements point to an improved generalization of the RR model via the pretraining and highlight its importance for domains where data is scarce.\"\n },\n {\n \"heading\": \"5 CONCLUSIONS\",\n \"text\": \"We have proposed a novel pretraining approach inspired by classic methods for unsupervised autoencoder pretraining and orthogonality constraints. In contrast to the classical methods, we employ a constrained reverse pass for the full non-linear network structure and include the original learning objective. We have shown for a wide range of scenarios, from mutual information, over transfer learning benchmarks to weather forecasting, that the proposed pretraining yields networks with better generalizing capabilities. Our training approach is general, easy to integrate, and imposes no requirements regarding network structure or training methods. Most importantly, our results show that unsupervised pretraining has not lost its relevance in today’s deep learning environment.\\nAs future work, we believe it will be exciting to evaluate our approach in additional contexts, e.g., for temporal predictions (Hochreiter & Schmidhuber, 1997; Cho et al., 2014), and for training explainable and interpretable models (Zeiler & Fergus, 2014; Chen et al., 2016; Du et al., 2018).\"\n },\n {\n \"heading\": \"A APPENDIX\",\n \"text\": \"To ensure reproducibility, source code and data for all tests will be published. Runtimes were measured on a machine with Nvidia GeForce GTX 1080 Ti GPUs and an Intel Core i7-6850K CPU.\"\n },\n {\n \"heading\": \"A.1 DISCUSSION OF RELATED WORK\",\n \"text\": \"Greedy layer-wise pretraining was first proposed by (Bengio et al., 2007), and influenced a large number of follow up works, providing a crucial method for enabling stable training runs of deeper networks. A detailed evaluation was performed by (Erhan et al., 2010), also highlighting cases were it can be detrimental. These problems were later on also detailed in other works, e.g., by (Alberti et al., 2017). The transferability of learned features was likewise a topic of interest for transfer learning applications (Yosinski et al., 2014). Sharing similarities with our approach, (Rasmus et al., 2015) combined supervised and unsupervised learning objectives, but focused on denoising autoencoders and a layer-wise approach without weight sharing. We demonstrate the importance of leveraging state-of-the-art methods for training deep networks, i.e. without decomposing or modifying the network structure. This not only improves performance, but also very significantly simplifies the adoption of the pretraining pass in new application settings.\\nExtending the classic viewpoint of unsupervised autoencoder pretraining, several prior methods employed ”hard orthogonal constraints” to improve weight orthogonality via singular value decomposition (SVD) at training time (Huang et al., 2018; Jia et al., 2017; Ozay & Okatani, 2016). Bansal et al. (Bansal et al., 2018) additionally investigated efficient formulations of the orthogonality constraints. In practice, these constraints are difficult to satisfy, and correspondingly only weakly imposed. In addition, these methods focus on improving performance for a known, given task. This means the training process only extracts features that the network considers useful for improving the performance of the current task, not necessarily improving generalization or transfer performance (Torrey & Shavlik, 2010). While our approach shares similarities with SVD-based constraints, it can be realized with a very efficient L2-based formulation, and takes the full input distribution into account.\\nRecovering all input information from hidden representations of a network is generally very difficult (Dinh et al., 2016; Mahendran & Vedaldi, 2016), due to the loss of information throughout the layer transformations. In this context, (Tishby & Zaslavsky, 2015) proposed the information bottleneck principle, which states that for an optimal representation, information unrelated to the current task is omitted. This highlights the common specialization of conventional training approaches.\\nReversed network architectures were proposed in previous work (Ardizzone et al., 2018; Jacobsen et al., 2018; Gomez et al., 2017), but mainly focus on how to make a network fully invertible via augmenting the network with special structures. As a consequence, the path from input to output is different from the reverse path that translates output to input. Besides, the augmented structures of these approaches can be challenging to apply to general network architectures. In contrast, our approach fully preserves an existing architecture for the backward path, and does not require any operations that were not part of the source network. As such, it can easily be applied in new settings, e.g., adversarial training (Goodfellow et al., 2014). While methods using reverse connections were previously proposed (Zhang et al., 2018a; Teng & Choromanska, 2019), these modules primarily focus on transferring information between layers for a given task, and on auto-encoder structures for domain adaptation, respectively.\"\n },\n {\n \"heading\": \"A.2 PRETRAINING AND SINGULAR VALUE DECOMPOSITION\",\n \"text\": \"In this section we give a more detailed derivation of our loss formulation, extending Section 3 of the main paper. As explained there, our loss formulation aims for minimizing\\nLRR = n∑\\nm=1\\n(MTmMmd i m − dim)2, (11)\\nwhere Mm ∈ Rs out m×s in m denotes the weight matrix of layer m, and data from the input data set Dm is denoted by dim ⊂ Rs in m , i = 1, 2, ..., t. Here t denotes the number of samples in the input data set.\\nMinimizing equation 11 is mathematically equivalent to\\nMTmMmd i m − dim = 0 (12)\\nfor all dim. Hence, perfectly fulfilling equation 11 would require all d i m to be eigenvectors of MTmMm with corresponding eigenvalues being 1. As in Sec. 3 of the main paper, we make use of an auxiliary orthonormal basis Bm : 〈 w1m, ...,w q m 〉 , for which q (with q ≤ t) denotes the number of linearly independent entries inDm. While Bm never has to be explicitly constructed for our method, it can, e.g., be obtained via Gram-Schmidt. The matrix consisting of the vectors in Bm is denoted by Dm.\\nSince the whm(h = 1, 2, ...q) necessarily can be expressed as linear combinations of d i m, equation 11 similarly requires whm to be eigenvectors of M T mMm with corresponding eigenvalues being 1, i.e.:\\nMTmMmw h m −whm = 0 (13)\\nWe denote the vector of coefficients to express dim via Dm with c i m, i.e. d i m = Dmc i m. Then equation 12 can be rewritten as:\\nMTmMmDmc i m −Dmcim = 0 (14)\\nVia an SVD of the matrix Mm in equation 14 we obtain\\nMTmMmDmcm −Dmcm\\n= q∑ h=1 MTmMmw h mcmh −whmcmh\\n= q∑ h=1 VmΣ T mΣmV T mw h mcmh −whmcmh\\n(15)\\nwhere the coefficient vector cm is accumulated over the training data set size t via cm = ∑t i=1 c i m. Here we assume that over the course of a typical training run eventually every single datum in Dm will contribute to LRRm . This form of the loss highlights that minimizing LRR requires an alignment of VmΣTmΣmV T mw h mcmh and w h mcmh .\\nBy construction, Σm contains the square roots of the eigenvalues of MTmMm as its diagonal entries. The matrix has rank r = rank(MTmMm), and since all eigenvalues are required to be 1 by equation 13, the multiplication with Σm in equation 15 effectively performs a selection of r column vectors from Vm. Hence, we can focus on the interaction between the basis vectors wm and the r active column vectors of Vm:\\nVmΣ T mΣmV T mw h mcmh −whmcmh\\n= cmh(VmΣ T mΣmV T mw h m −whm)\\n= cmh( r∑ f=1 (vfm) Twhmv f m −whm).\\n(16)\\nAs Vm is obtained via an SVD it contains r orthogonal eigenvectors of MTmMm. equation 13 requires w1m, ...,w q m to be eigenvectors of M T mMm, but since typically the dimension of the input data set is much larger than the dimension of the weight matrix, i.e. r ≤ q, in practice only r vectors from Bm can fulfill equation 13. This means the vectors v1m, ...,vrm in Vm are a subset of the orthonormal basis vectors Bm : 〈 w1m, ...,w q m 〉 with (whm)\\n2 = 1. Then for any whm we have{ (vfm) Twhm = 1, if v f m = w h m\\n(vfm) Twhm = 0, otherwise.\\n(17)\\nThus if Vm contains whm, we have r∑\\nf=1\\n(vfm) Twhmv f m = w h m, (18)\\nand we trivially fulfill the constraint\\ncmh( r∑ f=1 (vfm) Twhmv f m −whm) = 0. (19)\\nHowever, due to r being smaller than q in practice, Vm typically can not include all vectors from Bm. Thus, if Vm does not contain whm, we have (vfm)Twhm = 0 for every vector vfm in Vm, which means\\nr∑ f=1 (vfm) Twhmv f m = 0. (20)\\nAs a consequence, the constraint equation 12 is only partially fulfilled:\\ncmh( r∑ f=1 (vfm) Twhmv f m −whm) = −cmhwhm . (21)\\nAs the whm have unit length, the factors cm determine the contribution of a datum to the overall loss. A feature whm that appears multiple times in the input data will have a correspondingly larger factor in cm and hence will more strongly contribute to LRR. The L2 formulation of equation 11 leads to the largest contributors being minimized most strongly, and hence the repeating features of the data, i.e., dominant features, need to be represented in Vm to minimize the loss. Interestingly, this argumentation holds when additional loss terms are present, e.g., a loss term for classification. In such a case, the factors cm will be skewed towards those components that fulfill the additional loss terms, i.e. favor basis vectors whm that contain information for about the loss terms. This, e.g., leads to clear digit structures being embedded in the weight matrices for the MNIST example below.\\nIn summary, to minimize LRR, Vm is driven towards containing r orthogonal vectors whm which represent the most frequent features of the input data, i.e. the dominant features. It is worth emphasizing that above Bm is only an auxiliary basis, i.e., the derivation does not depend on any particular choice of Bm.\"\n },\n {\n \"heading\": \"A.3 EXAMPLES OF NETWORK ARCHITECTURES WITH PRETRAINING\",\n \"text\": \"While the proposed pretraining is significantly more easy to integrate into training pipelines than classic autoencoder pretraining, there are subtleties w.r.t. the order of the operations in the reverse pass that we clarify with examples in the following sections. To specify NN architectures, we use the following notation: C(k, l, q), and D(k, l, q) denote convolutional and deconvolutional operations, respectively, while fully connected layers are denoted with F (l), where k, l, q denote kernel size, output channels and stride size, respectively. The bias of a CNN layer is denoted with b. I/O(z) denote input/output, their dimensionality is given by z. Ir denotes the input of the reverse pass network. tanh, relu, lrelu denote hyperbolic tangent, ReLU, and leaky ReLU activation functions (AF), where we typically use a leaky tangent of 0.2 for the negative half-space. UP , MP and BN denote 2× nearest-neighbor up-sampling, max pooling with 2 × 2 filters and stride 2, and batch normalization, respectively.\\nBelow we provide additional examples how to realize the pretraining loss Lrr in a neural network architecture. As explained in the main document, the constraint equation 11 is formulated via\\nLrr = n∑\\nm=1\\nλm ∥∥∥dm − d′m∥∥∥2 F , (22)\\nwith dm, and λm denoting the vector of activated intermediate data in layer m from the forward pass, and a scaling factor, respectively. d ′\\nm denotes the activations of layer m from the reverse pass. E.g., let Lm() denote the operations of a layer m in the foward pass, and L′m() the corresponding operations for the reverse pass. Then dm+1 = Lm(dm), and d ′ m = L ′ m(d ′ m+1).\\nWhen equation 22 is minimized, we obtain activated intermediate content during the reverse pass that reconstructs the values computed in the forward pass, i.e. d ′ m+1 = dm+1 holds. Then d ′\\nm can be reconstructed from the incoming activations from the reverse pass, i.e., d ′\\nm+1, or from the output of layer m, i.e., dm+1. Using d ′ m+1 results in a global coupling of input and output throughout\\na m\\no un\\nt o f s\\nh a\\nre d\\nin fo\\nrm a\\ntio n\\nb et\\nw e\\ne n\\nla ye\\nr a\\nn d\\namount of shared information between layer ℒ and 𝑋\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\nmore information about both 𝑋&𝑌 more information about 𝑌 only\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\nless shared information between layer ℒ and 𝑋\\nle s s\\nsh ar\\ned in\\nfo rm\\nat io n b et w ee n la ye r ℒ a n d\\n𝑌\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\n0.9\\n0.92\\n0.94\\n0.96\\n0.98\\n1\\n1.02\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝐼( 𝒟\\n;𝑌 )\\n0\\n0.1\\n0.2\\n0.3\\n0.4\\n0.5\\n0.6\\n0.7\\n0.8\\n0.9\\n1\\n1.1\\n0 2 4 6 8 10 12\\n𝑂𝑟𝑡\\n𝑅𝑅 𝑅𝑅 𝑆𝑡𝑑\\n𝐼(𝑋; 𝒟 )\\n𝒟 𝒟 𝒟 𝒟 𝒟 𝒟\\n𝑆𝑡𝑑\\n𝐼( 𝒟\\n;𝑌 )\\nmore information about 𝑋 only\\n(a) Mutual Information Plane, How to Read (b) Mutual Information for Task A (c) After fine-tuning for A (d) After fine-tuning for B\\nno information\\nLayers of 𝑅𝑅 models exhibit strong MI with in- & output\\nall layers, i.e., the full loss variant. On the other hand, dm+1 yields a variant that ensures local reversibility of each layer, and yields a very similar performance, as we will demonstrate below. We employ this local loss for networks without a unique, i.e., bijective, connection between two layers. Intuitively, whe inputs cannot be reliably reconstructed from outputs.\\nFull Network Pretraining: An illustration of a CNN structure with AF and BN and a full loss is shown in figure 10. While the construction of the reverse pass is straight-forward for all standard operations, i.e., fully connected layers, convolutions, pooling, etc., slight adjustments are necessary for AF and BN. It is crucial for our formulation that dm and d ′\\nm contain the same latent space content in terms of range and dimensionality, such that they can be compared in the loss. Hence, we use the BN parameters and the AF of layer m − from the forward pass for layer m in the reverse pass. An example is shown in figure 14.\\nTo illustrate this setup, we consider an example network employing convolutions with mixed AFs, BN, and MP. Let the network receives a field of 322 scalar values as input. From this input, 20, 40, and 60 feature maps are extracted in the first three layers. Besides, the kernel sizes are decreased from 5× 5 to 3× 3. To clarify the structure, we use ReLU activation for the first convolution, while the second one uses a hyperbolic tangent, and the third one a sigmoid function. With the notation outlined above, the first three layers of the network are\\nI(32, 32, 1) = d1 → C1(5, 20, 1) + b1 → BN1 → relu → d2 →MP → C2(4, 40, 1) + b2 → BN2 → tanh → d3 →MP → C3(3, 60, 1) + b3 → BN3 → sigm → d4 → ...\\n(23)\\nThe reverse pass for evaluating the loss re-uses all weights of the forward pass and ensures that all intermediate vectors of activations, dm and d ′\\nm, have the same size and content in terms of normalization and non-linearity. We always consider states after activation for Lrr. Thus, dm denotes activations before pooling in the forward pass and d ′\\nm contains data after up-sampling in the reverse pass, in order to ensure matching dimensionality. Thus, the last three layers of the reverse network for computing Lrr take the form:\\n...→ d ′\\n4 → −b3 → D3(3, 40, 1)→ BN2 → tanh→ UP\\n→ d ′\\n3 → −b2 → D2(4, 20, 1)→ BN1 → relu→ UP\\n→ d ′\\n2 → −b1 → D1(5, 3, 1)\\n→ d ′\\n1 = O(32, 32, 1).\\n(24)\\nHere, the de-convolutions Dx in the reverse network share weights with Cx in the forward network. I.e., the 4× 4× 20× 40 weight matrix of C2 is reused in its transposed form as a 4× 4× 40× 20 matrix in D2. Additionally, it becomes apparent that AF and BN of layer 3 from the forward pass do not appear in the listing of the three last layers of the reverse pass. This is caused by the fact that both are required to establish the latent space of the fourth layer. Instead, d3 in our example represents the activations after the second layer (with BN2 and tanh), and hence the reverse pass for d ′ 3 reuses both functions. This ensures that dm and d ′\\nm contain the same latent space content in terms of range and dimensionality, and can be compared in equation 22.\\nFor the reverse pass, we additionally found it beneficial to employ an AF for the very last layer if the output space has suitable content. E.g., for inputs in the form of RGB data we employ an additional activation with a ReLU function for the output to ensure the network generates only positive values.\\nLocalized Pretraining: In the example above, we use a full pretraining with d ′\\nm+1 to reconstruct the activations d ′\\nm. The full structure establishes a slightly stronger relationship among the loss terms of different layers, and allows earlier layers to decrease the accumulated loss of later layers. However, if the architecture of the original network makes use of operations between layers that are not bijective, we instead use the local loss. E.g., this happens for residual connections with an addition or non-invertible pooling operations such as max-pooling. In the former, we cannot uniquely determine the b, c in a = b + c given only a. And unless special care is taken (Bruna et al., 2013), the source neuron of an output is not known for regular max-pooling operations. Note that our loss formulation has no problems with irreversible operations within a layer, e.g., most convolutional or fully-connected layers typically are not fully invertible. In all these cases the loss will drive the network towards a state that is as-invertible-as-possible for the given input data set. However, this requires a reliable vector of target activations in order to apply the constraints. If the connection betweeen layers is not bijective, we cannot reconstruct this target for the constraints, as in the examples given above.\\nIn such cases, we regard every layer as an individual unit to which we apply the constraints by building a localized reverse pass. For example, given a simple convolutional architecture with\\nd1 → C1(5, 20, 1) + b1 = d2 (25) in the forward pass, we calculate d ′\\n1 with\\n(d2 − b1)→ D1(5, 3, 1) = d ′\\n1, (26) We, e.g., use this local loss in the Resnet110 network below. It is important to note that despite being closer to regular autoencoder pretraining, this formulation still incorporates all non-linearities of the original network structure, and jointly trains full networks while taking into account the original learning objective.\\nA.4 MNIST AND PEAK TESTS\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\\n0.312\\n0.392\\n0.298\\n0.334\\n0.231\\n0.114\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n(a) One hidden layer, without BN, without AF\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\\n0.499\\n0.502\\n0.499\\n0.491\\n0.239\\n0.400\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n(b) One hidden layer, with BN, with ReLU AF\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.356\\n0.437\\n0.405\\n0.433\\n0.196\\n0.267\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n… … …\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.200\\n0.110\\n𝑅𝑅\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.233\\n0.202\\n𝑅𝑅\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × ,\\n(c) Two hidden layers, with BN, with ReLU AF\\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.175\\n0.181\\n𝑅𝑅\\n…\\n𝒗\\n0.502\\n0.491\\n0.400\\n𝒗\\n0.440\\n0.395\\n0.524\\n𝒗\\n0.401\\n0.377\\n0.417\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\\n0.395\\n0.528\\n0.420\\n0.465\\n0.225\\n0.441\\n(4)\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆\\n0.582\\n0.447\\n0.389\\n0.411\\n0.214\\n0.370\\n(5)\\n0.3190.500 0.495\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.377 0.426 0.408\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3360.407 0.412\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3330.462 0.443\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.2920.514 0.400\\nSTDEV LPIPS 0.002 0.006 0.114 0.020 0.017 0.209 STDEV LPIPS 0.009 0.050 0.115 STDEV LPIPS\\n0.094 0.032 0.153 STDEV LPIPS\\n0.095 0.016 0.110 STDEV LPIPS\\n0.202\\n0.217\\n𝑅𝑅\\n0.022\\n0.244\\n𝑅𝑅 0.243\\n0.001\\n0.219\\n𝑅𝑅 0.183\\n0.051\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.142\\n0.232\\n𝑅𝑅 0.187\\n0.064\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.306\\n0.240\\n𝑅𝑅\\n0.273\\n0.047\\nBelow we give details for the peak tests from Sec. 3 of the main paper and show additional tests with the MNIST data set.\\nPeak Test: For the Peak test we generated a data set of 110 images shown in figure 12. 55 images contain a peak located in the upper left corner of the image. The other 55 contain a peak located in the bottom right corner. We added random scribbles in the images to complicate the task. All 110 images were labeled with a one-hot encoding of the two possible positions of the peak. We use 100 images as training data set, and the remaining 10 for testing. All peak models are trained for 5000 epochs with a learning rate of 0.0001, with λ = 1e − 6 for RRA. To draw reliable conclusions, we show results for five repeated runs here. The neural network in this case contains one fully connected layer, with BN and ReLU activation. The results are shown in figure 13, with both peak modes being consistently embedded into the weight matrix of RRA, while regular and orthogonal training show primarily random singular vectors.\\nWe also use different network architectures in figure 14 to verify that the dominant features are successfully extracted when using more complex\\n16\\ntraining data test data\\nFigure 12: Data set used for the peak tests.\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.312\\n0.392\\n0.298\\n0.334\\n0.231\\n0.114\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n(a) One hidden layer, without BN, without AF\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.499\\n0.502\\n0.499\\n0.491\\n0.239\\n0.400\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n(b) One hidden layer, with BN, with ReLU AF\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.356\\n0.437\\n0.405\\n0.433\\n0.196\\n0.267\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × (c) Two hidden layers, with BN, with ReLU AF\\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n… … …\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.200\\n0.110\\n𝑅𝑅\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.233\\n0.202\\n𝑅𝑅\\n𝒅 ∈ ℝ × , 𝑀 ∈ ℝ × ,\\n(c) Two hidden layers, with BN, with ReLU AF\\n𝑀 ∈ ℝ × , 𝒅 ∈ ℝ ×\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.175\\n0.181\\n𝑅𝑅\\n… 𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.499\\n0.502\\n0.499\\n0.491\\n0.239\\n0.400\\n𝑅𝑒𝑝𝑒𝑎𝑡𝑒𝑑 𝑃𝑒𝑎𝑘 𝑇𝑒𝑠𝑡 𝑤𝑖𝑡ℎ 𝐵𝑁, 𝑎𝑛𝑑 𝑅𝑒𝐿𝑈 𝐴𝐹 (1)\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.412\\n0.440\\n0.420\\n0.395\\n0.229\\n0.524\\n(2)\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.414\\n0.401\\n0.448\\n0.377\\n0.255\\n0.417\\n(3)\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.395\\n0.528\\n0.420\\n0.465\\n0.225\\n0.441\\n(4)\\n𝒗\\n𝒗\\n𝐋𝐏𝐈𝐏𝐒𝐒𝐭𝐝 𝐋𝐏𝐈𝐏𝐒𝐎𝐫𝐭 𝐋𝐏𝐈𝐏𝐒𝑷𝒓𝒆 0.582\\n0.447\\n0.389\\n0.411\\n0.214\\n0.370\\n(5)\\n0.3190.500 0.495\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.377 0.426 0.408\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3360.407 0.412\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.3330.462 0.443\\n𝑆𝑡𝑑 𝑂𝑟𝑡 𝑃𝑟𝑒 Avg. LPIPS 0.2920.514 0.400\\nSTDEV LPIPS 0.002 0.006 0.114 0.020 0.017 0.209 STDEV LPIPS 0.009 0.050 0.115 STDEV LPIPS\\n0.094 0.032 0.153 STDEV LPIPS\\n0.095 0.016 0.110 STDEV LPIPS\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹 0.233\\n0.202\\n0.217\\n𝑅𝑅\\n0.022\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.243\\n0.244\\n𝑅𝑅 0.243\\n0.001\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.147\\n0.219\\n𝑅𝑅 0.183\\n0.051\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.142\\n0.232\\n𝑅𝑅 0.187\\n0.064\\n𝐋𝐏𝐈𝐏𝐒𝑹𝑹\\n0.306\\n0.240\\n𝑅𝑅\\n0.273\\n0.047\\nFigure 13: Five repeated tests with the peak data shown in Sec. 3 of the main paper. RRA robustly extracts dominant features from the data set. The two singular vectors strongly resemble the two peak modes of the training data. This is confirmed by the LPIPS measurements.\\nnetwork structures. Even for two layers with BN and ReLU activations, our pretraining clearly extracts the two modes of the training data. The visual resemblance is slightly reduced in this case, as the network has the freedom to embed the features in both layers. Across all three cases (for which we performed 5 runs each), our pretraining clearly outperforms regular training and the orthogonality constraint in terms of extracting and embedding the dominant structures of the training data set in the weight matrix.\\nMNIST Test: We additionally verify that the column vectors of Vm of models from RR training contain the dominant features of the input with MNIST tests, which employ a single fully connected layer, i.e. d2 = M1d1. In the first MNIST test, the training data consists only of 2 different images. All MNIST models are trained for 1000 epochs with a learning rate of 0.0001, and λ = 1e − 5 for RRA. After training, we compute the SVD for M1. SVDs of the weight matrices of trained models can be seen in figure 11. The LPIPS scores show that features embedded in the weights of RR are consistently closer to the training data set than all other methods, i.e., regular training Std, classic autoencoder pretraining Pre, and regularization via orthogonalization Ort. While the vectors of Std and Ort contain no recognizable structures.\\nOverall, our experiments confirm the motivation of our pretraining formulation. They additionally show that employing an SVD of the network weights after our pretraining yields a simple and convenient method to give humans intuition about the features learned by a network.\"\n },\n {\n \"heading\": \"B MUTUAL INFORMATION\",\n \"text\": \"This section gives details of the mutual information and disentangled representation tests from Sec. 4 of the main paper.\"\n },\n {\n \"heading\": \"B.1 MUTUAL INFORMATION TEST\",\n \"text\": \"Mutual information (MI) measures the dependence of two random variables, i.e., higher MI means that there is more shared information between two parameters. More formally, the mutual information I(X;Y ) of random variables X and Y measures how different the joint distribution of X and Y is w.r.t. the product of their marginal distributions, i.e., the Kullback-Leibler divergence I(X;Y ) = KL[P(X,Y )||PXPY ], where KL denotes the Kullback-Leibler divergence. Let I(X;Dm) denote the mutual information between the activations of a layer Dm and input X. Similarly I(Dm;Y ) denotes the MI between layer m and the output Y . We use MI planes in the main paper, which show I(X;Dm) and I(Dm;Y ) in a 2D graph for the activations of each layer Dm of a network after training. This visualizes how much information about input and output distribution is retained at each layer, and how these relationships change within the network. For regular training, the information bottleneck principle (Tishby & Zaslavsky, 2015) states that early layers contain more information about the input, i.e., show high values for I(X;Dm) and I(Dm;Y ). Hence in the MI plane visualizations, these layers are often visible at the top-right corner. Later layers typically share a large amount of information with the output after training, i.e. show large I(Dm;Y ) values, and correlate less with the input (low I(X;Dm)). Thus, they typically show up in the top-left corner of the MI plane graphs.\\nTraining Details: We use the same numerical studies as in (Shwartz-Ziv & Tishby, 2017) as task A, i.e. a regular feed-forward neural network with 6 fully-connected layers. The input variable X contains 12 binary digits that represent 12 uniformly distributed points on a 2D sphere. The learning objective is to discover binary decision rules which are invariant under O(3) rotations of the sphere. X has 4096 different patterns, which are divided into 64 disjoint orbits of the rotation group, forming a minimal sufficient partition for spherically symmetric rules (Kazhdan et al., 2003). To generate the input-output distribution P (X,Y ), We apply the stochastic rule p(y = 1|x) = Ψ(f(x) − θ), (x ∈ X, y ∈ Y ), where Ψ is a standard sigmoidal function Ψ(u) = 1/(1 + exp(−γu)), following (Shwartz-Ziv & Tishby, 2017). We then use a spherically symmetric real valued function of the pattern f(x), evaluated through its spherical harmonics power spectrum (Kazhdan et al., 2003), and compare with a threshold θ, which was selected to make p(y = 1) = ∑ x p(y = 1|x)p(x) ≈ 0.5, with uniform p(x). γ is high enough to keep the mutual information I(X;Y ) ≈ 0.99 bits.\\nFor the transfer learning taskB, we reverse output labels to check whether the model learned specific or generalizing features. E.g., if the output is [0,1] in the original data set, we swap the entries to [1,0]. 80% of the data (3277 data pairs) are used for training and rests (819 data pairs) are used for testing.\\nFor the MI comparison in Fig. 4 of the main paper, we discuss models before and after fine-tuning separately, in order to illustrate the effects of regularization. We include a model with greedy layerwise pretraining Pre, a regular model StdA, one with orthogonality constraints OrtA, and our regular model RRA, all before fine-tuning. For the model RRA all layers are constrained to be recovered in the backward pass. We additionally include the version RR1A, i.e. a model trained with only one loss term λ1|d1 − d ′\\n1|2, which means that only the input is constrained to be recovered. Thus, RR1A represents a simplified version of our approach which receives no constraints that intermediate results of the forward and backward pass should match. For OrtA, we used the Spectral Restricted Isometry Property (SRIP) regularization (Bansal et al., 2018),\\nLSRIP = βσ(WTW − I), (27)\\nwhere W is the kernel, I denotes an identity matrix, and β represents the regularization coefficient. σ(W ) = supz∈Rn,z 6=0 ‖Wz‖ ‖z‖ denotes the spectral norm of W .\\nAs explained in the main text, all layers of the first stage, i.e. from RRA, RR 1 A, OrtA , PreA and StdA are reused for training the fine-tuned models without regularization, i.e. RRAA, RR 1 AA, OrtAA , PreAA and StdAA. Likewise, all layers of the transfer task models RRAB, RR 1 AB, OrtAB , PreAB and StdAB are initialized from the models of the first training stage.\\nAnalysis of Results: We first compare the version only constraining input and output reconstruction (RR1A) and the full loss version RRA. Fig. 4(b) of the main paper shows that all points of RRA are located in a central region of the MI place, which means that all layers successfully encode information about the inputs as well as the outputs. This also indicates that every layer contains a similar amount of information about X and Y , and that the path from input to output is similar to the path from output to input. The points of RR1A, on the other hand, form a diagonal line. I.e., this network has different amounts of mutual information across its layers, and potentially a very different path for each direction. This difference in behavior is caused by the difference of the constraints in these two versions: RR1A is only constrained to be able to regenerate its input, while the full loss for RRA ensures that the network learns features which are beneficial for both directions. This test highlights the importance of the constraints throughout the depth of a network in our formulation. In contrast, the I(X;D) values of later layers for StdA and OrtA exhibit small values (points near the left side), while I(D;Y ) is high throughout. This indicates that the outputs were successfully encoded and that increasing amounts of information about the inputs are discarded. Hence, more specific features about the given output data-set are learned by StdA and OrtA. This shows that both models are highly specialized for the given task, and potentially perform worse when applied to new tasks. PreA only focuses on decreasing the reconstruction loss, which results in high I(X;D) values for early layers, and low I(D;Y ) values for later layers. During the fine-tuning phase for task A (i.e. regularizers being disabled), all models focus on the output and maximize I(D;Y ). There are differences in the distributions of the points along the y-axis, i.e., how much MI with the output is retained, as shown in Fig. 4(c) of the main paper. For model RRAA, the I(D;Y ) value is higher than for StdAA, OrtAA, PreAA and RR1AA, which means outputs of RRAA are more closely related to the outputs, i.e., the ground truth labels for task A. Thus, RRAA outperforms the other variants for the original task.\\nIn the fine-tuning phase for task B, StdAB stands out with very low accuracy in Fig. 5 of the main paper. This model from a regular training run has large difficulties to adapt to the new task. PreA aims at extracting features from inputs and reconstructed them. PreAB outperforms StdAB, which means features helpful for task B are extracted by PreA, however, it’s hard to guide the feature extracting process. Model OrtAB also performs worse than StdB. RRAB shows the best performance in this setting, demonstrating that our loss formulation yielded more generic features, improving the performance for related tasks such as the inverted outputs for B.\\nWe also analyze the two variants of our pretraining: the local variant lRRA and the full version RRA in terms of mutual information. figure 15 shows the MI planes for these two models, also\\nshowing RR1A for comparison. Despite the local nature of lRRA it manages to establish MI for the majority of the layers, as indicated by the cluster of layers in the center of the MI plane. Only the first layer moves towards the top right corner, and the second layer is affected slightly. I.e., these layers exhibit a stronger relationship with the distribution of the outputs. Despite this, the overall performance when fine-tuning or for the task transfer remains largely unaffected, e.g., the lRRA still clearly outperforms RR1A. This confirms our choice to use the full pretraining when network connectivity permits, and employ the local version in all other cases.\"\n },\n {\n \"heading\": \"B.2 DISENTANGLED REPRESENTATIONS\",\n \"text\": \"The InfoGAN approach (Chen et al., 2016) demonstrated the possibility to control the output of generative models via maximizing mutual information between outputs and structured latent variables. However, mutual information is very hard to estimate in practice (Walters-Williams & Li, 2009). The previous section and Fig. 4(b) of the main paper demonstrated that models from our pretraining (both RR1A and RRA) can increase the mutual information between network inputs and outputs. Intuitively, the pretraining explicitly constrains the model to recover an input given an output, which directly translates into an increase of mutual information between input and output distributions compared to regular training runs. For highlighting how our pretraining can yield disentangled representations (as discussed in the later paragraphs of Sec. 4 of the main text), we follow the experimental setup of InfoGAN (Chen et al., 2016): the input dimension of our network is 74, containing 1 ten-dimensional category code c1, 2 continuous latent codes c2, c3 ∼ U(−1, 1) and 62 noise variables. Here, U denotes a uniform distribution.\\nTraining Details: As InfoGAN focuses on structuring latent variables and thus only increases the mutual information between latent variables and the output, we also focus the pretraining on the corresponding latent variables. I.e., the goal is to maximize their mutual information with the output of the generative model. Hence, we train a model RR1 for which only latent dimensions c1, c2, c3 of the input layer are involved in the loss. We still employ a full reverse pass structure in the neural network architecture. c1 is a ten-dimensional category code, which is used for controlling the output digit category, while c2 and c3 are continuous latent codes, to represent (previously unknown) key properties of the digits, such as orientation or thickness. Building relationship between c1 and outputs is more difficult than for c2 or c3, since the 10 different digit outputs need to be encoded in a sinlge continuous variable c1. Thus, for the corresponding loss term for c1 we use a slightly larger λ factor (by 33%) than for c2 and c3. Details of our results are shown in figure 16. Models are trained using a GAN loss (Goodfellow et al., 2014) as the loss function for the outputs.\\nAnalysis of Results: In figure 16 we show additional results for the disentangling test case. It is visible that our pretraining of the RR1 model yields distinct and meaningful latent space dimensions for c1,2,3. While c1 controls the digit, c2,3 control the style and orientation of the digits. For comparison, a regular training run with model Std does result in meaningful or visible changes when adjusting the latent space dimensions. This illustrates how strongly the pretraining can shape the\\nUnder review as a conference paper at ICLR 2021\\nVarying 𝑐\\nV arying in put no ise\\nV arying in put no ise\\nVarying 𝑐\\nV arying in put no ise\\nVarying 𝑐\\nV arying in put no ise\\nVarying 𝑐\\nlatent space, and in addition to an intuitive embedding of dominant features, yield a disentangled representation.\"\n },\n {\n \"heading\": \"C DETAILS OF EXPERIMENTAL RESULTS\",\n \"text\": \"C.1 TEXTURE-SHAPE BENCHMARK\\n0.208\\n0.085\\n0.159\\n0.442\\n0.158\\n0.331\\n0.470\\n0.203\\n0.372\\n0.547\\n0.237\\n0.408\\nS T Y L I Z E D D A T A E D G E D A T A F I L L E D D A T A\\nEdge data setStylized data set Filled data set\\nAccuracy Comparisons of 𝑂𝑟𝑡 , 𝑆𝑡𝑑 and 𝑅𝑅\\n𝑆𝑡𝑑 𝑅𝑅\\n𝑂𝑟𝑡\\nEdge data\\nFilled data\\nStylized data\\n(𝑎) (𝑏) (𝑐) (𝑑)\\n𝑃𝑟𝑒\\nTraining Details: All training data of the texture-shape tests were obtained from (Geirhos et al., 2018). The stylized data set contains 1280 images, 1120 images are used as training data, and 160 as test data. Both edge and filled data sets contain 160 images each, all of which are used for testing only. All three sets (stylized, edge, and filled) contain data for 16 different classes.\\nAnalysis of Results: For a detailed comparison, we list per-class accuracy of stylized data training runs for OrtTS, StdTS, PreTS and RRTS in figure 17. RRTS out-\\nperforms the other three models for most of the classes. RRTS requires an additional 41.86% for training compared to StdTS, but yields a 23.76% higher performance. (Training times for these models are given in the supplementary document.) All models saturated, i.e. training StdTS or OrtTS longer does not increase classification accuracy any further. We also investigated how much we can reduce model size when using our pretraining in comparison to the baselines. A reduced model only uses 67.94% of the parameters, while still outperforming OrtTS.\"\n },\n {\n \"heading\": \"C.2 GENERATIVE ADVERSARIAL MODELS\",\n \"text\": \"Training Details: The data set of smoke simulation was generated with a Navier-Stokes solver from an open-source library (Thuerey & Pfaff, 2018). We generated 20 randomized simulations with 120 frames each, with 10% of the data being used for training. The low-resolution data were down-sampled from the high-resolution data by a factor of 4. Data augmentation, such as flipping and rotation was used in addition. As outlined in the main text, we consider building an autoencoder model for the synthetic data as task B1, and a generating samples from a real-world smoke data set as task B2. The smoke capture data set for B2 contains 2500 smoke images from the ScalarFlow data set (Eckert et al., 2019), and we again used 10% of these images as training data set.\\nTask A: We use a fully convolutional CNN-based architecture for generator and discriminator networks. Note that the inputs of the discriminator contain high resolution data (64, 64, 1), as well as low resolution (16, 16, 1), which is up-sampled to (64, 64, 1) and concatenated with the high resolution data. In line with previous work (Xie et al., 2018), RRA and StdA are trained with a\\nnon-saturating GAN loss, feature space loss and L2 loss as base loss function. All generator layers are involved in the pretraining loss. As greedy layer-wise autoencoder pretraining is not compatible with adversarial training, we pretrain PreA for reconstructing the high resolution data instead.\\nTask B1: All encoder layers are initialized from RRA and StdA when training RRAB1 and StdAB1 . It is worth noting that the reverse pass of the generator is also constrained when training PreA and RRA. So both encoder and decoder are initialized with parameters from PreA and RRA when training PreAB1 and RRAB1 , respectively. This is not possible for a regular network like StdAB1 , as the weights obtained with a normal training run are not suitable to be transposed. Hence, the deconvolutions of StdAB1 are initialized randomly.\\nTask B2: As the data set for the task B2 is substantially different and contains RBG images (instead of single channel gray-scale images), we choose the following setups for the RRA, PreA and StdA models: parameters from all six layers of StdA and RRA are reused for initializing decoder part of StdAB2 and RRAB2 , parameters from all six layers of PreA are reused for initializing the encoder part of PreAB2 . Specially, when initializing the last layer of PreAB2 , StdAB2 and RRAB2 , we copy and stack the parameters from the last layer of PreA, StdA and RRA, respectively, into three channels to match the dimenions of the outputs for taskB2. Here, the encoder part of RRAB2 and the decoder of PreAB2 are not initialized with RRA and PreA, due to the significant gap between training data sets of taskB1 and taskB2. Our experiments show that only initializing the decoder part of RRAB2 (avg. loss:1.56e7, std. dev.:3.81e5) outperforms initializing both encoder and decoder (avg. loss:1.82e7± 2.07e6), and only initializing the encoder part of PreAB2 (avg. loss:4.41e7 ± 6.36e6) outperforms initializing both encoder and decoder (avg. loss:9.42e7 ± 6.11e7). We believe the reason is that initializing both encoder and decoder part makes it more difficult to adjust the parameters for new data set that is very different from the data set of the source task.\\nAnalysis of Results: Example outputs of PreAB1 , StdAB1 and RRAB1 are shown in figure 18. It is clearly visible that RRAB1 gives the best performance among these models. We similarly illustrate the behavior of the transfer learning task B2 for images of real-world fluids. This example\\n387 1057 2070 1990 2005 2189\\nlikewise uses an autoencoder structure. Visual comparisons are provided in figure 19, where RRAB2 generates results that are closer to the reference. Overall, these results demonstrate the benefits of our pretraining for GANs, and indicate its potential to obtain more generic features from synthetic data sets that can be used for tasks involving real-world data.\"\n },\n {\n \"heading\": \"C.3 WEATHER FORECASTING\",\n \"text\": \"Training Details: The weather forecasting scenario discussed in the main text follows the methodology of the WeatherBench benchmark (Rasp et al., 2020). This benchmark contains 40 years of data from the ERA5 reanalysis project Hersbach et al. (2020) which was re-sampled to a 5.625◦ resolution, yielding 32× 64 grid points in ca. two-hour intervals. Data from the year of 1979 to 2015 (i.e., 162114 samples) are used for training, the year of 2016 for validation. The last two years (2017 and 2018) are used as test data. All RMSE measurements are latitude-weighted to account for area distortions from the spherical projection.\\nThe neural networks for the forecasting tasks employ a ResNet architecture with 19 layers, all of which contain 128 features with 3 × 3 kernels (apart from 7 × 7 in the first layer). All layers use batch normalization, leaky ReLU activation (tangent 0.3), and dropout with strength 0.1. As inputs, the model receives feature-wise concatenated data from the WeatherBench data for 3 consecutive time steps, i.e., t, t − 6h, and t − 12h, yielding 117 channels in total. The last convolution jointly generates all three output fields, i.e., pressure at 500 hPa (Z500), temperature at 850 hPa (T850), and the 2-meter temperature (T2M).\\nAnalysis of Results: In addition to the quantitative results for both years of test data given in the main text, figure 20 and 21 contain additional example visualizations from the test data set. A visualization of the spatial error distribution w.r.t. ground truth result in also shown in figure 21. It becomes apparent that our pretraining achieves reduced errors across the whole range of samples. Both temperature targets contain a larger number of smaller scale features than the pressure fields. While the gains from our pretraining approach are not huge (on the order of 3% in both cases), they represent important steps forward. The learning objective is highly non-trivial, and the improvements were achieved with the same limited set of training data. Being very easy to integrate into existing training pipelines, these results indicate that the proposed pretraining methodology has the potential to yield improved learning results for a wide range of problem settings.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":191,"cells":{"paper_id":{"kind":"string","value":"SP:70fc08b1b6161c770b5019272c2eaa0d2e3c39ee"},"summaries":{"kind":"list like","value":["This paper raises and studies concerns about the generalization of 3D human motion prediction approaches across unseen motion categories. The authors address this problem by augmenting existing architectures with a VAE framework. More precisely, an encoder network that is responsible for summarizing the seed sequence is shared by two decoders for the reconstruction of the seed motion and prediction of the future motion. Hence, the encoder is trained by using both the ELBO of a VAE and the objective of the original motion prediction task. "],"string":"[\n \"This paper raises and studies concerns about the generalization of 3D human motion prediction approaches across unseen motion categories. The authors address this problem by augmenting existing architectures with a VAE framework. More precisely, an encoder network that is responsible for summarizing the seed sequence is shared by two decoders for the reconstruction of the seed motion and prediction of the future motion. Hence, the encoder is trained by using both the ELBO of a VAE and the objective of the original motion prediction task. \"\n]","isConversational":false},"abstractText":{"kind":"string","value":"The task of predicting human motion is complicated by the natural heterogeneity and compositionality of actions, necessitating robustness to distributional shifts as far as out-of-distribution (OoD). Here we formulate a new OoD benchmark based on the Human3.6M and CMU motion capture datasets, and introduce a hybrid framework for hardening discriminative architectures to OoD failure by augmenting them with a generative model. When applied to current state-of-theart discriminative models, we show that the proposed approach improves OoD robustness without sacrificing in-distribution performance, and can theoretically facilitate model interpretability. We suggest human motion predictors ought to be constructed with OoD challenges in mind, and provide an extensible general framework for hardening diverse discriminative architectures to extreme distributional shift."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Emre Aksan","Manuel Kaufmann","Otmar Hilliges"],"title":"Structured prediction helps 3d human motion modelling","venue":"In Proceedings of the IEEE International Conference on Computer Vision,","year":2019},{"authors":["Alexandre Alahi","Kratarth Goel","Vignesh Ramanathan","Alexandre Robicquet","Li Fei-Fei","Silvio Savarese"],"title":"Social lstm: Human trajectory prediction in crowded spaces","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2016},{"authors":["Dario Amodei","Chris Olah","Jacob Steinhardt","Paul Christiano","John Schulman","Dan Mané"],"title":"Concrete problems in ai safety","venue":"arXiv preprint arXiv:1606.06565,","year":2016},{"authors":["Apratim Bhattacharyya","Mario Fritz","Bernt Schiele"],"title":"Long-term on-board prediction of people in traffic scenes under uncertainty","venue":"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,","year":2018},{"authors":["Anthony Bourached","Parashkev Nachev"],"title":"Unsupervised videographic analysis of rodent behaviour","venue":"arXiv preprint arXiv:1910.11065,","year":2019},{"authors":["Judith Butepage","Michael J Black","Danica Kragic","Hedvig Kjellstrom"],"title":"Deep representation learning for human motion prediction and classification","venue":"In Proceedings of the IEEE conference on computer vision and pattern recognition,","year":2017},{"authors":["Yujun Cai","Lin Huang","Yiwei Wang","Tat-Jen Cham","Jianfei Cai","Junsong Yuan","Jun Liu","Xu Yang","Yiheng Zhu","Xiaohui Shen"],"title":"Learning progressive joint propagation for human motion prediction","venue":"In Proceedings of the European Conference on Computer Vision (ECCV),","year":2020},{"authors":["Zhe Cao","Gines Hidalgo","Tomas Simon","Shih-En Wei","Yaser Sheikh"],"title":"Openpose: realtime multiperson 2d pose estimation using part affinity fields","venue":"arXiv preprint arXiv:1812.08008,","year":2018},{"authors":["Chien-Yen Chang","Belinda Lange","Mi Zhang","Sebastian Koenig","Phil Requejo","Noom Somboon","Alexander A Sawchuk","Albert A Rizzo"],"title":"Towards pervasive physical rehabilitation using microsoft kinect","venue":null,"year":2012},{"authors":["Nutan Chen","Justin Bayer","Sebastian Urban","Patrick Van Der Smagt"],"title":"Efficient movement representation by embedding dynamic movement primitives in deep autoencoders","venue":"In 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids),","year":2015},{"authors":["Erik Daxberger","José Miguel Hernández-Lobato"],"title":"Bayesian variational autoencoders for unsupervised out-of-distribution detection","venue":"arXiv preprint arXiv:1912.05651,","year":2019},{"authors":["Katerina Fragkiadaki","Sergey Levine","Panna Felsen","Jitendra Malik"],"title":"Recurrent network models for human dynamics","venue":"In Proceedings of the IEEE International Conference on Computer Vision, pp","year":2015},{"authors":["Evelien E Geertsema","Roland D Thijs","Therese Gutter","Ben Vledder","Johan B Arends","Frans S Leijten","Gerhard H Visser","Stiliyan N Kalitzin"],"title":"Automated video-based detection of nocturnal convulsive seizures in a residential care","venue":"setting. 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Journal of neuroengineering and rehabilitation,","year":2014},{"authors":["A. say"],"title":"It may be useful to know that the patient’s deviation from a classical case of A, is in the direction of condition, say, B. We trained the augmented GCN model discussed in the main text with all actions, for both datasets. We use Uniform Manifold Approximation and Projection (UMAP) (McInnes et al., 2018) to project the latent space of the trained GCN models onto 2 dimensions for all samples in the dataset","venue":null,"year":2018}],"string":"[\n {\n \"authors\": [\n \"Emre Aksan\",\n \"Manuel Kaufmann\",\n \"Otmar Hilliges\"\n ],\n \"title\": \"Structured prediction helps 3d human motion modelling\",\n \"venue\": \"In Proceedings of the IEEE International Conference on Computer Vision,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Alexandre Alahi\",\n \"Kratarth Goel\",\n \"Vignesh Ramanathan\",\n \"Alexandre Robicquet\",\n \"Li Fei-Fei\",\n \"Silvio Savarese\"\n ],\n \"title\": \"Social lstm: Human trajectory prediction in crowded spaces\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Dario Amodei\",\n \"Chris Olah\",\n \"Jacob Steinhardt\",\n \"Paul Christiano\",\n \"John Schulman\",\n \"Dan Mané\"\n ],\n \"title\": \"Concrete problems in ai safety\",\n \"venue\": \"arXiv preprint arXiv:1606.06565,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Apratim Bhattacharyya\",\n \"Mario Fritz\",\n \"Bernt Schiele\"\n ],\n \"title\": \"Long-term on-board prediction of people in traffic scenes under uncertainty\",\n \"venue\": \"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Anthony Bourached\",\n \"Parashkev Nachev\"\n ],\n \"title\": \"Unsupervised videographic analysis of rodent behaviour\",\n \"venue\": \"arXiv preprint arXiv:1910.11065,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Judith Butepage\",\n \"Michael J Black\",\n \"Danica Kragic\",\n \"Hedvig Kjellstrom\"\n ],\n \"title\": \"Deep representation learning for human motion prediction and classification\",\n \"venue\": \"In Proceedings of the IEEE conference on computer vision and pattern recognition,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yujun Cai\",\n \"Lin Huang\",\n \"Yiwei Wang\",\n \"Tat-Jen Cham\",\n \"Jianfei Cai\",\n \"Junsong Yuan\",\n \"Jun Liu\",\n \"Xu Yang\",\n \"Yiheng Zhu\",\n \"Xiaohui Shen\"\n ],\n \"title\": \"Learning progressive joint propagation for human motion prediction\",\n \"venue\": \"In Proceedings of the European Conference on Computer Vision (ECCV),\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Zhe Cao\",\n \"Gines Hidalgo\",\n \"Tomas Simon\",\n \"Shih-En Wei\",\n \"Yaser Sheikh\"\n ],\n \"title\": \"Openpose: realtime multiperson 2d pose estimation using part affinity fields\",\n \"venue\": \"arXiv preprint arXiv:1812.08008,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Chien-Yen Chang\",\n \"Belinda Lange\",\n \"Mi Zhang\",\n \"Sebastian Koenig\",\n \"Phil Requejo\",\n \"Noom Somboon\",\n \"Alexander A Sawchuk\",\n \"Albert A Rizzo\"\n ],\n \"title\": \"Towards pervasive physical rehabilitation using microsoft kinect\",\n \"venue\": null,\n \"year\": 2012\n },\n {\n \"authors\": [\n \"Nutan Chen\",\n \"Justin Bayer\",\n \"Sebastian Urban\",\n \"Patrick Van Der Smagt\"\n ],\n \"title\": \"Efficient movement representation by embedding dynamic movement primitives in deep autoencoders\",\n \"venue\": \"In 2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids),\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Erik Daxberger\",\n \"José Miguel Hernández-Lobato\"\n ],\n \"title\": \"Bayesian variational autoencoders for unsupervised out-of-distribution detection\",\n \"venue\": \"arXiv preprint arXiv:1912.05651,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Katerina Fragkiadaki\",\n \"Sergey Levine\",\n \"Panna Felsen\",\n \"Jitendra Malik\"\n ],\n \"title\": \"Recurrent network models for human dynamics\",\n \"venue\": \"In Proceedings of the IEEE International Conference on Computer Vision, pp\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Evelien E Geertsema\",\n \"Roland D Thijs\",\n \"Therese Gutter\",\n \"Ben Vledder\",\n \"Johan B Arends\",\n \"Frans S Leijten\",\n \"Gerhard H Visser\",\n \"Stiliyan N Kalitzin\"\n ],\n \"title\": \"Automated video-based detection of nocturnal convulsive seizures in a residential care\",\n \"venue\": \"setting. Epilepsia,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Anand Gopalakrishnan\",\n \"Ankur Mali\",\n \"Dan Kifer\",\n \"Lee Giles\",\n \"Alexander G Ororbia\"\n ],\n \"title\": \"A neural temporal model for human motion prediction\",\n \"venue\": \"In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"F Sebastian Grassia\"\n ],\n \"title\": \"Practical parameterization of rotations using the exponential map\",\n \"venue\": \"Journal of graphics tools,\",\n \"year\": 1998\n },\n {\n \"authors\": [\n \"Will Grathwohl\",\n \"Kuan-Chieh Wang\",\n \"Jörn-Henrik Jacobsen\",\n \"David Duvenaud\",\n \"Mohammad Norouzi\",\n \"Kevin Swersky\"\n ],\n \"title\": \"Your classifier is secretly an energy based model and you should treat it like one\",\n \"venue\": null,\n \"year\": 1912\n },\n {\n \"authors\": [\n \"Liang-Yan Gui\",\n \"Yu-Xiong Wang\",\n \"Xiaodan Liang\",\n \"José MF Moura\"\n ],\n \"title\": 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Learning,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Ahmadreza Reza Rofougaran\",\n \"Maryam Rofougaran\",\n \"Nambirajan Seshadri\",\n \"Brima B Ibrahim\",\n \"John Walley\",\n \"Jeyhan Karaoguz\"\n ],\n \"title\": \"Game console and gaming object with motion prediction modeling and methods for use therewith\",\n \"venue\": \"US Patent 9,943,760\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Akihiko Shirai\",\n \"Erik Geslin\",\n \"Simon Richir\"\n ],\n \"title\": \"Wiimedia: motion analysis methods and applications using a consumer video game controller\",\n \"venue\": \"In Proceedings of the 2007 ACM SIGGRAPH symposium on Video games,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Ilya Sutskever\",\n \"Geoffrey E Hinton\",\n \"Graham W Taylor\"\n ],\n \"title\": \"The recurrent temporal restricted boltzmann machine\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Graham W Taylor\",\n \"Geoffrey E Hinton\",\n \"Sam T Roweis\"\n ],\n \"title\": \"Modeling human motion using binary\",\n \"venue\": null,\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Mao Wei\",\n \"Liu Miaomiao\",\n \"Salzemann Mathieu\"\n ],\n \"title\": \"History repeats itself: Human motion prediction\",\n \"venue\": \"rehabilitation. Journal of neuroengineering and rehabilitation,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"A. say\"\n ],\n \"title\": \"It may be useful to know that the patient’s deviation from a classical case of A, is in the direction of condition, say, B. We trained the augmented GCN model discussed in the main text with all actions, for both datasets. We use Uniform Manifold Approximation and Projection (UMAP) (McInnes et al., 2018) to project the latent space of the trained GCN models onto 2 dimensions for all samples in the dataset\",\n \"venue\": null,\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Human motion is naturally intelligible as a time-varying graph of connected joints constrained by locomotor anatomy and physiology. Its prediction allows the anticipation of actions with applications across healthcare (Geertsema et al., 2018; Kakar et al., 2005), physical rehabilitation and training (Chang et al., 2012; Webster & Celik, 2014), robotics (Koppula & Saxena, 2013b;a; Gui et al., 2018b), navigation (Paden et al., 2016; Alahi et al., 2016; Bhattacharyya et al., 2018; Wang et al., 2019), manufacture (Švec et al., 2014), entertainment (Shirai et al., 2007; Rofougaran et al., 2018; Lau & Chan, 2008), and security (Kim & Paik, 2010; Ma et al., 2018).\nThe favoured approach to predicting movements over time has been purely inductive, relying on the history of a specific class of movement to predict its future. For example, state space models (Koller & Friedman, 2009) enjoyed early success for simple, common or cyclic motions (Taylor et al., 2007; Sutskever et al., 2009; Lehrmann et al., 2014). The range, diversity and complexity of human motion has encouraged a shift to more expressive, deep neural network architectures (Fragkiadaki et al., 2015; Butepage et al., 2017; Martinez et al., 2017; Li et al., 2018; Aksan et al., 2019; Mao et al., 2019; Li et al., 2020b; Cai et al., 2020), but still within a simple inductive framework.\nThis approach would be adequate were actions both sharply distinct and highly stereotyped. But their complex, compositional nature means that within one category of action the kinematics may vary substantially, while between two categories they may barely differ. Moreover, few real-world tasks restrict the plausible repertoire to a small number of classes–distinct or otherwise–that could be explicitly learnt. Rather, any action may be drawn from a great diversity of possibilities–both kinematic and teleological–that shape the characteristics of the underlying movements. This has two crucial implications. First, any modelling approach that lacks awareness of the full space of motion possibilities will be vulnerable to poor generalisation and brittle performance in the face of kinematic anomalies. Second, the very notion of In-Distribution (ID) testing becomes moot, for the relations between different actions and their kinematic signatures are plausibly determinable only across the entire domain of action. A test here arguably needs to be Out-of-Distribution (OoD) if it is to be considered a robust test at all.\nThese considerations are amplified by the nature of real-world applications of kinematic modelling, such as anticipating arbitrary deviations from expected motor behaviour early enough for an automatic intervention to mitigate them. Most urgent in the domain of autonomous driving (Bhattacharyya et al., 2018; Wang et al., 2019), such safety concerns are of the highest importance, and\nare best addressed within the fundamental modelling framework. Indeed, Amodei et al. (2016) cites the ability to recognize our own ignorance as a safety mechanism that must be a core component in safe AI. Nonetheless, to our knowledge, current predictive models of human kinematics neither quantify OoD performance nor are designed with it in mind. There is therefore a need for two frameworks, applicable across the domain of action modelling: one for hardening a predictive model to anomalous cases, and another for quantifying OoD performance with established benchmark datasets. General frameworks are here desirable in preference to new models, for the field is evolving so rapidly greater impact can be achieved by introducing mechanisms that can be applied to a breadth of candidate architectures, even if they are demonstrated in only a subset. Our approach here is founded on combining a latent variable generative model with a standard predictive model, illustrated with the current state-of-the-art discriminative architecture (Mao et al., 2019; Wei et al., 2020), a strategy that has produced state-of-the-art in the medical imaging domain Myronenko (2018). Our aim is to achieve robust performance within a realistic, low-volume, high-heterogeneity data regime by providing a general mechanism for enhancing a discriminative architecture with a generative model.\nIn short, our contributions to the problem of achieving robustness to distributional shift in human motion prediction are as follows:\n1. We provide a framework to benchmark OoD performance on the most widely used opensource motion capture datasets: Human3.6M (Ionescu et al., 2013), and CMU-Mocap1, and evaluate state-of-the-art models on it.\n2. We present a framework for hardening deep feed-forward models to OoD samples. We show that the hardened models are fast to train, and exhibit substantially improved OoD performance with minimal impact on ID performance.\nWe begin section 2 with a brief review of human motion prediction with deep neural networks, and of OoD generalisation using generative models. In section 3, we define a framework for benchmarking OoD performance using open-source multi-action datasets. We introduce in section 4 the discriminative models that we harden using a generative branch to achieve a state-of-the-art (SOTA) OoD benchmark. We then turn in section 5 to the architecture of the generative model and the overall objective function. Section 6 presents our experiments and results. We conclude in section 7 with a summary of our results, current limitations, and caveats, and future directions for developing robust and reliable OoD performance and a quantifiable awareness of unfamiliar behaviour."},{"heading":"2 RELATED WORK","text":"Deep-network based human motion prediction. Historically, sequence-to-sequence prediction using Recurrent Neural Networks (RNNs) have been the de facto standard for human motion prediction (Fragkiadaki et al., 2015; Jain et al., 2016; Martinez et al., 2017; Pavllo et al., 2018; Gui et al., 2018a; Guo & Choi, 2019; Gopalakrishnan et al., 2019; Li et al., 2020b). Currently, the SOTA is dominated by feed forward models (Butepage et al., 2017; Li et al., 2018; Mao et al., 2019; Wei et al., 2020). These are inherently faster and easier to train than RNNs. The jury is still out, however, on the optimal way to handle temporality for human motion prediction. Meanwhile, recent trends have overwhelmingly shown that graph-based approaches are an effective means to encode the spatial dependencies between joints (Mao et al., 2019; Wei et al., 2020), or sets of joints (Li et al., 2020b). In this study, we consider the SOTA models that have graph-based approaches with a feed forward mechanism as presented by (Mao et al., 2019), and the subsequent extension which leverages motion attention, Wei et al. (2020). We show that these may be augmented to improve robustness to OoD samples.\nGenerative models for Out-of-Distribution prediction and detection. Despite the power of deep neural networks for prediction in complex domains (LeCun et al., 2015), they face several challenges that limits their suitability for safety-critical applications. Amodei et al. (2016) list robustness to distributional shift as one of the five major challenges to AI safety. Deep generative models, have been used extensively for detection of OoD inputs and have been shown to generalise\n1t http://mocap.cs.cmu.edu/\nwell in such scenarios (Hendrycks & Gimpel, 2016; Liang et al., 2017; Hendrycks et al., 2018). While recent work has showed some failures in simple OoD detection using density estimates from deep generative models (Nalisnick et al., 2018; Daxberger & Hernández-Lobato, 2019), they remain a prime candidate for anomaly detection (Kendall & Gal, 2017; Grathwohl et al., 2019; Daxberger & Hernández-Lobato, 2019).\nMyronenko (2018) use a Variational Autoencoder (VAE) (Kingma & Welling, 2013) to regularise an encoder-decoder architecture with the specific aim of better generalisation. By simultaneously using the encoder as the recognition model of the VAE, the model is encouraged to base its segmentations on a complete picture of the data, rather than on a reductive representation that is more likely to be fitted to the training data. Furthermore, the original loss and the VAE’s loss are combined as a weighted sum such that the discriminator’s objective still dominates. Further work may also reveal useful interpretability of behaviour (via visualisation of the latent space as in Bourached & Nachev (2019)), generation of novel motion (Motegi et al., 2018), or reconstruction of missing joints as in Chen et al. (2015)."},{"heading":"3 QUANTIFYING OUT-OF-DISTRIBUTION PERFORMANCE OF HUMAN MOTION PREDICTORS","text":"Even a very compact representation of the human body such as OpenPose’s 17 joint parameterisation Cao et al. (2018) explodes to unmanageable complexity when a temporal dimension is introduced of the scale and granularity necessary to distinguish between different kinds of action: typically many seconds, sampled at hundredths of a second. Moreover, though there are anatomical and physiological constraints on the space of licit joint configurations, and their trajectories, the repertoire of possibility remains vast, and the kinematic demarcations of teleologically different actions remain indistinct. Thus, no practically obtainable dataset may realistically represent the possible distance between instances. To simulate OoD data we first need ID data that can be varied in its quantity and heterogeneity, closely replicating cases where a particular kinematic morphology may be rare, and therefore undersampled, and cases where kinematic morphologies are both highly variable within a defined class and similar across classes. Such replication needs to accentuate the challenging aspects of each scenario.\nWe therefore propose to evaluate OoD performance where only a single action, drawn from a single action distribution, is available for training and hyperparameter search, and testing is carried out on the remaining classes. In appendix A, to show that the action categories we have chosen can be distinguished at the time scales on which our trajectories are encoded, we train a simple classifier and show it separates the selected ID action from the others with high accuracy (100% precision and recall for the CMU dataset). Performance over the remaining set of actions may thus be considered OoD."},{"heading":"4 BACKGROUND","text":"Here we describe the current SOTA model proposed by Mao et al. (2019) (GCN). We then describe the extension by Wei et al. (2020) (attention-GCN) which antecedes the GCN prediction model with motion attention."},{"heading":"4.1 PROBLEM FORMULATION","text":"We are given a motion sequence X1:N = (x1,x2,x3, · · · ,xN ) consisting of N consecutive human poses, where xi ∈ RK , with K the number of parameters describing each pose. The goal is to predict the poses XN+1:N+T for the subsequent T time steps."},{"heading":"4.2 DCT-BASED TEMPORAL ENCODING","text":"The input is transformed using Discrete Cosine Transformations (DCT). In this way each resulting coefficient encodes information of the entire sequence at a particular temporal frequency. Furthermore, the option to remove high or low frequencies is provided. Given a joint, k, the position of k over N time steps is given by the trajectory vector: xk = [xk,1, . . . , xk,N ] where we convert to a\nDCT vector of the form: Ck = [Ck,1, . . . , Ck,N ] where Ck,l represents the lth DCT coefficient. For δl1 ∈ RN = [1, 0, · · · , 0], these coefficients may be computed as\nCk,l =\n√ 2\nN N∑ n=1 xk,n 1√ 1 + δl1 cos ( π 2N (2n− 1)(l − 1) ) . (1)\nIf no frequencies are cropped, the DCT is invertible via the Inverse Discrete Cosine Transform (IDCT):\nxk,l =\n√ 2\nN N∑ l=1 Ck,l 1√ 1 + δl1 cos ( π 2N (2n− 1)(l − 1) ) . (2)\nMao et al. use the DCT transform with a graph convolutional network architecture to predict the output sequence. This is achieved by having an equal length input-output sequence, where the input is the DCT transformation of xk = [xk,1, . . . , xk,N , xk,N+1, . . . , xk,N+T ], here [xk,1, . . . , xk,N ] is the observed sequence and [xk,N+1, . . . , xk,N+T ] are replicas of xk,N (ie xk,n = xk,N for n ≥ N ). The target is now simply the ground truth xk."},{"heading":"4.3 GRAPH CONVOLUTIONAL NETWORK","text":"Suppose C ∈ RK×(N+T ) is defined on a graph with k nodes and N +T dimensions, then we define a graph convolutional network to respect this structure. First we define a Graph Convolutional Layer (GCL) that, as input, takes the activation of the previous layer (A[l−1]), where l is the current layer.\nGCL(A[l−1]) = SA[l−1]W + b (3)\nwhere A[0] = C ∈ RK×(N+T ), and S ∈ RK×K is a layer-specific learnable normalised graph laplacian that represents connections between joints, W ∈ Rn[l−1]×n[l] are the learnable inter-layer weightings and b ∈ Rn[l] are the learnable biases where n[l] are the number of hidden units in layer l."},{"heading":"4.4 NETWORK STRUCTURE AND LOSS","text":"The network consists of 12 Graph Convolutional Blocks (GCBs), each containing 2 GCLs with skip (or residual) connections, see figure 7. Additionally, there is one GCL at the beginning of the network, and one at the end. n[l] = 256, for each layer, l. There is one final skip connection from the DCT inputs to the DCT outputs, which greatly reduces train time. The model has around 2.6M parameters. Hyperbolic tangent functions are used as the activation function. Batch normalisation is applied before each activation.\nThe outputs are converted back to their original coordinate system using the IDCT (equation 2) to be compared to the ground truth. The loss used for joint angles is the average l1 distance between the ground-truth joint angles, and the predicted ones. Thus, the joint angle loss is:\n`a = 1\nK(N + T ) N+T∑ n=1 K∑ k=1 |x̂k,n − xk,n| (4)\nwhere x̂k,n is the predicted kth joint at timestep n and xk,n is the corresponding ground truth.\nThis is separately trained on 3D joint coordinate prediction making use of the Mean Per Joint Position Error (MPJPE), as proposed in Ionescu et al. (2013) and used in Mao et al. (2019); Wei et al. (2020). This is defined, for each training example, as\n`m = 1\nJ(N + T ) N+T∑ n=1 J∑ j=1 ‖p̂j,n − pj,n‖2 (5)\nwhere p̂j,n ∈ R3 denotes the predicted jth joint position in frame n. And pj,n is the corresponding ground truth, while J is the number of joints in the skeleton."},{"heading":"4.5 MOTION ATTENTION EXTENSION","text":"Wei et al. (2020) extend this model by summing multiple DCT transformations from different sections of the motion history with weightings learned via an attention mechanism. For this extension, the above model (the GCN) along with the anteceding motion attention is trained end-to-end. We refer to this as the attention-GCN."},{"heading":"5 OUR APPROACH","text":"Myronenko (2018) augment an encoder-decoder discriminative model by using the encoder as a recognition model for a Variational Autoencoder (VAE), (Kingma & Welling, 2013; Rezende et al., 2014). Myronenko (2018) show this to be a very effective regulariser. Here, we also use a VAE, but for conjugacy with the discriminator, we use graph convolutional layers in the decoder. This can be compared to the Variational Graph Autoencoder (VGAE), proposed by Kipf & Welling (2016). However, Kipf & Welling’s application is a link prediction task in citation networks and thus it is desired to model only connectivity in the latent space. Here we model connectivity, position, and temporal frequency. To reflect this distinction, the layers immediately before, and after, the latent space are fully connected creating a homogenous latent space.\nThe generative model sets a precedence for information that can be modelled causally, while leaving elements of the discriminative machinery, such as skip connections, to capture correlations that remain useful for prediction but are not necessarily persuant to the objective of the generative model. In addition to performing the role of regularisation in general, we show that we gain robustness to distributional shift across similar, but different, actions that are likely to share generative properties. The architecture may be considered with the visual aid in figure 1."},{"heading":"5.1 VARIATIONAL AUTOENCODER (VAE) BRANCH AND LOSS","text":"Here we define the first 6 GCB blocks as our VAE recognition model, with a latent variable z ∈ RK×nz = N(µz, σz), where µz ∈ RK×nz , σz ∈ RK×nz . nz = 8, or 32 depending on training stability.\nThe KL divergence between the latent space distribution and a spherical Gaussian N(0, I) is given by:\n`l = KL(q(Z|C)||q(Z)) = 1\n2 nz∑ 1 ( µz 2 + σz 2 − 1− log((σz)2) ) . (6)\nThe decoder part of the VAE has the same structure as the discriminative branch; 6 GCBs. We parametrise the output neurons as µ ∈ RK×(N+T ), and log(σ2) ∈ RK×(N+T ). We can now model the reconstruction of inputs as samples of a maximum likelihood of a Gaussian distribution which constitutes the second term of the negative Variational Lower Bound (VLB) of the VAE:\n`G = log(p(C|Z)) = − 1\n2 N+T∑ n=1 K∑ l=1 ( log(σ2k,l) + log(2π) + |Ck,l − µk,l|2 elog(σ 2 k,l) ) , (7)\nwhere Ck,l are the DCT coefficients of the ground truth."},{"heading":"5.2 TRAINING","text":"We train the entire network together with the additional of the negative VLB:\n` = 1\n(N + T )K N+T∑ n=1 K∑ k=1\n|x̂k,n − xk,n|︸ ︷︷ ︸ Discriminitive loss −λ (`G − `l)︸ ︷︷ ︸ VLB . (8)\nHere λ is a hyperparameter of the model. The overall network is ≈ 3.4M parameters. The number of parameters varies slightly as per the number of joints, K, since this is reflected in the size of the graph in each layer (k = 48 for H3.6M, K = 64 for CMU joint angles, and K = J = 75 for CMU Cartesian coordinates). Furthermore, once trained, the generative model is not required for prediction and hence for this purpose is as compact as the original models."},{"heading":"6 EXPERIMENTS","text":""},{"heading":"6.1 DATASETS AND EXPERIMENTAL SETUP","text":"Human3.6M (H3.6M) The H3.6M dataset (Ionescu et al., 2011; 2013), so called as it contains a selection of 3.6 million 3D human poses and corresponding images, consists of seven actors each performing 15 actions, such as walking, eating, discussion, sitting, and talking on the phone. Martinez et al. (2017); Mao et al. (2019); Li et al. (2020b) all follow the same training and evaluation procedure: training their motion prediction model on 6 (5 for train and 1 for cross-validation) of the actors, for each action, and evaluate metrics on the final actor, subject 5. For easy comparison to these ID baselines, we maintain the same train; cross-validation; and test splits. However, we use the single, most well-defined action (see appendix A), walking, for train and cross-validation, and we report test error on all the remaining actions from subject 5. In this way we conduct all parameter selection based on ID performance.\nCMU motion capture (CMU-mocap) The CMU dataset consists of 5 general classes of actions. Similarly to (Li et al., 2018; 2020a; Mao et al., 2019) we use 8 detailed actions from these classes: ’basketball’, ’basketball signal’, ’directing traffic’ ’jumping, ’running’, ’soccer’, ’walking’, and ’window washing’. We use two representations, a 64-dimensional vector that gives an exponential map representation (Grassia, 1998) of the joint angle, and a 75-dimensional vector that gives the 3D Cartesian coordinates of 25 joints. We do not tune any hyperparameters on this dataset and use only a train and test set with the same split as is common in the literature (Martinez et al., 2017; Mao et al., 2019).\nModel configuration We implemented the model in PyTorch (Paszke et al., 2017) using the ADAM optimiser (Kingma & Ba, 2014). The learning rate was set to 0.0005 for all experiments where, unlike Mao et al. (2019); Wei et al. (2020), we did not decay the learning rate as it was hypothesised that the dynamic relationship between the discriminative and generative loss would make this redundant. The batch size was 16. For numerical stability, gradients were clipped to a maximum `2-norm of 1 and log(σ̂2) and values were clamped between -20 and 3. Code for all experiments is available at the following anonymized link: https://anonymous.4open.science/r/11a7a2b5-da1343f8-80de-51e526913dd2/\nBaseline comparison Both Mao et al. (2019) (GCN), and Wei et al. (2020) (attention-GCN) use this same Graph Convolutional Network (GCN) architecture with DCT inputs. In particular, Wei et al. (2020) increase the amount of history accounted for by the GCN by adding a motion attention mechanism to weight the DCT coefficients from different sections of the history prior to being inputted to the GCN. We compare against both of these baselines on OoD actions. For attentionGCN we leave the attention mechanism preceding the GCN unchanged such that the generative branch of the model is reconstructing the weighted DCT inputs to the GCN, and the whole network is end-to-end differentiable.\nHyperparameter search Since a new term has been introduced to the loss function, it was necessary to determine a sensible weighting between the discriminative and generative models. In Myronenko (2018), this weighting was arbitrarily set to 0.1. It is natural that the optimum value here will relate to the other regularisation parameters in the model. Thus, we conducted random hyperparameter search for pdrop and λ in the ranges pdrop = [0, 0.5] on a linear scale, and λ = [10, 0.00001] on a logarithmic scale. For fair comparison we also conducted hyperparameter search on GCN, for values of the dropout probability (pdrop) between 0.1 and 0.9. For each model, 25 experiments were run and the optimum values were selected on the lowest ID validation error. The hyperparameter search was conducted only for the GCN model on short-term predictions for the H3.6M dataset and used for all future experiments hence demonstrating generalisability of the architecture."},{"heading":"6.2 RESULTS","text":"Consistent with the literature we report short-term (< 500ms) and long-term (> 500ms) predictions. In comparison to GCN, we take short term history into account (10 frames, 400ms) for both\ndatasets to predict both short- and long-term motion. In comparison to attention-GCN, we take long term history (50 frames, 2 seconds) to predict the next 10 frames, and predict futher into the future by recursively applying the predictions as input to the model as in Wei et al. (2020). In this way a single short term prediction model may produce long term predictions.\nWe use Euclidean distance between the predicted and ground-truth joint angles for the Euler angle representation. For 3D joint coordinate representation we use the MPJPE as used for training (equation 5). Table 1 reports the joint angle error for the short term predictions on the H3.6M dataset. Here we found the optimum hyperparameters to be pdrop = 0.5 for GCN, and λ = 0.003, with pdrop = 0.3 for our augmentation of GCN. The latter of which was used for all future experiments, where for our augmentation of attention-GCN we removed dropout altogether. On average, our model performs convincingly better both ID and OoD. Here the generative branch works well as both a regulariser for small datasets and by creating robustness to distributional shifts. We see similar and consistent results for long-term predictions in table 2.\nFrom tables 3 and 4, we can see that the superior OoD performance generalises to the CMU dataset with the same hyperparameter settings with a similar trend of the difference being larger for longer predictions for both joint angles and 3D joint coordinates. For each of these experiments nz = 8.\nTable 5, shows that the effectiveness of the generative branch generalises to the very recent motion attention architecture. For attention-GCN we used nz = 32. Here, interestingly short term predictions are poor but long term predictions are consistently better. This supports our assertion that information relevant to generative mechanisms are more intrinsic to the causal model and thus, here, when the predicted output is recursively used, more useful information is available for the future predictions.\nWalking (ID) Eating (OoD) Smoking (OoD) Average (of 14 for OoD) milliseconds 560 720 880 1000 560 720 880 1000 560 720 880 1000 560 720 880 1000 att-GCN (OoD) 55.4 60.5 65.2 68.7 87.6 103.6 113.2 120.3 81.7 93.7 102.9 108.7 112.1 129.6 140.3 147.8 ours (OoD) 58.7 60.6 65.5 69.1 81.7 94.4 102.7 109.3 80.6 89.9 99.2 104.1 113.1 127.7 137.9 145.3\nTable 5: Long-term prediction of 3D joint positions on H3.6M. Here, ours is also trained with the attention-GCN model. Full table in appendix, table 9."},{"heading":"7 CONCLUSION","text":"We draw attention to the need for robustness to distributional shifts in predicting human motion, and propose a framework for its evaluation based on major open source datasets. We demonstrate that state-of-the-art discriminative architectures can be hardened to extreme distributional shifts by augmentation with a generative model, combining low in-distribution predictive error with maximal generalisability. The introduction of a surveyable latent space further provides a mechanism for model perspicuity and interpretability, and explicit estimates of uncertainty facilitate the detection of anomalies: both characteristics are of substantial value in emerging applications of motion prediction, such as autonomous driving, where safety is paramount. Our investigation argues for wider use of generative models in behavioural modelling, and shows it can be done with minimal or no performance penalty, within hybrid architectures of potentially diverse constitution."},{"heading":"B FULL RESULTS","text":""},{"heading":"C LATENT SPACE OF THE VAE","text":"One of the advantages of having a generative model involved is that we have a latent variable which represents a distribution over deterministic encodings of the data. We considered the question of whether or not the VAE was learning anything interpretable with its latent variable as was the case in Kipf & Welling (2016).\nThe purpose of this investigation was two-fold. First to determine if the generative model was learning a comprehensive internal state, or just a non-linear average state as is common to see in the training of VAE like architectures. The result of this should suggest a key direction of future work. Second, an interpretable latent space may be of paramount usefulness for future applications of human motion prediction. Namely, if dimensionality reduction of the latent space to an inspectable number of dimensions yields actions, or behaviour that are close together if kinematically or teleolgically similar, as in Bourached & Nachev (2019), then human experts may find unbounded potential application for a interpretation that is both quantifiable and qualitatively comparable to all other classes within their domain of interest. For example, a medical doctor may consider a patient to have unusual symptoms for condition, say, A. It may be useful to know that the patient’s deviation from a classical case of A, is in the direction of condition, say, B.\nWe trained the augmented GCN model discussed in the main text with all actions, for both datasets. We use Uniform Manifold Approximation and Projection (UMAP) (McInnes et al., 2018) to project the latent space of the trained GCN models onto 2 dimensions for all samples in the dataset for each dataset independently. From figure 6 we can see that for both models the 2D project relatively closely resembles a spherical gaussian. Further, we can see from figure 6b that the action walking does not occupy a discernible domain of the latent space. This result is further verified by using the same classifier as used in appendix A, which achieved no better than chance when using the latent variables as input rather than the raw data input.\nThis result implies that the benefit observed in the main text is by using the generative model is significant even if the generative model has poor performance itself. In this case we can be sure that the reconstructions are at least not good enough to distinguish between actions. It is hence natural for future work to investigate if the improvement on OoD performance is greater if trained in such a way as to ensure that the generative model performs well. There are multiple avenues through which such an objective might be achieve. Pre-training the generative model being one of the salient candidates.\nD ARCHITECTURE DIAGRAMS\nG C L (\n2 0 K ) =\nG C B (\n2 0 K ) ="}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Human motion is naturally intelligible as a time-varying graph of connected joints constrained by locomotor anatomy and physiology. Its prediction allows the anticipation of actions with applications across healthcare (Geertsema et al., 2018; Kakar et al., 2005), physical rehabilitation and training (Chang et al., 2012; Webster & Celik, 2014), robotics (Koppula & Saxena, 2013b;a; Gui et al., 2018b), navigation (Paden et al., 2016; Alahi et al., 2016; Bhattacharyya et al., 2018; Wang et al., 2019), manufacture (Švec et al., 2014), entertainment (Shirai et al., 2007; Rofougaran et al., 2018; Lau & Chan, 2008), and security (Kim & Paik, 2010; Ma et al., 2018).\\nThe favoured approach to predicting movements over time has been purely inductive, relying on the history of a specific class of movement to predict its future. For example, state space models (Koller & Friedman, 2009) enjoyed early success for simple, common or cyclic motions (Taylor et al., 2007; Sutskever et al., 2009; Lehrmann et al., 2014). The range, diversity and complexity of human motion has encouraged a shift to more expressive, deep neural network architectures (Fragkiadaki et al., 2015; Butepage et al., 2017; Martinez et al., 2017; Li et al., 2018; Aksan et al., 2019; Mao et al., 2019; Li et al., 2020b; Cai et al., 2020), but still within a simple inductive framework.\\nThis approach would be adequate were actions both sharply distinct and highly stereotyped. But their complex, compositional nature means that within one category of action the kinematics may vary substantially, while between two categories they may barely differ. Moreover, few real-world tasks restrict the plausible repertoire to a small number of classes–distinct or otherwise–that could be explicitly learnt. Rather, any action may be drawn from a great diversity of possibilities–both kinematic and teleological–that shape the characteristics of the underlying movements. This has two crucial implications. First, any modelling approach that lacks awareness of the full space of motion possibilities will be vulnerable to poor generalisation and brittle performance in the face of kinematic anomalies. Second, the very notion of In-Distribution (ID) testing becomes moot, for the relations between different actions and their kinematic signatures are plausibly determinable only across the entire domain of action. A test here arguably needs to be Out-of-Distribution (OoD) if it is to be considered a robust test at all.\\nThese considerations are amplified by the nature of real-world applications of kinematic modelling, such as anticipating arbitrary deviations from expected motor behaviour early enough for an automatic intervention to mitigate them. Most urgent in the domain of autonomous driving (Bhattacharyya et al., 2018; Wang et al., 2019), such safety concerns are of the highest importance, and\\nare best addressed within the fundamental modelling framework. Indeed, Amodei et al. (2016) cites the ability to recognize our own ignorance as a safety mechanism that must be a core component in safe AI. Nonetheless, to our knowledge, current predictive models of human kinematics neither quantify OoD performance nor are designed with it in mind. There is therefore a need for two frameworks, applicable across the domain of action modelling: one for hardening a predictive model to anomalous cases, and another for quantifying OoD performance with established benchmark datasets. General frameworks are here desirable in preference to new models, for the field is evolving so rapidly greater impact can be achieved by introducing mechanisms that can be applied to a breadth of candidate architectures, even if they are demonstrated in only a subset. Our approach here is founded on combining a latent variable generative model with a standard predictive model, illustrated with the current state-of-the-art discriminative architecture (Mao et al., 2019; Wei et al., 2020), a strategy that has produced state-of-the-art in the medical imaging domain Myronenko (2018). Our aim is to achieve robust performance within a realistic, low-volume, high-heterogeneity data regime by providing a general mechanism for enhancing a discriminative architecture with a generative model.\\nIn short, our contributions to the problem of achieving robustness to distributional shift in human motion prediction are as follows:\\n1. We provide a framework to benchmark OoD performance on the most widely used opensource motion capture datasets: Human3.6M (Ionescu et al., 2013), and CMU-Mocap1, and evaluate state-of-the-art models on it.\\n2. We present a framework for hardening deep feed-forward models to OoD samples. We show that the hardened models are fast to train, and exhibit substantially improved OoD performance with minimal impact on ID performance.\\nWe begin section 2 with a brief review of human motion prediction with deep neural networks, and of OoD generalisation using generative models. In section 3, we define a framework for benchmarking OoD performance using open-source multi-action datasets. We introduce in section 4 the discriminative models that we harden using a generative branch to achieve a state-of-the-art (SOTA) OoD benchmark. We then turn in section 5 to the architecture of the generative model and the overall objective function. Section 6 presents our experiments and results. We conclude in section 7 with a summary of our results, current limitations, and caveats, and future directions for developing robust and reliable OoD performance and a quantifiable awareness of unfamiliar behaviour.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Deep-network based human motion prediction. Historically, sequence-to-sequence prediction using Recurrent Neural Networks (RNNs) have been the de facto standard for human motion prediction (Fragkiadaki et al., 2015; Jain et al., 2016; Martinez et al., 2017; Pavllo et al., 2018; Gui et al., 2018a; Guo & Choi, 2019; Gopalakrishnan et al., 2019; Li et al., 2020b). Currently, the SOTA is dominated by feed forward models (Butepage et al., 2017; Li et al., 2018; Mao et al., 2019; Wei et al., 2020). These are inherently faster and easier to train than RNNs. The jury is still out, however, on the optimal way to handle temporality for human motion prediction. Meanwhile, recent trends have overwhelmingly shown that graph-based approaches are an effective means to encode the spatial dependencies between joints (Mao et al., 2019; Wei et al., 2020), or sets of joints (Li et al., 2020b). In this study, we consider the SOTA models that have graph-based approaches with a feed forward mechanism as presented by (Mao et al., 2019), and the subsequent extension which leverages motion attention, Wei et al. (2020). We show that these may be augmented to improve robustness to OoD samples.\\nGenerative models for Out-of-Distribution prediction and detection. Despite the power of deep neural networks for prediction in complex domains (LeCun et al., 2015), they face several challenges that limits their suitability for safety-critical applications. Amodei et al. (2016) list robustness to distributional shift as one of the five major challenges to AI safety. Deep generative models, have been used extensively for detection of OoD inputs and have been shown to generalise\\n1t http://mocap.cs.cmu.edu/\\nwell in such scenarios (Hendrycks & Gimpel, 2016; Liang et al., 2017; Hendrycks et al., 2018). While recent work has showed some failures in simple OoD detection using density estimates from deep generative models (Nalisnick et al., 2018; Daxberger & Hernández-Lobato, 2019), they remain a prime candidate for anomaly detection (Kendall & Gal, 2017; Grathwohl et al., 2019; Daxberger & Hernández-Lobato, 2019).\\nMyronenko (2018) use a Variational Autoencoder (VAE) (Kingma & Welling, 2013) to regularise an encoder-decoder architecture with the specific aim of better generalisation. By simultaneously using the encoder as the recognition model of the VAE, the model is encouraged to base its segmentations on a complete picture of the data, rather than on a reductive representation that is more likely to be fitted to the training data. Furthermore, the original loss and the VAE’s loss are combined as a weighted sum such that the discriminator’s objective still dominates. Further work may also reveal useful interpretability of behaviour (via visualisation of the latent space as in Bourached & Nachev (2019)), generation of novel motion (Motegi et al., 2018), or reconstruction of missing joints as in Chen et al. (2015).\"\n },\n {\n \"heading\": \"3 QUANTIFYING OUT-OF-DISTRIBUTION PERFORMANCE OF HUMAN MOTION PREDICTORS\",\n \"text\": \"Even a very compact representation of the human body such as OpenPose’s 17 joint parameterisation Cao et al. (2018) explodes to unmanageable complexity when a temporal dimension is introduced of the scale and granularity necessary to distinguish between different kinds of action: typically many seconds, sampled at hundredths of a second. Moreover, though there are anatomical and physiological constraints on the space of licit joint configurations, and their trajectories, the repertoire of possibility remains vast, and the kinematic demarcations of teleologically different actions remain indistinct. Thus, no practically obtainable dataset may realistically represent the possible distance between instances. To simulate OoD data we first need ID data that can be varied in its quantity and heterogeneity, closely replicating cases where a particular kinematic morphology may be rare, and therefore undersampled, and cases where kinematic morphologies are both highly variable within a defined class and similar across classes. Such replication needs to accentuate the challenging aspects of each scenario.\\nWe therefore propose to evaluate OoD performance where only a single action, drawn from a single action distribution, is available for training and hyperparameter search, and testing is carried out on the remaining classes. In appendix A, to show that the action categories we have chosen can be distinguished at the time scales on which our trajectories are encoded, we train a simple classifier and show it separates the selected ID action from the others with high accuracy (100% precision and recall for the CMU dataset). Performance over the remaining set of actions may thus be considered OoD.\"\n },\n {\n \"heading\": \"4 BACKGROUND\",\n \"text\": \"Here we describe the current SOTA model proposed by Mao et al. (2019) (GCN). We then describe the extension by Wei et al. (2020) (attention-GCN) which antecedes the GCN prediction model with motion attention.\"\n },\n {\n \"heading\": \"4.1 PROBLEM FORMULATION\",\n \"text\": \"We are given a motion sequence X1:N = (x1,x2,x3, · · · ,xN ) consisting of N consecutive human poses, where xi ∈ RK , with K the number of parameters describing each pose. The goal is to predict the poses XN+1:N+T for the subsequent T time steps.\"\n },\n {\n \"heading\": \"4.2 DCT-BASED TEMPORAL ENCODING\",\n \"text\": \"The input is transformed using Discrete Cosine Transformations (DCT). In this way each resulting coefficient encodes information of the entire sequence at a particular temporal frequency. Furthermore, the option to remove high or low frequencies is provided. Given a joint, k, the position of k over N time steps is given by the trajectory vector: xk = [xk,1, . . . , xk,N ] where we convert to a\\nDCT vector of the form: Ck = [Ck,1, . . . , Ck,N ] where Ck,l represents the lth DCT coefficient. For δl1 ∈ RN = [1, 0, · · · , 0], these coefficients may be computed as\\nCk,l =\\n√ 2\\nN N∑ n=1 xk,n 1√ 1 + δl1 cos ( π 2N (2n− 1)(l − 1) ) . (1)\\nIf no frequencies are cropped, the DCT is invertible via the Inverse Discrete Cosine Transform (IDCT):\\nxk,l =\\n√ 2\\nN N∑ l=1 Ck,l 1√ 1 + δl1 cos ( π 2N (2n− 1)(l − 1) ) . (2)\\nMao et al. use the DCT transform with a graph convolutional network architecture to predict the output sequence. This is achieved by having an equal length input-output sequence, where the input is the DCT transformation of xk = [xk,1, . . . , xk,N , xk,N+1, . . . , xk,N+T ], here [xk,1, . . . , xk,N ] is the observed sequence and [xk,N+1, . . . , xk,N+T ] are replicas of xk,N (ie xk,n = xk,N for n ≥ N ). The target is now simply the ground truth xk.\"\n },\n {\n \"heading\": \"4.3 GRAPH CONVOLUTIONAL NETWORK\",\n \"text\": \"Suppose C ∈ RK×(N+T ) is defined on a graph with k nodes and N +T dimensions, then we define a graph convolutional network to respect this structure. First we define a Graph Convolutional Layer (GCL) that, as input, takes the activation of the previous layer (A[l−1]), where l is the current layer.\\nGCL(A[l−1]) = SA[l−1]W + b (3)\\nwhere A[0] = C ∈ RK×(N+T ), and S ∈ RK×K is a layer-specific learnable normalised graph laplacian that represents connections between joints, W ∈ Rn[l−1]×n[l] are the learnable inter-layer weightings and b ∈ Rn[l] are the learnable biases where n[l] are the number of hidden units in layer l.\"\n },\n {\n \"heading\": \"4.4 NETWORK STRUCTURE AND LOSS\",\n \"text\": \"The network consists of 12 Graph Convolutional Blocks (GCBs), each containing 2 GCLs with skip (or residual) connections, see figure 7. Additionally, there is one GCL at the beginning of the network, and one at the end. n[l] = 256, for each layer, l. There is one final skip connection from the DCT inputs to the DCT outputs, which greatly reduces train time. The model has around 2.6M parameters. Hyperbolic tangent functions are used as the activation function. Batch normalisation is applied before each activation.\\nThe outputs are converted back to their original coordinate system using the IDCT (equation 2) to be compared to the ground truth. The loss used for joint angles is the average l1 distance between the ground-truth joint angles, and the predicted ones. Thus, the joint angle loss is:\\n`a = 1\\nK(N + T ) N+T∑ n=1 K∑ k=1 |x̂k,n − xk,n| (4)\\nwhere x̂k,n is the predicted kth joint at timestep n and xk,n is the corresponding ground truth.\\nThis is separately trained on 3D joint coordinate prediction making use of the Mean Per Joint Position Error (MPJPE), as proposed in Ionescu et al. (2013) and used in Mao et al. (2019); Wei et al. (2020). This is defined, for each training example, as\\n`m = 1\\nJ(N + T ) N+T∑ n=1 J∑ j=1 ‖p̂j,n − pj,n‖2 (5)\\nwhere p̂j,n ∈ R3 denotes the predicted jth joint position in frame n. And pj,n is the corresponding ground truth, while J is the number of joints in the skeleton.\"\n },\n {\n \"heading\": \"4.5 MOTION ATTENTION EXTENSION\",\n \"text\": \"Wei et al. (2020) extend this model by summing multiple DCT transformations from different sections of the motion history with weightings learned via an attention mechanism. For this extension, the above model (the GCN) along with the anteceding motion attention is trained end-to-end. We refer to this as the attention-GCN.\"\n },\n {\n \"heading\": \"5 OUR APPROACH\",\n \"text\": \"Myronenko (2018) augment an encoder-decoder discriminative model by using the encoder as a recognition model for a Variational Autoencoder (VAE), (Kingma & Welling, 2013; Rezende et al., 2014). Myronenko (2018) show this to be a very effective regulariser. Here, we also use a VAE, but for conjugacy with the discriminator, we use graph convolutional layers in the decoder. This can be compared to the Variational Graph Autoencoder (VGAE), proposed by Kipf & Welling (2016). However, Kipf & Welling’s application is a link prediction task in citation networks and thus it is desired to model only connectivity in the latent space. Here we model connectivity, position, and temporal frequency. To reflect this distinction, the layers immediately before, and after, the latent space are fully connected creating a homogenous latent space.\\nThe generative model sets a precedence for information that can be modelled causally, while leaving elements of the discriminative machinery, such as skip connections, to capture correlations that remain useful for prediction but are not necessarily persuant to the objective of the generative model. In addition to performing the role of regularisation in general, we show that we gain robustness to distributional shift across similar, but different, actions that are likely to share generative properties. The architecture may be considered with the visual aid in figure 1.\"\n },\n {\n \"heading\": \"5.1 VARIATIONAL AUTOENCODER (VAE) BRANCH AND LOSS\",\n \"text\": \"Here we define the first 6 GCB blocks as our VAE recognition model, with a latent variable z ∈ RK×nz = N(µz, σz), where µz ∈ RK×nz , σz ∈ RK×nz . nz = 8, or 32 depending on training stability.\\nThe KL divergence between the latent space distribution and a spherical Gaussian N(0, I) is given by:\\n`l = KL(q(Z|C)||q(Z)) = 1\\n2 nz∑ 1 ( µz 2 + σz 2 − 1− log((σz)2) ) . (6)\\nThe decoder part of the VAE has the same structure as the discriminative branch; 6 GCBs. We parametrise the output neurons as µ ∈ RK×(N+T ), and log(σ2) ∈ RK×(N+T ). We can now model the reconstruction of inputs as samples of a maximum likelihood of a Gaussian distribution which constitutes the second term of the negative Variational Lower Bound (VLB) of the VAE:\\n`G = log(p(C|Z)) = − 1\\n2 N+T∑ n=1 K∑ l=1 ( log(σ2k,l) + log(2π) + |Ck,l − µk,l|2 elog(σ 2 k,l) ) , (7)\\nwhere Ck,l are the DCT coefficients of the ground truth.\"\n },\n {\n \"heading\": \"5.2 TRAINING\",\n \"text\": \"We train the entire network together with the additional of the negative VLB:\\n` = 1\\n(N + T )K N+T∑ n=1 K∑ k=1\\n|x̂k,n − xk,n|︸ ︷︷ ︸ Discriminitive loss −λ (`G − `l)︸ ︷︷ ︸ VLB . (8)\\nHere λ is a hyperparameter of the model. The overall network is ≈ 3.4M parameters. The number of parameters varies slightly as per the number of joints, K, since this is reflected in the size of the graph in each layer (k = 48 for H3.6M, K = 64 for CMU joint angles, and K = J = 75 for CMU Cartesian coordinates). Furthermore, once trained, the generative model is not required for prediction and hence for this purpose is as compact as the original models.\"\n },\n {\n \"heading\": \"6 EXPERIMENTS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"6.1 DATASETS AND EXPERIMENTAL SETUP\",\n \"text\": \"Human3.6M (H3.6M) The H3.6M dataset (Ionescu et al., 2011; 2013), so called as it contains a selection of 3.6 million 3D human poses and corresponding images, consists of seven actors each performing 15 actions, such as walking, eating, discussion, sitting, and talking on the phone. Martinez et al. (2017); Mao et al. (2019); Li et al. (2020b) all follow the same training and evaluation procedure: training their motion prediction model on 6 (5 for train and 1 for cross-validation) of the actors, for each action, and evaluate metrics on the final actor, subject 5. For easy comparison to these ID baselines, we maintain the same train; cross-validation; and test splits. However, we use the single, most well-defined action (see appendix A), walking, for train and cross-validation, and we report test error on all the remaining actions from subject 5. In this way we conduct all parameter selection based on ID performance.\\nCMU motion capture (CMU-mocap) The CMU dataset consists of 5 general classes of actions. Similarly to (Li et al., 2018; 2020a; Mao et al., 2019) we use 8 detailed actions from these classes: ’basketball’, ’basketball signal’, ’directing traffic’ ’jumping, ’running’, ’soccer’, ’walking’, and ’window washing’. We use two representations, a 64-dimensional vector that gives an exponential map representation (Grassia, 1998) of the joint angle, and a 75-dimensional vector that gives the 3D Cartesian coordinates of 25 joints. We do not tune any hyperparameters on this dataset and use only a train and test set with the same split as is common in the literature (Martinez et al., 2017; Mao et al., 2019).\\nModel configuration We implemented the model in PyTorch (Paszke et al., 2017) using the ADAM optimiser (Kingma & Ba, 2014). The learning rate was set to 0.0005 for all experiments where, unlike Mao et al. (2019); Wei et al. (2020), we did not decay the learning rate as it was hypothesised that the dynamic relationship between the discriminative and generative loss would make this redundant. The batch size was 16. For numerical stability, gradients were clipped to a maximum `2-norm of 1 and log(σ̂2) and values were clamped between -20 and 3. Code for all experiments is available at the following anonymized link: https://anonymous.4open.science/r/11a7a2b5-da1343f8-80de-51e526913dd2/\\nBaseline comparison Both Mao et al. (2019) (GCN), and Wei et al. (2020) (attention-GCN) use this same Graph Convolutional Network (GCN) architecture with DCT inputs. In particular, Wei et al. (2020) increase the amount of history accounted for by the GCN by adding a motion attention mechanism to weight the DCT coefficients from different sections of the history prior to being inputted to the GCN. We compare against both of these baselines on OoD actions. For attentionGCN we leave the attention mechanism preceding the GCN unchanged such that the generative branch of the model is reconstructing the weighted DCT inputs to the GCN, and the whole network is end-to-end differentiable.\\nHyperparameter search Since a new term has been introduced to the loss function, it was necessary to determine a sensible weighting between the discriminative and generative models. In Myronenko (2018), this weighting was arbitrarily set to 0.1. It is natural that the optimum value here will relate to the other regularisation parameters in the model. Thus, we conducted random hyperparameter search for pdrop and λ in the ranges pdrop = [0, 0.5] on a linear scale, and λ = [10, 0.00001] on a logarithmic scale. For fair comparison we also conducted hyperparameter search on GCN, for values of the dropout probability (pdrop) between 0.1 and 0.9. For each model, 25 experiments were run and the optimum values were selected on the lowest ID validation error. The hyperparameter search was conducted only for the GCN model on short-term predictions for the H3.6M dataset and used for all future experiments hence demonstrating generalisability of the architecture.\"\n },\n {\n \"heading\": \"6.2 RESULTS\",\n \"text\": \"Consistent with the literature we report short-term (< 500ms) and long-term (> 500ms) predictions. In comparison to GCN, we take short term history into account (10 frames, 400ms) for both\\ndatasets to predict both short- and long-term motion. In comparison to attention-GCN, we take long term history (50 frames, 2 seconds) to predict the next 10 frames, and predict futher into the future by recursively applying the predictions as input to the model as in Wei et al. (2020). In this way a single short term prediction model may produce long term predictions.\\nWe use Euclidean distance between the predicted and ground-truth joint angles for the Euler angle representation. For 3D joint coordinate representation we use the MPJPE as used for training (equation 5). Table 1 reports the joint angle error for the short term predictions on the H3.6M dataset. Here we found the optimum hyperparameters to be pdrop = 0.5 for GCN, and λ = 0.003, with pdrop = 0.3 for our augmentation of GCN. The latter of which was used for all future experiments, where for our augmentation of attention-GCN we removed dropout altogether. On average, our model performs convincingly better both ID and OoD. Here the generative branch works well as both a regulariser for small datasets and by creating robustness to distributional shifts. We see similar and consistent results for long-term predictions in table 2.\\nFrom tables 3 and 4, we can see that the superior OoD performance generalises to the CMU dataset with the same hyperparameter settings with a similar trend of the difference being larger for longer predictions for both joint angles and 3D joint coordinates. For each of these experiments nz = 8.\\nTable 5, shows that the effectiveness of the generative branch generalises to the very recent motion attention architecture. For attention-GCN we used nz = 32. Here, interestingly short term predictions are poor but long term predictions are consistently better. This supports our assertion that information relevant to generative mechanisms are more intrinsic to the causal model and thus, here, when the predicted output is recursively used, more useful information is available for the future predictions.\\nWalking (ID) Eating (OoD) Smoking (OoD) Average (of 14 for OoD) milliseconds 560 720 880 1000 560 720 880 1000 560 720 880 1000 560 720 880 1000 att-GCN (OoD) 55.4 60.5 65.2 68.7 87.6 103.6 113.2 120.3 81.7 93.7 102.9 108.7 112.1 129.6 140.3 147.8 ours (OoD) 58.7 60.6 65.5 69.1 81.7 94.4 102.7 109.3 80.6 89.9 99.2 104.1 113.1 127.7 137.9 145.3\\nTable 5: Long-term prediction of 3D joint positions on H3.6M. Here, ours is also trained with the attention-GCN model. Full table in appendix, table 9.\"\n },\n {\n \"heading\": \"7 CONCLUSION\",\n \"text\": \"We draw attention to the need for robustness to distributional shifts in predicting human motion, and propose a framework for its evaluation based on major open source datasets. We demonstrate that state-of-the-art discriminative architectures can be hardened to extreme distributional shifts by augmentation with a generative model, combining low in-distribution predictive error with maximal generalisability. The introduction of a surveyable latent space further provides a mechanism for model perspicuity and interpretability, and explicit estimates of uncertainty facilitate the detection of anomalies: both characteristics are of substantial value in emerging applications of motion prediction, such as autonomous driving, where safety is paramount. Our investigation argues for wider use of generative models in behavioural modelling, and shows it can be done with minimal or no performance penalty, within hybrid architectures of potentially diverse constitution.\"\n },\n {\n \"heading\": \"B FULL RESULTS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"C LATENT SPACE OF THE VAE\",\n \"text\": \"One of the advantages of having a generative model involved is that we have a latent variable which represents a distribution over deterministic encodings of the data. We considered the question of whether or not the VAE was learning anything interpretable with its latent variable as was the case in Kipf & Welling (2016).\\nThe purpose of this investigation was two-fold. First to determine if the generative model was learning a comprehensive internal state, or just a non-linear average state as is common to see in the training of VAE like architectures. The result of this should suggest a key direction of future work. Second, an interpretable latent space may be of paramount usefulness for future applications of human motion prediction. Namely, if dimensionality reduction of the latent space to an inspectable number of dimensions yields actions, or behaviour that are close together if kinematically or teleolgically similar, as in Bourached & Nachev (2019), then human experts may find unbounded potential application for a interpretation that is both quantifiable and qualitatively comparable to all other classes within their domain of interest. For example, a medical doctor may consider a patient to have unusual symptoms for condition, say, A. It may be useful to know that the patient’s deviation from a classical case of A, is in the direction of condition, say, B.\\nWe trained the augmented GCN model discussed in the main text with all actions, for both datasets. We use Uniform Manifold Approximation and Projection (UMAP) (McInnes et al., 2018) to project the latent space of the trained GCN models onto 2 dimensions for all samples in the dataset for each dataset independently. From figure 6 we can see that for both models the 2D project relatively closely resembles a spherical gaussian. Further, we can see from figure 6b that the action walking does not occupy a discernible domain of the latent space. This result is further verified by using the same classifier as used in appendix A, which achieved no better than chance when using the latent variables as input rather than the raw data input.\\nThis result implies that the benefit observed in the main text is by using the generative model is significant even if the generative model has poor performance itself. In this case we can be sure that the reconstructions are at least not good enough to distinguish between actions. It is hence natural for future work to investigate if the improvement on OoD performance is greater if trained in such a way as to ensure that the generative model performs well. There are multiple avenues through which such an objective might be achieve. Pre-training the generative model being one of the salient candidates.\\nD ARCHITECTURE DIAGRAMS\\nG C L (\\n2 0 K ) =\\nG C B (\\n2 0 K ) =\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":192,"cells":{"paper_id":{"kind":"string","value":"SP:8f1c7fabe235bdf095007948007509102dd0c126"},"summaries":{"kind":"list like","value":["The authors address the problem of discrete keypoint matching. For an input pair of images, the task is to match the unannotated (but given as part of the input) keypoints. The main contribution is identifying the bottleneck of the current SOTA algorithm: a fixed connectivity construction given by Delauney triangulation. By replacing this with an end-to-end learnable algorithm, they outperform SOTA with a decent margin."],"string":"[\n \"The authors address the problem of discrete keypoint matching. For an input pair of images, the task is to match the unannotated (but given as part of the input) keypoints. The main contribution is identifying the bottleneck of the current SOTA algorithm: a fixed connectivity construction given by Delauney triangulation. By replacing this with an end-to-end learnable algorithm, they outperform SOTA with a decent margin.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Graph matching (GM) has been traditionally modeled as a deterministic optimization problem characterized by an affinity matrix under pre-defined graph topology. Though there have been several attempts on learning more effective node-level affinity/representation for matching, they still heavily rely on the initial graph structure/topology which is typically obtained through heuristic ways (e.g. Delaunay or k-nearest) and will not be adjusted during the learning process to adapt to problem-specific patterns. We argue that such a mechanism for learning on the fixed topology may restrict the potential of a GM solver for specific tasks, and propose to learn latent graph topology in replacement of the fixed topology as input. To this end, we devise two types of latent graph generation procedures in a deterministic and generative fashion, respectively. Particularly, the generative procedure emphasizes the across-graph consistency and thus can be viewed as a matching-guided generative model. 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Torre\"\n ],\n \"title\": \"Factorized graph matching\",\n \"venue\": \"IEEE PAMI,\",\n \"year\": 2016\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Being a long standing NP-hard problem (Loiola et al., 2007), graph matching (GM) has received persistent attention from the machine learning and optimization communities for many years. Concretely, for two graphs with n nodes for each, graph matching seeks to solve1:\nmax z\nz>Mz s.t. Z ∈ {0, 1}n×n, Hz = 1 (1)\nwhere the affinity matrix M ∈ Rn 2×n2\n+ encodes node (diagonal elements) and edge (off-diagonal) affinities/similarities and z is the column-wise vectorization form of the permutation matrix Z. H is a selection matrix ensuring each row and column of Z summing to 1. 1 is a column vector filled with 1. Eq. (1) is the so-called quadratic assignment problem (QAP) (Cho et al., 2010). Maximizing Eq. (1) amounts to maximizing the sum of the similarity induced by matching vector Z. While Eq. (1) does not encode the topology of graphs, Zhou & Torre (2016) further propose to factorize M to explicitly incorporate topology matrix, where a connectivity matrix A ∈ {0, 1}n×n is used to indicate the topology of a single graph (Aij = 1 if there exists an edge between nodes i and j; Aij = 0 otherwise). To ease the computation, Eq. (1) is typically relaxed by letting z ∈ [0, 1]n 2 and keeping other parts of Eq. (1) intact. Traditional solvers to such relaxed problem generally fall into the categories of iterative update (Cho et al., 2010; Jiang et al., 2017) or numerical continuation (Zhou & Torre, 2016; Yu et al., 2018), where the solvers are developed under two key assumptions: 1) Affinity M is pre-computed with some non-negative metrics, e.g. Gaussian kernel, L2-distance or Manhattan distance; 2) Graph topology is pre-defined as input either in dense (Schellewald & Schnörr, 2005) or sparse (Zhou & Torre, 2016) fashion. There have been several successful attempts towards adjusting the first assumption by leveraging the power of deep networks to learn more effective graph representation for GM (Wang et al., 2019a; Yu et al., 2020; Fey et al., 2020). However, to our best knowledge, there is little previous work questioning and addressing the problem regarding the second assumption in the context of learning-based graph matching2. For example, existing\n1Without loss of generality, we discuss graph matching under the setting of equal number of nodes without outliers. The unequal case can be readily handled by introducing extra constraints or dummy nodes. Bipartite matching and graph isomorphism are subsets of this quadratic formulation (Loiola et al., 2007).\n2There are some loosely related works (Du et al., 2019; 2020) on network alignment and link prediction without learning, which will be discussed in detail in the related works.\nstandard pipeline of keypoint matching in computer vision will construct initial topology by Delaunay triangulation or k-nearest neighbors. Then this topology will be freezed throughout the subsequent learning and matching procedures. In this sense, the construction of graph topology is peeled from matching task as a pre-processing stage. More examples can be found beyond the vision communities such as in social network alignment (Zhang & Tong, 2016; Heimann et al., 2018; Xiong & Yan, 2020) assuming fixed network structure for individual node matching in two networks.\nWe argue that freezing graph topology for matching can hinder the capacity of graph matching solvers. For a pre-defined graph topology, the linked nodes sometimes result in less meaningful interaction, especially under the message-passing mechanism in graph neural networks (Kipf & Welling, 2017). We give a schematic demonstration in Fig. 1. Though some earlier attempts (Cho & Lee, 2012; Cho et al., 2013) seek to adjust the graph topology under traditional non-deep learning setting, such procedures cannot be readily integrated into end-to-end deep learning frameworks due to undifferentiable nature. Building upon the hypothesis that there exists some latent topology better than heuristically created one for GM, our aim is to learn it (or its distribution) for GM. Indeed, jointly solving matching and graph topology learning can be intimidating due to the combinatorial nature, which calls for more advanced approaches.\nIn this paper, we propose an end-to-end framework to jointly learn the latent graph topology and perform GM, termed as deep latent graph matching (DLGM). We leverage the power of graph generative model to automatically produce graph topology from given features and their geometric relations, under specific locality prior. Different from generative learning on singleton graphs (Kipf & Welling, 2016; Bojchevski et al., 2018), our graph generative learning is performed in a pairwise fashion, leading to a novel matching-guided generative paradigm. The source code will be made publicly available.\nContributions: 1) We explore a new direction for more flexible GM by actively learning latent topology, in contrast to previous works using fixed topology as input; 2) Under this setting, we propose a deterministic optimization approach to learn graph topology for matching; 3) We further present a generative way to produce latent topology under a probabilistic interpretation by ExpectationMaximization. This framework can also adapt to other problems where graph topology is the latent structure to infer; 4) Our method achieves state-ofthe-art performance on public benchmarks."},{"heading":"2 RELATED WORKS","text":"In this section, we first discuss existing works for graph topology and matching updating whose motivation is a bit similar to ours while the technique is largely different. Then we discuss relevant works in learning graph matching and generative graph models from the technical perspective.\nTopology updating and matching. There are a few works for joint graph topology updating and matching, in the context of network alignment. Specifically, given two initial networks for matching, Du et al. (2019) show how to alternatively perform link prediction within each network and node matching across networks based on the observation that these two tasks can benefit to each other. In their extension (Du et al., 2020), a skip-gram embedding framework is further established under the same problem setting. In fact, these works involve a random-walk based node embedding updating and classification based link prediction modules and the whole algorithm runs in a one-shot optimization fashion. There is neither explicit training dataset nor trained matching model (except\nfor the link classifier), which bears less flavor of machine learning. In contrast, our method involves training an explicit model for topology recovery and matching solving. Specifically, our deterministic technique (see Sec. 3.4.1) solves graph topology and matching in one-shot, while the proposed generative method alternatively estimates the topology and matching (see Sec. 3.4.2). Our approach allows to fully leverage multiple training samples in many applications like computer vision to boost the performance on test set. Moreover, the combinatorial nature of the matching problem is not addressed in (Du et al., 2019; 2020), and they adopt a greedy selection strategy instead. While we develop a principled combinatorial learning approach to this challenge. Also their methods rely on a considerable amount of seed matchings, yet this paper directly learns the latent topology from scratch which is more challenging and seldom studied.\nLearning of graph matching. Early non-deep learning-based methods seek to learn effective metric (e.g. weighted Euclid distance) for node and edge features or affinity kernel (e.g. Gaussian kernel) in a parametric fashion (Caetano et al., 2009; Cho et al., 2013). Recent deep graph matching methods have shown how to extracte more dedicated feature representation. The work (Zanfir & Sminchisescu, 2018) adopts VGG16 (Simonyan & Zisserman, 2014) as the backbone for feature extraction on images. Other efforts have been witnessed in developing more advanced pipelines, where graph embedding (Wang et al., 2019a; Yu et al., 2020; Fey et al., 2020) and geometric learning (Zhang & Lee, 2019; Fey et al., 2020) are involved. Rolı́nek et al. (2020) study the way of incorporating traditional non-differentiable combinatorial solvers, by introducing a differentiatiable blackbox GM solver (Pogancic et al., 2020). Recent works in tackling combinatorial problem with deep learning (Huang et al., 2019; Kool & Welling, 2018) also inspire developing combinatorial deep solvers, for GM problems formulated by both Koopmans-Beckmann’s QAP (Nowak et al., 2018; Wang et al., 2019a) and Lawler’s QAP (Wang et al., 2019b). Specifically, Wang et al. (2019a) devise a permutation loss for supervised learning, with an improvement in Yu et al. (2020) via Hungarian attention. Wang et al. (2019b) solve the most general Lawler’s QAP with graph embedding technique.\nGenerative graph model. Early generative models for graph can date back to (Erdos & Renyi, 1959), in which edges are generated with fixed probability. Recently, Kipf & Welling (2016) present a graph generative model by re-parameterizing the edge probability from Gaussian noise. Johnson (2017) propose to generate graph in an incremental fashion, and in each iteration a portion of the graph is produced. Gómez-Bombarelli et al. (2018) utilized recurrent neural network to generate graph from a sequence of molecule representation. Adversarial graph generation is considered in (Pan et al., 2018; Wang et al., 2018; Bojchevski et al., 2018). Specifically, Wang et al. (2018); Bojchevski et al. (2018) seek to unify graph generative model and generative adversarial networks. In parallel, reinforcement learning has been adopted to generate discrete graphs (De Cao & Kipf, 2018)."},{"heading":"3 LEARNING LATENT TOPOLOGY FOR GM","text":"In this section, we describe details of the proposed framework with two specific algorithms derived from deterministic and generative perspectives, respectively. Both algorithms are motivated by the hypothesis that there exists some latent topology more suitable for matching rather than a fixed one. Note the proposed deterministic algorithm performs a standard forward-backward pass to jointly learn the topology and matching, while our generative algorithm consists of an alternative optimization procedure between estimating latent topology and learning matching under an Expectation-Maximization (EM) interpretation. In general, the generative algorithm assumes that a latent topology is sampled from a latent distribution, where the expected matching accuracy sufficing this distribution is maximized. Therefore, we expect to learn a topology generator sufficing such distribution. We reformulate GM into Bayesian fashion for consistent discussion in Sec. 3.1, detail deterministic/generative latent module in Sec. 3.2 and discuss the loss functions from a probabilistic perspective in Sec. 3.3. We finally elaborate on the holistic framework and the optimization procedure for both algorithms (deterministic and generative) in Sec. 3.4."},{"heading":"3.1 PROBLEM DEFINITION AND BACKGROUND","text":"GM problem can be viewed as a Bayesian variant of Eq. (1). In general, let G(s) and G(t) represent the initial source and target graphs for matching, respectively. We represent graph as G := {X,E,A}, where X ∈ Rn×d1 is the representation of n nodes with dimension d1. E ∈ Rm×d2 are features of\nm edges and A ∈ {0, 1}n×n is initial connectivity (i.e. topology) matrix by heuristics e.g. Delaunay triangulation. For notational brevity, we assume d1 and d2 keep intact after updating the features across each convolutional layers of GNN (i.e., feature dimensions of both nodes and edges will not change after each layer’s update). Denote the matching Z ∈ {0, 1}n×n between two graphs, where Zij = 1 indicates a correspondence exists between node i in G(s) and node j in G(t), and Zij = 0 otherwise. Given training samples {Zk,G(s)k ,G (t) k } with k = 1, 2, ..., N , the objective of learning-based GM aims to maximize the likelihood:\nmax θ ∏ k Pθ ( Zk|G(s)k ,G (t) k ) (2)\nwhere θ denotes model parameters. Pθ(·) measures the probability of matching Zk given the k-th pair, and is instantiated via a network parameterized by θ.\nBeing a generic module for producing latent topology, our method can be flexibly integrated into existing deep GM frameworks. We build up our method based on state-of-the-art (Rolı́nek et al., 2020), which utilizes SplineCNN (Fey et al., 2018) for node/edge representation learning. SplineCNN is a specific graph neural networks which updates a node representation via a weighted summation of its neighbors. The update rule at node i of a standard SplineCNN reads:\n(x ∗ g)(i) = 1 |N (i)| d1∑ l=1 ∑ j∈N (i) xl(j) · gl(e(i, j)) (3)\nwhere xl(j) performs the convolution on node j and outputs a d1-dimensional feature. gl(·) delivers the message weight given the edge feature e(i, j). N (i) refers to i’s neighboring nodes Summation over neighbors follows the topology A. Since our algorithm learns to generate topology, we need to explicitly express Eq. (3) in a differentiable way w.r.t. A. To this end, we rewrite Eq. (3) as:\n(x ∗ g|A) = ( ◦G)X̂ (4)\nwhere  is the normalized connectivity with each row normalized by the degree |N (i)| (see Eq. (3)) of the corresponding node i. G and X̂ correspond to outputs of gl(·) and xl(·) operators, respectively. (· ◦ ·) is the Hadamard product. With Eq. (4), we thus can perform back-propagation on connectivity/topology A. See more details in Appendix A.2."},{"heading":"3.2 LATENT TOPOLOGY LEARNING","text":"Existing learning-based graph matching algorithms consider A to be fixed throughout the computation without questioning if the input topology is optimal or not. This can be problematic since input graph construction is heuristic, and it never takes into account how suitable it is for the subsequent GM task. In our framework, instead of utilizing a fixed pre-defined topology, we consider to produce latent topology under two settings: 1) a deterministic and 2) a generative way. The former is often more efficient while the latter method can be more accurate at the cost of exploring more latent topology. Note both methods produce discrete topology to verify our hypothesis about the existence of more suitable discrete latent topology for GM problem. The followings describe two deep structures.\nDeterministic learning: Given input features X and initial topology A, the deterministic way of generating latent topology A ∈ {0, 1}n×n is3:\nAij = Rounding(sigmoid(y > i Wyj)) with Y = GCN(X,A) (5)\nwhere GCN(·) is the graph convolutional networks (GCN) (Kipf & Welling, 2017) and yi corresponds to the feature of node i in feature map Y. W is the learnable parameter matrix. Note function Rounding(·) is undifferentiable, and will be discussed in Sec. 3.4.1. Generative learning: We reparameterize the representation as:\nP (yi|X,A) = N (yi|µi,diag(σ2)) (6) 3We consider the case when only node feature E and topology A are necessary. Edge feature E can be\nreadily integrated as another input.\nwith µ = GCNµ(X,A) and σ = GCNσ(X,A) are two GCNs producing mean and covariance. It is equivalent to sampling a random vector from i.i.d. uniform distribution s ∼ U(0,1), then applying y = µ+ s · σ, where (·) is element-wise product. Similar as Eq. (5) by introducing learnable parameter W, the generative latent topology is sampled following i.i.d. distribution over each edge (i, j):\nP (A|Y) = ∏ i ∏ j P (Aij |yi,yj) with P (Aij = 1|yi,yj) = sigmoid(y>i Wyj) (7)\nSince sigmoid(·) maps any input into (0, 1), Eq. (7) can be interpreted as the probability of sampling edge (i, j). As the sampling procedure is undifferentiable, we apply Gumbel-softmax trick (Jang et al., 2017) as another reparameterization procedure. As such, a latent graph topology A can be sampled fully from distribution P (A) and the procedure becomes differentiable."},{"heading":"3.3 LOSS FUNCTIONS","text":"In this section, we explain three loss functions and the behind motivation: matching loss, locality loss and consistency loss. The corresponding probabilistic interpretation of each loss function can be found in Sec. 3.4.2. These functions are selectively activated in DLGM-D and DLGM-G (see Sec. 3.4). In DLGM-G, different loss functions are activated in inference and learning steps.\ni) Matching loss. This common term measures how the predicted matching Ẑ diverges from groundtruth Z. Following Rolı́nek et al. (2020), we adopt Hamming distance on node-wise matching:\nLM = Hamming(Ẑ,Z) (8)\nii) Locality loss. This loss is devised to account for the general prior that the produced/learnt graph topology should advocate local connection rather than distant one, since two nodes may have less meaningful interaction once they are too distant from each other. In this sense, locality loss serves as a prior or regularizer in GM. As shown in multiple GM methods (Yu et al., 2018; Wang et al., 2019a; Fey et al., 2020), Delaunay triangulation is an effective way to deliver good locality. Therefore in our method, the locality loss is the Hamming distance between the initial topology A (obtained from Delaunay) and predicted topology A for both source graph and target graph:\nLL = Hamming(A(s),A(s)) + Hamming(A(t),A(t)) (9)\nWe emphasize that locality loss serves as a prior for latent graph. It focuses on advocating locality, but not reconstructing the initial Delaunay triangulation (as in Graph VAE (Kipf & Welling, 2016)).\niii) Consistency loss. One can imagine that a GM solver is likely to deliver better performance if two graphs in a training pair are similar. In particular, we anticipate the latent topology A(s) and A(t) to be isomorphic under a specific matching, since isomorphic topological structures tend to be easier to match. Driven by this consideration, we devise the consistency loss which measures the level of isomorphism between latent topology A(s) and A(t):\nLC(·|Z) = |Z>A(s)Z−A(t)|+ |ZA(t)Z> −A(s)| (10)\nNote Z does not necessarily refer to the ground-truth, but can be any predicted matching. In this sense, latent topology A(s) and A(t) can be generated jointly given the matching Z as guidance information. This term can also serve as a consistency prior or regularization. We given a schematic example showing the merit of introducing consistency loss in Fig. 2(b)."},{"heading":"3.4 FRAMEWORK","text":"A schematic diagram of our framework is given in Fig. 2(a) which consists of a singleton pipeline for processing a single image. It consists of three essential modules: a feature backbone (NB), a latent topology module (NG) and a feature refinement module (NR). Specifically, module NG corresponds to Sec. 3.2 with deterministic or generative implementations. Note the geometric relation of keypoints provide some prior for generating topology A. We employ VGG16 (Simonyan & Zisserman, 2014)\nas NB and feed the produced node feature X and edge feature E to NG. NB also produces a global feature for each image. After generating the latent topology A, we pass over X and E together with A to NR (SplineCNN (Fey et al., 2018)). The holistic pipeline handling pairwise graph inputs can be found in Fig. 4 in Appendix A.1 which consists of two copies of singleton pipeline processing source and target data (in a Siamese fashion), respectively. Then the outputs of two singleton pipelines are formulated into affinity matrix, followed by a differentiable Blackbox GM solver (Pogancic et al., 2020) with message-passing mechanism (Swoboda et al., 2017). Note once without NG, the holistic pipeline with only NB +NR is identical to the method in (Rolı́nek et al., 2020). Readers are referred to this strong baseline (Rolı́nek et al., 2020) for more mutual algorithmic details.\n3.4.1 OPTIMIZATION WITH DETERMINISTIC LATENT GRAPH\nWe show how to optimize with deterministic latent graph module, where the topology A is produced by Eq. (5). The objective of matching conditioned on the produced latent topology A becomes:\nmax ∏ k P ( Zk|A(s)k ,A (t) k ,G (s) k ,G (t) k ) (11)\nEq. (11) can be optimized with standard back-propagation with three loss terms activated, except for the Rounding function (see Eq. (5)), which makes the procedure undifferentiable. To address this, we use straight-through operator (Bengio et al., 2013) which performs a standard rounding during the forward pass but approximates it with the gradient of identity during the backward pass on [0, 1]:\n∂Rounding(x)/∂x = 1 (12)\nThough there exist some unbiased gradient estimators (e.g., REINFORCE (Williams, 1992)), the biased straight-through estimator proved to be more efficient and has been successfully applied in several applications (Chung et al., 2017; Campos et al., 2018). All the network modules (NG +NB + NR) are simultaneously learned during the training. All three losses are activated in the learning procedure (see Sec. 3.3), which are applied on the predicted matching Ẑ, the latent topology A(s) and A(t). We term the algorithm under this setting DLGM-D.\n3.4.2 OPTIMIZATION WITH GENERATIVE LATENT GRAPH\nSee more details in Appendix A.3. In this setting, the source and target latent topology A(s) and A(t) are sampled according to Eq. (6) and (7). The objective becomes:\nmax ∏ k ∫ A (s) k ,A (t) k Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) (13)\nUnfortunately, directly optimizing Eq. (13) is difficult due to the integration over A which is intractable. Instead, we maximize the evidence lower bound (ELBO) (Bishop, 2006) as follows:\nlogPθ(Z|G(s),G(t)) ≥ EQφ(A(s),A(t)|G(s),G(t)) [ logPθ(Z,A (s),A(t)|G(s),G(t))− logQφ(A(s),A(t)|G(s),G(t)) ] (14)\nwhere Qφ(A(s),A(t)|G(s),G(t)) can be any joint distribution of A(s) and A(t) given the input graphs G(s) and G(t). Equality of Eq. (14) holds when Qφ(A(s),A(t)|G(s),G(t)) = Pθ(A\n(s),A(t)|Z,G(s),G(t)). For tractability, we rationally introduce the independence by assuming that we can use an identical latent topology module Qφ (corresponding to NG in Fig. 2(a)) to separately handle each input graph:\nQφ(A (s),A(t)|G(s),G(t)) = Qφ(A(s)|G(s))Qφ(A(t)|G(t)) (15)\nwhich can greatly simplify the model complexity. Then we can utilize a neural network to model Qφ (similar to modeling Pθ). The optimization of Eq. (14) is studied in (Neal & Hinton, 1998), known as the Expectation-Maximization (EM) algorithm. Optimization of Eq. (14) alternates between E-step and M-step. During E-step (inference), Pθ is fixed and the algorithm seeks to find an optimal Qφ to approximate the true posterior distribution (see Appendix A.3 for explanation):\nPθ(A (s),A(t)|Z,G(s),G(t)) (16)\nDuring M-step (learning), Qφ is instead fixed and algorithm alters to maximize the likelihood: EQφ(A(s)|G(s)),Qφ(A(t)|G(t)) [ logPθ(Z,A (s),A(t)|G(s),G(t)) ] ∝ −LM (17)\nWe give more details on the inference and learning steps as follows.\nInference. This step focuses on deriving posterior distribution Pθ(A(s),A(t)|Z,G(s),G(t)) using its approximation Qφ. To this end, we fix the parameters θ in modules NB and NR, and only update the parameters φ in module NG corresponding to Qφ. As stated in Sec. 3.2, we employ the Gumbel-softmax trick for sampling discrete A (Jang et al., 2017). To this end, we can formulate a 2D vector aij = [P (Aij = 1), 1− P (Aij = 1)]>. Then the sampling becomes:\nsoftmax (log(aij) + hij ; τ) (18) where hij is a random 2D vector from Gumbel distribution, and τ is a small temperature parameter. We further impose prior on latent topology A given A through locality loss:\nlog ∏ i,j P (Aij |Aij) ∝ −LL(A,A) (19)\nwhich is to preserve the locality in initial topology A. It should also be noted that Z is the predicted matching from current Pθ, as Qφ is an approximation. Besides, we also anticipate two generated topology A(s) and A(t) from a graph pair should be similar (isomorphic) given matching Z:\nlogP ( A(s),A(t)|Z ) ∝ −LC ( A(s),A(t)|Z ) (20)\nIn summary, we activate locality loss and consistency loss during the inference step, where the latter loss is conditioned with the predicted matching rather than the ground-truth. Note that the inference step involves twice re-parameterization tricks corresponding to Eq. (6) and (18), respectively. While the first generates the continuous topology distribution under edge independence assumption, the second performs discrete sampling sufficing the generated topology distribution.\nLearning. This step optimizes Pθ by fixing Qφ. We sample discrete graph topologies As completely from the probability of edge P (Aij = 1). Once latent topology As are sampled, we feed them to module NR together with the node-level features from NB . Only NB and NR are updated in this step, and only matching loss LM is activated. Remark. Note for each pair of graphs in training, we use an identical random vector s for generating both graphs’ topology (see Eq. (6)). We pretrain the network Pθ before alternativly training Pθ and Qφ. During pretraining, we activate NB + NR modules and LM loss during pretraining, and feed the network the topology obtained from Delaunay as the latent topology. After pretraining, the optimization will switch between inference and learning steps until convergence. We term the setting of generative latent graph matching as DLGM-G and summarize it in Alg. 1.\nmethod aero bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tv Ave BBGM-max 35.5 68.6 46.7 36.1 85.4 58.1 25.6 51.7 27.3 51.0 46.0 46.7 48.9 58.9 29.6 93.6 42.6 35.3 70.7 79.5 51.9\nBBGM 42.7 70.9 57.5 46.6 85.8 64.1 51.0 63.8 42.4 63.7 47.9 61.5 63.4 69.0 46.1 94.2 57.4 39.0 78.0 82.7 61.4 DLGM-D (ours) 42.5 71.8 57.8 46.8 86.9 70.3 53.4 66.7 53.8 67.6 64.7 64.6 65.2 70.1 47.9 95.5 59.6 47.7 77.7 82.6 63.9 DLGM-G (ours) 43.8 72.9 58.5 47.4 86.4 71.2 53.1 66.9 54.6 67.8 64.9 65.7 66.9 70.8 47.4 96.5 61.4 48.4 77.5 83.9 64.8"},{"heading":"4 EXPERIMENT","text":"We conduct experiments on datasets including Pascal VOC with Berkeley annotation (Everingham et al., 2010; Bourdev & Malik, 2009), Willow ObjectClass (Cho et al., 2013) and SPair-71K (Min et al., 2019). We report the per-category and average performance. The objective of all experiments is to maximize the average matching accuracy. Both our DLGM-D and DLGM-G are tested.\nPeer methods. We conduct comparison experiments against the following algorithms: 1) GMN (Zanfir & Sminchisescu, 2018), which is a seminal work incorporating graph matching into deep learning framework equipped with a spectral solver (Egozi et al., 2013); 2) PCA (Wang et al., 2019a). This method treats graph matching as feature matching problem and employs GCN (Kipf & Welling, 2017) to learn better features; 3) CIE1/GAT-H (Yu et al., 2020). This paper develops a novel embedding and attention mechanism, where GAT-H is the version by replacing the basic embedding block with Graph Attention Networks (Veličković et al., 2018); 4) DGMC (Fey et al., 2020). This method devises a post-processing step by emphasizing the neighborhood similarity; 5) BBGM (Rolı́nek et al., 2020). It integrates a differentiable linear combinatorial solver (Pogancic et al., 2020) into a deep learning framework and achieves state-of-the-art performance.\nResults on Pascal VOC. The dataset (Everingham et al., 2010; Bourdev & Malik, 2009) consists of 7,020 training images and 1,682 testing images with 20 classes in total, together with the object bounding boxing for each. Following the data preparation in (Wang et al., 2019a), each object within the bounding box is cropped and resized to 256× 256. The number of nodes per graph ranges from 6 to 23. We further follow (Rolı́nek et al., 2020) under two evaluating metrics: 1) Accuracy: this is the standard metric evaluated on the keypoints by filtering out the outliers; 2) F1-score: this metric is evaluated without keypoint filtering, being the harmonic mean of precision and recall.\nExperimental results on the two setting are shown in Tab. 1 and Tab. 2. The proposed method under either settings of DLGM-D and DLGM-G outperforms counterparts by accuracy and f1-score. DLGM-G generally outperforms DLGM-D. Discussion can be found in Appendix A.5.\nQuality of generated topology. We further show the consistency/locality curve vs epoch in Fig. 3, since both consistency and locality losses can somewhat reflect the quality of topology generation. It shows that both locality and consistency losses descend during the training. Note that the consistency loss with Delaunay triangulation (green dashed line) is far more larger than our generated ones (blue/red dashed line). This clearly supports the claim that our method generates similar (more isomorphic) typologies, as well as preserving locality.\nResults on Willow Object. The benchmark (Cho et al., 2013) consists of 256 images in 5 categories, where two categories (car and motorbike) are subsets selected from Pascal VOC. Following the preparation protocol in Wang et al. (2019a), we crop the image within the object bounding box and resize it to 256 × 256. Since the dataset is relatively small, we conduct the experiment to verify the transfer ability of different methods under two settings: 1) trained on Pascal VOC and directly applied to Willow (Pt); 2) trained on Pascal VOC then finetuned on Willow (Wt). Results under the two settings are shown in Tab. 3. Since this dataset is relatively small, further improvement is difficult. It is shown both DLGM-D and DLGM-G have good transfer ability.\nResults on SPair-71K. The dataset (Min et al., 2019) is much larger than Pascal VOC and WillowObject since it consists of 70,958 image pairs collected from Pascal VOC 2012 and Pascal 3D+ (53,340 for training, 5,384 for validation and 12,234 for testing). It improves Pascal VOC by removing ambiguous categories sofa and dining table. This dataset is considered to contain more difficult matching instances and higher annotation quality. Results are summarized in Tab. 4. Our method consistently improves the matching performance, agreeing with the results in Pascal VOC and Willow."},{"heading":"5 CONCLUSION","text":"Graph matching involves two essential factors: the affinity model and topology. By incorporating learning paradigm for affinity/feature, the performance of matching on public datasets has significantly been improved. However, there has been little previous work exploring more effective topology for matching. In this paper, we argue that learning a more effective graph topology can significantly improve the matching, thus being essential. To this end, we propose to incorporate a latent topology module under an end-to-end deep network framework that learns to produce better graph topology. We also present the interpretation and optimization of topology module in both deterministic and generative perspectives, respectively. Experimental results show that, by learning the latent graph, the matching performance can be consistently and significantly enhanced on several public datasets."},{"heading":"A APPENDIX","text":"A.1 HOLISTIC PIPELINE\nWe show the holistic pipeline of our framework in Fig. 4 consisting of two “singleton pipelines” (see introduction part of Sec. 3 for more details). In general, the holistic pipeline follows the convention in a series of deep graph matching methods by utilizing an identical singleton pipeline to extract features, then exploits the produced features to perform matching (Yu et al., 2020; Wang et al., 2019a; Fey et al., 2020; Rolı́nek et al., 2020). Except for the topology module NG, all others parts of our network are the same as those in Rolı́nek et al. (2020).\nA.2 SPLINECNN\nSplineCNN is a method to perform graph-based representation learning via convolution operators defined based on B-splines (Fey et al., 2018). The initial input to SplineCNN is G = {X,E,A}, where X ∈ Gn×d1 and A ∈ {0, 1}n×n indicate node features and topology, respectively (same as in Sec. 3.1). E ∈ [0, 1]n×n×d2 is so-called pseudo-coordinates and can be viewed as n2 × d2dimensional edge features for a fully connected graph (in case m = n2, see Sec. 3.1). Let normalized edge feature e(i, j) = Ei,j,: ∈ [0, 1]d2 if a directed edge (i, j) exists (Ai,j = 1), and 0 otherwise (Ai,j = 0). Note topology A fully carries the information of N (i) which defines the neighborhood\nof node i. During the learning, X and E will be updated while topology A will not. Therefore SplineCNN is a geometric graph embedding method without adjusting the latent graph topology.\nB-spline is employed as basic kernel in SplineCNN, where a basis function has only support on a specific real-valued interval (Piegl & Tiller, 2012). Let ((Nq1,i)1≤i≤k1 , ..., (N q d,i)1≤i≤kd2 ) be d2 B-spline bases with degree q. The kernel size is defined in k = (k1, ..., kd2). In SplineCNN, the continuous kernel function gl : [a1, b1]× ...× [ad2 , bd2 ]→ G is defined as:\ngl(e) = ∑ p∈P wp,l ·Bp(e) (21)\nwhere P = (Nq1,i)i × ... × (N q d,i)i is the B-spline bases (Piegl & Tiller, 2012) and wp,l is the trainable parameter corresponding to the lth node feature in X, with Bp being the product of the basis functions in P:\nBp = d∏ i=1 Nqi,pi(ei) (22)\nwhere e is the pseudo-coordinate in E. Then, given the kernel function g = (g1, ..., gd1) and the node feature X ∈ Gn×d1 , one layer of the convolution at node i in SplineCNN reads (same as Eq. (3)):\n(x ∗ g)(i) = 1 |N (i)| d1∑ l=1 ∑ j∈N (i) xl(j) · gl(e(i, j)) (23)\nwhere xl(j) indicates the convolved node feature value of node j at lth dimension. This formulation can be tensorized into Eq. (4) with explicit topology matrix A. In this sense, we can back-propagate the gradient of A. Reader are referred to Fey et al. (2018) for more comprehensive understanding of this method.\nA.3 DERIVATION OF DLGM-D\nWe give more details of the optimization on DLGM-D in this section. This part also interprets some basic formulation conversion (e.g. from Eq. (2) to its Bayesian form). First, we assume there is no latent topology A(s) and A(s) at the current stage. In this case, the objective of GM is simply:\nmax ∏ k Pθ ( Zk|G(s)k ,G (t) k ) (24)\nwhere Pθ measures the probability of a matching Zk given graph pair G(s)k and G (t) k . If we impose the latent topology A(s) and A(t), as well as some distribution over them, then Eq. (24) can be equivalently expressed as:\nmax ∏ k Pθ ( Zk|G(s)k ,G (t) k ) = max ∏ k ∫ A (s) k ,A (t) k Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) (25)\nwhere Pθ ( Zk|G(s)k ,G (t) k ) is the marginal distribution of Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) with respect\nto Zk, since A (s) k and A (t) k are integrated over some distribution. Herein we can impose another distribution of the topology Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) characterized by parameter φ, then we have:\nlog ∫ A\n(s) k ,A (t) k\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = log\n∫ A\n(s) k ,A (t) k\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ(A(s)k ,A(t)k |G(s)k ,G(t)k ) Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k )\n= log E Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) ≥E\nQφ(A (s) k ,A (t) k |G (s) k ,G (t) k )\n[ logPθ(Z,A (s),A(t)|G(s),G(t))− logQφ(A(s),A(t)|G(s),G(t)) ]\n(26)\nwhere the final step is derived from Jensen’s inequality. Since optimizating Eq. 25 is difficult, we can alter to maximize the right hand side of inequality of Eq. (26) instead, which is the Evidence Lower Bound (ELBO) (Bishop, 2006). Since two input graphs are handled separately by two identical subroutines (see Fig. 2a), we can then impose the independence of topology A(s)k and A (t) k : Qφ(A (s),A(t)|G(s),G(t)) = Qφ(A(s)|G(s))Qφ(A(t)|G(t)). In this sense, we can utilize the same parameter φ to characterize two identical neural networks (generators) for modeling Qφ.\nAssuming θ is fixed, ELBO is determined by Qφ. According to Jensen’s inequality, equality of Eq. (26) holds when:\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ ( A (s) k ,A (t) k |G (s) k ,G (t) k\n) = c (27) where c 6= 0 is a constant. We then have:∫\nA (s) k ,A (t) k\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = c ∫ A\n(s) k ,A (t) k\nQφ\n( A\n(s) k ,A (t) k |G (s) k ,G (t) k\n) (28)\nAs Qφ is a distribution, we have:∫ A\n(s) k ,A (t) k\nQφ\n( A\n(s) k ,A (t) k |G (s) k ,G (t) k\n) = 1 (29)\nTherefore, we have: ∫ A\n(s) k ,A (t) k\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = c (30)\nWe now have:\nQφ\n( A\n(s) k ,A (t) k |G (s) k ,G (t) k\n) = Pθ\n( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) c\n= Pθ\n( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) ∫ A\n(s) k ,A (t) k\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k\n) Pθ ( Zk|G(s)k ,G (t) k\n) =Pθ ( A (s) k ,A (t) k |Zk,G (s) k ,G (t) k )\n(31)\nEq. (31) shows that, once θ is fixed, maximizing ELBO amounts to finding a distribution Qφ approximating the posterior probability Pθ ( A (s) k ,A (t) k |Zk,G (s) k ,G (t) k ) . This can be done by training the generatorQφ to produce latent topology A given graph pair and the matching Z. This corresponds to the Inference part in Sec. 3.4.2.\nA.4 ABLATION STUDY\nIn this part, we evaluate the performance of DLGM-D and DLGM-G by selectively deactivating different loss functions (refer Sec. 3.3 for more details of the functions). We also conduct the test on DLGM-G using different sample size of the generator. This ablation test is conducted on Pascal VOC dataset and average accuracy is reported in Tab. 5.\nWe first test the performance of both settings of DLGM by selectively activate the designated loss functions. Experimental results are summarized in Tab. 5a. As matching loss LM is essential for GM task, we constantly activate this loss for all settings. We see that the proposed novel losses LC and LL can consistently enhance the matching performance. Besides, DLGM-G indeed delivers better performance than DLGM-D under fair comparison.\nWe then test the impact of sample size from the generator Qφ under DLGM-G. Experimental results are summarized in Tab. 5b. We see that along with the increasing sample size, the average accuracy ascends. The performance becomes stable when the sample size reaches over 16.\nTable 5: Ablation test on Pascal VOC dataset. (a) Selectively deactivating loss functions on Pascal VOC. LM , LC and LL are selectively activated in DLGM-D and DLGM-G. “full” indicates all loss functions are activated. Average accuracy (%) is reported. (b) Average matching accuracy under different sampling sizes from the generator Qφ with “full” DLGM-G setting.\n(a) On losses\nmethod Ave DLGM-D (LM + LC) 79.8 DLGM-D (LM + LL) 79.5 DLGM-G (LM + LC) 80.9 DLGM-G (LM + LL) 80.4\nDLGM-D (full) 82.9 DLGM-G (full) 83.8\n(b) On sample size\n#Sample Ave 1 82.5 2 83.2 4 83.2 8 83.5\n16 83.8 32 83.7\nClass\naero & bike\nbird & boat\nbottle & bus\ncar & cat\nchair & cow\ntable & dog\nhorse & mbike\nperson & plant\nsheep & sofa\ntrain & tv\nTable 6: Matching examples of DLGM-G on 20 classes of Pascal VOC. The coloring of graphs and matchings follows the principle of Fig. 1 in the manuscript. Zoom in for better view.\nA.5 MORE VISUAL EXAMPLES AND ANALYSIS\nWe show more visual examples of matchings and generated topology using DLGM-G on Pascal VOC in Tab. 6 and Tab. 7, respectively. Each table follows distinct coloring regulation which will be detailed as follows:\n• Tab. 6. For each class, the left and right images corresponds to Delaunay triangulation. The image in the middle refers to the predicted matching and generated graph topology. Cyan solid and dashed lines correspond to correct and wrong matchings, respectively. Green dashed lines are the ground-truth matchings that are missed by our model. • Tab. 7. In this table, the leftmost and the rightmost columns correspond to original topology\nconstructed using Delaunay triangulation. The two columns in the middle are the generated topology using our method given Delaunay triangulation as prior. Blue edges are the edges that Delaunay and generated ones have in common. Green edges corresponds to the ones that are in Delaunay but not in generated topology, while red edges are the ones that are generated but not in Delaunay.\nWe give some analysis for the following questions."},{"heading":"In what case a different graph is generated?","text":"Since there are some generated graphs are identical to Delaunay, this question may naturally arise. We observe that, DLGM tends to produce an identical graph to Delaunay when objects are rarely with distortion and graphs are simple (e.g. tv, bottle and plant in Tab. 6 and last two rows in Tab. 7). However, when Delaunay is not sufficient to reveal the complex geometric relation or objects are with large distortion and feature diversity (e.g. cow and cat in Tab. 6 and person in Tab. 7), DLGM will resort to generating new topology with richer and stronger hint for graph matching. In other words, DLGM somewhat finds a way to identify if current instance pair is difficult or easy to match, and learns an adaptive strategy to handle these two cases.\nWhy DLGM-G delivers better performance than DLGM-D?\nIn general, DLGM-D is a deterministic gradient-based method. That is, the solution trajectory of DLGM-D almost follows the gradient direction at each iteration (with some variance from minibatch). Though it is assured to reach a local optima, only following gradient is too greedy since generated graph is coupled with predicted matching. Besides, as the topology is discrete, the optimal continuous solution will have a large objective score gap to its nearest discrete sampled solution once the landspace of the neural network is too sharp. On the other hand, DLGM-G performs discrete sampling under feasible graph distribution at each iteration, which generally but not fully follows the gradient direction. This procedure can thus find better discrete direction with probability, hence better exploring the searching space. This behavior is similar to Reinforcement Learning, but with much higher efficiency. Additionally, EM framework can guarantee the convergence (Bishop, 2006)."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Being a long standing NP-hard problem (Loiola et al., 2007), graph matching (GM) has received persistent attention from the machine learning and optimization communities for many years. Concretely, for two graphs with n nodes for each, graph matching seeks to solve1:\\nmax z\\nz>Mz s.t. Z ∈ {0, 1}n×n, Hz = 1 (1)\\nwhere the affinity matrix M ∈ Rn 2×n2\\n+ encodes node (diagonal elements) and edge (off-diagonal) affinities/similarities and z is the column-wise vectorization form of the permutation matrix Z. H is a selection matrix ensuring each row and column of Z summing to 1. 1 is a column vector filled with 1. Eq. (1) is the so-called quadratic assignment problem (QAP) (Cho et al., 2010). Maximizing Eq. (1) amounts to maximizing the sum of the similarity induced by matching vector Z. While Eq. (1) does not encode the topology of graphs, Zhou & Torre (2016) further propose to factorize M to explicitly incorporate topology matrix, where a connectivity matrix A ∈ {0, 1}n×n is used to indicate the topology of a single graph (Aij = 1 if there exists an edge between nodes i and j; Aij = 0 otherwise). To ease the computation, Eq. (1) is typically relaxed by letting z ∈ [0, 1]n 2 and keeping other parts of Eq. (1) intact. Traditional solvers to such relaxed problem generally fall into the categories of iterative update (Cho et al., 2010; Jiang et al., 2017) or numerical continuation (Zhou & Torre, 2016; Yu et al., 2018), where the solvers are developed under two key assumptions: 1) Affinity M is pre-computed with some non-negative metrics, e.g. Gaussian kernel, L2-distance or Manhattan distance; 2) Graph topology is pre-defined as input either in dense (Schellewald & Schnörr, 2005) or sparse (Zhou & Torre, 2016) fashion. There have been several successful attempts towards adjusting the first assumption by leveraging the power of deep networks to learn more effective graph representation for GM (Wang et al., 2019a; Yu et al., 2020; Fey et al., 2020). However, to our best knowledge, there is little previous work questioning and addressing the problem regarding the second assumption in the context of learning-based graph matching2. For example, existing\\n1Without loss of generality, we discuss graph matching under the setting of equal number of nodes without outliers. The unequal case can be readily handled by introducing extra constraints or dummy nodes. Bipartite matching and graph isomorphism are subsets of this quadratic formulation (Loiola et al., 2007).\\n2There are some loosely related works (Du et al., 2019; 2020) on network alignment and link prediction without learning, which will be discussed in detail in the related works.\\nstandard pipeline of keypoint matching in computer vision will construct initial topology by Delaunay triangulation or k-nearest neighbors. Then this topology will be freezed throughout the subsequent learning and matching procedures. In this sense, the construction of graph topology is peeled from matching task as a pre-processing stage. More examples can be found beyond the vision communities such as in social network alignment (Zhang & Tong, 2016; Heimann et al., 2018; Xiong & Yan, 2020) assuming fixed network structure for individual node matching in two networks.\\nWe argue that freezing graph topology for matching can hinder the capacity of graph matching solvers. For a pre-defined graph topology, the linked nodes sometimes result in less meaningful interaction, especially under the message-passing mechanism in graph neural networks (Kipf & Welling, 2017). We give a schematic demonstration in Fig. 1. Though some earlier attempts (Cho & Lee, 2012; Cho et al., 2013) seek to adjust the graph topology under traditional non-deep learning setting, such procedures cannot be readily integrated into end-to-end deep learning frameworks due to undifferentiable nature. Building upon the hypothesis that there exists some latent topology better than heuristically created one for GM, our aim is to learn it (or its distribution) for GM. Indeed, jointly solving matching and graph topology learning can be intimidating due to the combinatorial nature, which calls for more advanced approaches.\\nIn this paper, we propose an end-to-end framework to jointly learn the latent graph topology and perform GM, termed as deep latent graph matching (DLGM). We leverage the power of graph generative model to automatically produce graph topology from given features and their geometric relations, under specific locality prior. Different from generative learning on singleton graphs (Kipf & Welling, 2016; Bojchevski et al., 2018), our graph generative learning is performed in a pairwise fashion, leading to a novel matching-guided generative paradigm. The source code will be made publicly available.\\nContributions: 1) We explore a new direction for more flexible GM by actively learning latent topology, in contrast to previous works using fixed topology as input; 2) Under this setting, we propose a deterministic optimization approach to learn graph topology for matching; 3) We further present a generative way to produce latent topology under a probabilistic interpretation by ExpectationMaximization. This framework can also adapt to other problems where graph topology is the latent structure to infer; 4) Our method achieves state-ofthe-art performance on public benchmarks.\"\n },\n {\n \"heading\": \"2 RELATED WORKS\",\n \"text\": \"In this section, we first discuss existing works for graph topology and matching updating whose motivation is a bit similar to ours while the technique is largely different. Then we discuss relevant works in learning graph matching and generative graph models from the technical perspective.\\nTopology updating and matching. There are a few works for joint graph topology updating and matching, in the context of network alignment. Specifically, given two initial networks for matching, Du et al. (2019) show how to alternatively perform link prediction within each network and node matching across networks based on the observation that these two tasks can benefit to each other. In their extension (Du et al., 2020), a skip-gram embedding framework is further established under the same problem setting. In fact, these works involve a random-walk based node embedding updating and classification based link prediction modules and the whole algorithm runs in a one-shot optimization fashion. There is neither explicit training dataset nor trained matching model (except\\nfor the link classifier), which bears less flavor of machine learning. In contrast, our method involves training an explicit model for topology recovery and matching solving. Specifically, our deterministic technique (see Sec. 3.4.1) solves graph topology and matching in one-shot, while the proposed generative method alternatively estimates the topology and matching (see Sec. 3.4.2). Our approach allows to fully leverage multiple training samples in many applications like computer vision to boost the performance on test set. Moreover, the combinatorial nature of the matching problem is not addressed in (Du et al., 2019; 2020), and they adopt a greedy selection strategy instead. While we develop a principled combinatorial learning approach to this challenge. Also their methods rely on a considerable amount of seed matchings, yet this paper directly learns the latent topology from scratch which is more challenging and seldom studied.\\nLearning of graph matching. Early non-deep learning-based methods seek to learn effective metric (e.g. weighted Euclid distance) for node and edge features or affinity kernel (e.g. Gaussian kernel) in a parametric fashion (Caetano et al., 2009; Cho et al., 2013). Recent deep graph matching methods have shown how to extracte more dedicated feature representation. The work (Zanfir & Sminchisescu, 2018) adopts VGG16 (Simonyan & Zisserman, 2014) as the backbone for feature extraction on images. Other efforts have been witnessed in developing more advanced pipelines, where graph embedding (Wang et al., 2019a; Yu et al., 2020; Fey et al., 2020) and geometric learning (Zhang & Lee, 2019; Fey et al., 2020) are involved. Rolı́nek et al. (2020) study the way of incorporating traditional non-differentiable combinatorial solvers, by introducing a differentiatiable blackbox GM solver (Pogancic et al., 2020). Recent works in tackling combinatorial problem with deep learning (Huang et al., 2019; Kool & Welling, 2018) also inspire developing combinatorial deep solvers, for GM problems formulated by both Koopmans-Beckmann’s QAP (Nowak et al., 2018; Wang et al., 2019a) and Lawler’s QAP (Wang et al., 2019b). Specifically, Wang et al. (2019a) devise a permutation loss for supervised learning, with an improvement in Yu et al. (2020) via Hungarian attention. Wang et al. (2019b) solve the most general Lawler’s QAP with graph embedding technique.\\nGenerative graph model. Early generative models for graph can date back to (Erdos & Renyi, 1959), in which edges are generated with fixed probability. Recently, Kipf & Welling (2016) present a graph generative model by re-parameterizing the edge probability from Gaussian noise. Johnson (2017) propose to generate graph in an incremental fashion, and in each iteration a portion of the graph is produced. Gómez-Bombarelli et al. (2018) utilized recurrent neural network to generate graph from a sequence of molecule representation. Adversarial graph generation is considered in (Pan et al., 2018; Wang et al., 2018; Bojchevski et al., 2018). Specifically, Wang et al. (2018); Bojchevski et al. (2018) seek to unify graph generative model and generative adversarial networks. In parallel, reinforcement learning has been adopted to generate discrete graphs (De Cao & Kipf, 2018).\"\n },\n {\n \"heading\": \"3 LEARNING LATENT TOPOLOGY FOR GM\",\n \"text\": \"In this section, we describe details of the proposed framework with two specific algorithms derived from deterministic and generative perspectives, respectively. Both algorithms are motivated by the hypothesis that there exists some latent topology more suitable for matching rather than a fixed one. Note the proposed deterministic algorithm performs a standard forward-backward pass to jointly learn the topology and matching, while our generative algorithm consists of an alternative optimization procedure between estimating latent topology and learning matching under an Expectation-Maximization (EM) interpretation. In general, the generative algorithm assumes that a latent topology is sampled from a latent distribution, where the expected matching accuracy sufficing this distribution is maximized. Therefore, we expect to learn a topology generator sufficing such distribution. We reformulate GM into Bayesian fashion for consistent discussion in Sec. 3.1, detail deterministic/generative latent module in Sec. 3.2 and discuss the loss functions from a probabilistic perspective in Sec. 3.3. We finally elaborate on the holistic framework and the optimization procedure for both algorithms (deterministic and generative) in Sec. 3.4.\"\n },\n {\n \"heading\": \"3.1 PROBLEM DEFINITION AND BACKGROUND\",\n \"text\": \"GM problem can be viewed as a Bayesian variant of Eq. (1). In general, let G(s) and G(t) represent the initial source and target graphs for matching, respectively. We represent graph as G := {X,E,A}, where X ∈ Rn×d1 is the representation of n nodes with dimension d1. E ∈ Rm×d2 are features of\\nm edges and A ∈ {0, 1}n×n is initial connectivity (i.e. topology) matrix by heuristics e.g. Delaunay triangulation. For notational brevity, we assume d1 and d2 keep intact after updating the features across each convolutional layers of GNN (i.e., feature dimensions of both nodes and edges will not change after each layer’s update). Denote the matching Z ∈ {0, 1}n×n between two graphs, where Zij = 1 indicates a correspondence exists between node i in G(s) and node j in G(t), and Zij = 0 otherwise. Given training samples {Zk,G(s)k ,G (t) k } with k = 1, 2, ..., N , the objective of learning-based GM aims to maximize the likelihood:\\nmax θ ∏ k Pθ ( Zk|G(s)k ,G (t) k ) (2)\\nwhere θ denotes model parameters. Pθ(·) measures the probability of matching Zk given the k-th pair, and is instantiated via a network parameterized by θ.\\nBeing a generic module for producing latent topology, our method can be flexibly integrated into existing deep GM frameworks. We build up our method based on state-of-the-art (Rolı́nek et al., 2020), which utilizes SplineCNN (Fey et al., 2018) for node/edge representation learning. SplineCNN is a specific graph neural networks which updates a node representation via a weighted summation of its neighbors. The update rule at node i of a standard SplineCNN reads:\\n(x ∗ g)(i) = 1 |N (i)| d1∑ l=1 ∑ j∈N (i) xl(j) · gl(e(i, j)) (3)\\nwhere xl(j) performs the convolution on node j and outputs a d1-dimensional feature. gl(·) delivers the message weight given the edge feature e(i, j). N (i) refers to i’s neighboring nodes Summation over neighbors follows the topology A. Since our algorithm learns to generate topology, we need to explicitly express Eq. (3) in a differentiable way w.r.t. A. To this end, we rewrite Eq. (3) as:\\n(x ∗ g|A) = ( ◦G)X̂ (4)\\nwhere  is the normalized connectivity with each row normalized by the degree |N (i)| (see Eq. (3)) of the corresponding node i. G and X̂ correspond to outputs of gl(·) and xl(·) operators, respectively. (· ◦ ·) is the Hadamard product. With Eq. (4), we thus can perform back-propagation on connectivity/topology A. See more details in Appendix A.2.\"\n },\n {\n \"heading\": \"3.2 LATENT TOPOLOGY LEARNING\",\n \"text\": \"Existing learning-based graph matching algorithms consider A to be fixed throughout the computation without questioning if the input topology is optimal or not. This can be problematic since input graph construction is heuristic, and it never takes into account how suitable it is for the subsequent GM task. In our framework, instead of utilizing a fixed pre-defined topology, we consider to produce latent topology under two settings: 1) a deterministic and 2) a generative way. The former is often more efficient while the latter method can be more accurate at the cost of exploring more latent topology. Note both methods produce discrete topology to verify our hypothesis about the existence of more suitable discrete latent topology for GM problem. The followings describe two deep structures.\\nDeterministic learning: Given input features X and initial topology A, the deterministic way of generating latent topology A ∈ {0, 1}n×n is3:\\nAij = Rounding(sigmoid(y > i Wyj)) with Y = GCN(X,A) (5)\\nwhere GCN(·) is the graph convolutional networks (GCN) (Kipf & Welling, 2017) and yi corresponds to the feature of node i in feature map Y. W is the learnable parameter matrix. Note function Rounding(·) is undifferentiable, and will be discussed in Sec. 3.4.1. Generative learning: We reparameterize the representation as:\\nP (yi|X,A) = N (yi|µi,diag(σ2)) (6) 3We consider the case when only node feature E and topology A are necessary. Edge feature E can be\\nreadily integrated as another input.\\nwith µ = GCNµ(X,A) and σ = GCNσ(X,A) are two GCNs producing mean and covariance. It is equivalent to sampling a random vector from i.i.d. uniform distribution s ∼ U(0,1), then applying y = µ+ s · σ, where (·) is element-wise product. Similar as Eq. (5) by introducing learnable parameter W, the generative latent topology is sampled following i.i.d. distribution over each edge (i, j):\\nP (A|Y) = ∏ i ∏ j P (Aij |yi,yj) with P (Aij = 1|yi,yj) = sigmoid(y>i Wyj) (7)\\nSince sigmoid(·) maps any input into (0, 1), Eq. (7) can be interpreted as the probability of sampling edge (i, j). As the sampling procedure is undifferentiable, we apply Gumbel-softmax trick (Jang et al., 2017) as another reparameterization procedure. As such, a latent graph topology A can be sampled fully from distribution P (A) and the procedure becomes differentiable.\"\n },\n {\n \"heading\": \"3.3 LOSS FUNCTIONS\",\n \"text\": \"In this section, we explain three loss functions and the behind motivation: matching loss, locality loss and consistency loss. The corresponding probabilistic interpretation of each loss function can be found in Sec. 3.4.2. These functions are selectively activated in DLGM-D and DLGM-G (see Sec. 3.4). In DLGM-G, different loss functions are activated in inference and learning steps.\\ni) Matching loss. This common term measures how the predicted matching Ẑ diverges from groundtruth Z. Following Rolı́nek et al. (2020), we adopt Hamming distance on node-wise matching:\\nLM = Hamming(Ẑ,Z) (8)\\nii) Locality loss. This loss is devised to account for the general prior that the produced/learnt graph topology should advocate local connection rather than distant one, since two nodes may have less meaningful interaction once they are too distant from each other. In this sense, locality loss serves as a prior or regularizer in GM. As shown in multiple GM methods (Yu et al., 2018; Wang et al., 2019a; Fey et al., 2020), Delaunay triangulation is an effective way to deliver good locality. Therefore in our method, the locality loss is the Hamming distance between the initial topology A (obtained from Delaunay) and predicted topology A for both source graph and target graph:\\nLL = Hamming(A(s),A(s)) + Hamming(A(t),A(t)) (9)\\nWe emphasize that locality loss serves as a prior for latent graph. It focuses on advocating locality, but not reconstructing the initial Delaunay triangulation (as in Graph VAE (Kipf & Welling, 2016)).\\niii) Consistency loss. One can imagine that a GM solver is likely to deliver better performance if two graphs in a training pair are similar. In particular, we anticipate the latent topology A(s) and A(t) to be isomorphic under a specific matching, since isomorphic topological structures tend to be easier to match. Driven by this consideration, we devise the consistency loss which measures the level of isomorphism between latent topology A(s) and A(t):\\nLC(·|Z) = |Z>A(s)Z−A(t)|+ |ZA(t)Z> −A(s)| (10)\\nNote Z does not necessarily refer to the ground-truth, but can be any predicted matching. In this sense, latent topology A(s) and A(t) can be generated jointly given the matching Z as guidance information. This term can also serve as a consistency prior or regularization. We given a schematic example showing the merit of introducing consistency loss in Fig. 2(b).\"\n },\n {\n \"heading\": \"3.4 FRAMEWORK\",\n \"text\": \"A schematic diagram of our framework is given in Fig. 2(a) which consists of a singleton pipeline for processing a single image. It consists of three essential modules: a feature backbone (NB), a latent topology module (NG) and a feature refinement module (NR). Specifically, module NG corresponds to Sec. 3.2 with deterministic or generative implementations. Note the geometric relation of keypoints provide some prior for generating topology A. We employ VGG16 (Simonyan & Zisserman, 2014)\\nas NB and feed the produced node feature X and edge feature E to NG. NB also produces a global feature for each image. After generating the latent topology A, we pass over X and E together with A to NR (SplineCNN (Fey et al., 2018)). The holistic pipeline handling pairwise graph inputs can be found in Fig. 4 in Appendix A.1 which consists of two copies of singleton pipeline processing source and target data (in a Siamese fashion), respectively. Then the outputs of two singleton pipelines are formulated into affinity matrix, followed by a differentiable Blackbox GM solver (Pogancic et al., 2020) with message-passing mechanism (Swoboda et al., 2017). Note once without NG, the holistic pipeline with only NB +NR is identical to the method in (Rolı́nek et al., 2020). Readers are referred to this strong baseline (Rolı́nek et al., 2020) for more mutual algorithmic details.\\n3.4.1 OPTIMIZATION WITH DETERMINISTIC LATENT GRAPH\\nWe show how to optimize with deterministic latent graph module, where the topology A is produced by Eq. (5). The objective of matching conditioned on the produced latent topology A becomes:\\nmax ∏ k P ( Zk|A(s)k ,A (t) k ,G (s) k ,G (t) k ) (11)\\nEq. (11) can be optimized with standard back-propagation with three loss terms activated, except for the Rounding function (see Eq. (5)), which makes the procedure undifferentiable. To address this, we use straight-through operator (Bengio et al., 2013) which performs a standard rounding during the forward pass but approximates it with the gradient of identity during the backward pass on [0, 1]:\\n∂Rounding(x)/∂x = 1 (12)\\nThough there exist some unbiased gradient estimators (e.g., REINFORCE (Williams, 1992)), the biased straight-through estimator proved to be more efficient and has been successfully applied in several applications (Chung et al., 2017; Campos et al., 2018). All the network modules (NG +NB + NR) are simultaneously learned during the training. All three losses are activated in the learning procedure (see Sec. 3.3), which are applied on the predicted matching Ẑ, the latent topology A(s) and A(t). We term the algorithm under this setting DLGM-D.\\n3.4.2 OPTIMIZATION WITH GENERATIVE LATENT GRAPH\\nSee more details in Appendix A.3. In this setting, the source and target latent topology A(s) and A(t) are sampled according to Eq. (6) and (7). The objective becomes:\\nmax ∏ k ∫ A (s) k ,A (t) k Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) (13)\\nUnfortunately, directly optimizing Eq. (13) is difficult due to the integration over A which is intractable. Instead, we maximize the evidence lower bound (ELBO) (Bishop, 2006) as follows:\\nlogPθ(Z|G(s),G(t)) ≥ EQφ(A(s),A(t)|G(s),G(t)) [ logPθ(Z,A (s),A(t)|G(s),G(t))− logQφ(A(s),A(t)|G(s),G(t)) ] (14)\\nwhere Qφ(A(s),A(t)|G(s),G(t)) can be any joint distribution of A(s) and A(t) given the input graphs G(s) and G(t). Equality of Eq. (14) holds when Qφ(A(s),A(t)|G(s),G(t)) = Pθ(A\\n(s),A(t)|Z,G(s),G(t)). For tractability, we rationally introduce the independence by assuming that we can use an identical latent topology module Qφ (corresponding to NG in Fig. 2(a)) to separately handle each input graph:\\nQφ(A (s),A(t)|G(s),G(t)) = Qφ(A(s)|G(s))Qφ(A(t)|G(t)) (15)\\nwhich can greatly simplify the model complexity. Then we can utilize a neural network to model Qφ (similar to modeling Pθ). The optimization of Eq. (14) is studied in (Neal & Hinton, 1998), known as the Expectation-Maximization (EM) algorithm. Optimization of Eq. (14) alternates between E-step and M-step. During E-step (inference), Pθ is fixed and the algorithm seeks to find an optimal Qφ to approximate the true posterior distribution (see Appendix A.3 for explanation):\\nPθ(A (s),A(t)|Z,G(s),G(t)) (16)\\nDuring M-step (learning), Qφ is instead fixed and algorithm alters to maximize the likelihood: EQφ(A(s)|G(s)),Qφ(A(t)|G(t)) [ logPθ(Z,A (s),A(t)|G(s),G(t)) ] ∝ −LM (17)\\nWe give more details on the inference and learning steps as follows.\\nInference. This step focuses on deriving posterior distribution Pθ(A(s),A(t)|Z,G(s),G(t)) using its approximation Qφ. To this end, we fix the parameters θ in modules NB and NR, and only update the parameters φ in module NG corresponding to Qφ. As stated in Sec. 3.2, we employ the Gumbel-softmax trick for sampling discrete A (Jang et al., 2017). To this end, we can formulate a 2D vector aij = [P (Aij = 1), 1− P (Aij = 1)]>. Then the sampling becomes:\\nsoftmax (log(aij) + hij ; τ) (18) where hij is a random 2D vector from Gumbel distribution, and τ is a small temperature parameter. We further impose prior on latent topology A given A through locality loss:\\nlog ∏ i,j P (Aij |Aij) ∝ −LL(A,A) (19)\\nwhich is to preserve the locality in initial topology A. It should also be noted that Z is the predicted matching from current Pθ, as Qφ is an approximation. Besides, we also anticipate two generated topology A(s) and A(t) from a graph pair should be similar (isomorphic) given matching Z:\\nlogP ( A(s),A(t)|Z ) ∝ −LC ( A(s),A(t)|Z ) (20)\\nIn summary, we activate locality loss and consistency loss during the inference step, where the latter loss is conditioned with the predicted matching rather than the ground-truth. Note that the inference step involves twice re-parameterization tricks corresponding to Eq. (6) and (18), respectively. While the first generates the continuous topology distribution under edge independence assumption, the second performs discrete sampling sufficing the generated topology distribution.\\nLearning. This step optimizes Pθ by fixing Qφ. We sample discrete graph topologies As completely from the probability of edge P (Aij = 1). Once latent topology As are sampled, we feed them to module NR together with the node-level features from NB . Only NB and NR are updated in this step, and only matching loss LM is activated. Remark. Note for each pair of graphs in training, we use an identical random vector s for generating both graphs’ topology (see Eq. (6)). We pretrain the network Pθ before alternativly training Pθ and Qφ. During pretraining, we activate NB + NR modules and LM loss during pretraining, and feed the network the topology obtained from Delaunay as the latent topology. After pretraining, the optimization will switch between inference and learning steps until convergence. We term the setting of generative latent graph matching as DLGM-G and summarize it in Alg. 1.\\nmethod aero bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tv Ave BBGM-max 35.5 68.6 46.7 36.1 85.4 58.1 25.6 51.7 27.3 51.0 46.0 46.7 48.9 58.9 29.6 93.6 42.6 35.3 70.7 79.5 51.9\\nBBGM 42.7 70.9 57.5 46.6 85.8 64.1 51.0 63.8 42.4 63.7 47.9 61.5 63.4 69.0 46.1 94.2 57.4 39.0 78.0 82.7 61.4 DLGM-D (ours) 42.5 71.8 57.8 46.8 86.9 70.3 53.4 66.7 53.8 67.6 64.7 64.6 65.2 70.1 47.9 95.5 59.6 47.7 77.7 82.6 63.9 DLGM-G (ours) 43.8 72.9 58.5 47.4 86.4 71.2 53.1 66.9 54.6 67.8 64.9 65.7 66.9 70.8 47.4 96.5 61.4 48.4 77.5 83.9 64.8\"\n },\n {\n \"heading\": \"4 EXPERIMENT\",\n \"text\": \"We conduct experiments on datasets including Pascal VOC with Berkeley annotation (Everingham et al., 2010; Bourdev & Malik, 2009), Willow ObjectClass (Cho et al., 2013) and SPair-71K (Min et al., 2019). We report the per-category and average performance. The objective of all experiments is to maximize the average matching accuracy. Both our DLGM-D and DLGM-G are tested.\\nPeer methods. We conduct comparison experiments against the following algorithms: 1) GMN (Zanfir & Sminchisescu, 2018), which is a seminal work incorporating graph matching into deep learning framework equipped with a spectral solver (Egozi et al., 2013); 2) PCA (Wang et al., 2019a). This method treats graph matching as feature matching problem and employs GCN (Kipf & Welling, 2017) to learn better features; 3) CIE1/GAT-H (Yu et al., 2020). This paper develops a novel embedding and attention mechanism, where GAT-H is the version by replacing the basic embedding block with Graph Attention Networks (Veličković et al., 2018); 4) DGMC (Fey et al., 2020). This method devises a post-processing step by emphasizing the neighborhood similarity; 5) BBGM (Rolı́nek et al., 2020). It integrates a differentiable linear combinatorial solver (Pogancic et al., 2020) into a deep learning framework and achieves state-of-the-art performance.\\nResults on Pascal VOC. The dataset (Everingham et al., 2010; Bourdev & Malik, 2009) consists of 7,020 training images and 1,682 testing images with 20 classes in total, together with the object bounding boxing for each. Following the data preparation in (Wang et al., 2019a), each object within the bounding box is cropped and resized to 256× 256. The number of nodes per graph ranges from 6 to 23. We further follow (Rolı́nek et al., 2020) under two evaluating metrics: 1) Accuracy: this is the standard metric evaluated on the keypoints by filtering out the outliers; 2) F1-score: this metric is evaluated without keypoint filtering, being the harmonic mean of precision and recall.\\nExperimental results on the two setting are shown in Tab. 1 and Tab. 2. The proposed method under either settings of DLGM-D and DLGM-G outperforms counterparts by accuracy and f1-score. DLGM-G generally outperforms DLGM-D. Discussion can be found in Appendix A.5.\\nQuality of generated topology. We further show the consistency/locality curve vs epoch in Fig. 3, since both consistency and locality losses can somewhat reflect the quality of topology generation. It shows that both locality and consistency losses descend during the training. Note that the consistency loss with Delaunay triangulation (green dashed line) is far more larger than our generated ones (blue/red dashed line). This clearly supports the claim that our method generates similar (more isomorphic) typologies, as well as preserving locality.\\nResults on Willow Object. The benchmark (Cho et al., 2013) consists of 256 images in 5 categories, where two categories (car and motorbike) are subsets selected from Pascal VOC. Following the preparation protocol in Wang et al. (2019a), we crop the image within the object bounding box and resize it to 256 × 256. Since the dataset is relatively small, we conduct the experiment to verify the transfer ability of different methods under two settings: 1) trained on Pascal VOC and directly applied to Willow (Pt); 2) trained on Pascal VOC then finetuned on Willow (Wt). Results under the two settings are shown in Tab. 3. Since this dataset is relatively small, further improvement is difficult. It is shown both DLGM-D and DLGM-G have good transfer ability.\\nResults on SPair-71K. The dataset (Min et al., 2019) is much larger than Pascal VOC and WillowObject since it consists of 70,958 image pairs collected from Pascal VOC 2012 and Pascal 3D+ (53,340 for training, 5,384 for validation and 12,234 for testing). It improves Pascal VOC by removing ambiguous categories sofa and dining table. This dataset is considered to contain more difficult matching instances and higher annotation quality. Results are summarized in Tab. 4. Our method consistently improves the matching performance, agreeing with the results in Pascal VOC and Willow.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"Graph matching involves two essential factors: the affinity model and topology. By incorporating learning paradigm for affinity/feature, the performance of matching on public datasets has significantly been improved. However, there has been little previous work exploring more effective topology for matching. In this paper, we argue that learning a more effective graph topology can significantly improve the matching, thus being essential. To this end, we propose to incorporate a latent topology module under an end-to-end deep network framework that learns to produce better graph topology. We also present the interpretation and optimization of topology module in both deterministic and generative perspectives, respectively. Experimental results show that, by learning the latent graph, the matching performance can be consistently and significantly enhanced on several public datasets.\"\n },\n {\n \"heading\": \"A APPENDIX\",\n \"text\": \"A.1 HOLISTIC PIPELINE\\nWe show the holistic pipeline of our framework in Fig. 4 consisting of two “singleton pipelines” (see introduction part of Sec. 3 for more details). In general, the holistic pipeline follows the convention in a series of deep graph matching methods by utilizing an identical singleton pipeline to extract features, then exploits the produced features to perform matching (Yu et al., 2020; Wang et al., 2019a; Fey et al., 2020; Rolı́nek et al., 2020). Except for the topology module NG, all others parts of our network are the same as those in Rolı́nek et al. (2020).\\nA.2 SPLINECNN\\nSplineCNN is a method to perform graph-based representation learning via convolution operators defined based on B-splines (Fey et al., 2018). The initial input to SplineCNN is G = {X,E,A}, where X ∈ Gn×d1 and A ∈ {0, 1}n×n indicate node features and topology, respectively (same as in Sec. 3.1). E ∈ [0, 1]n×n×d2 is so-called pseudo-coordinates and can be viewed as n2 × d2dimensional edge features for a fully connected graph (in case m = n2, see Sec. 3.1). Let normalized edge feature e(i, j) = Ei,j,: ∈ [0, 1]d2 if a directed edge (i, j) exists (Ai,j = 1), and 0 otherwise (Ai,j = 0). Note topology A fully carries the information of N (i) which defines the neighborhood\\nof node i. During the learning, X and E will be updated while topology A will not. Therefore SplineCNN is a geometric graph embedding method without adjusting the latent graph topology.\\nB-spline is employed as basic kernel in SplineCNN, where a basis function has only support on a specific real-valued interval (Piegl & Tiller, 2012). Let ((Nq1,i)1≤i≤k1 , ..., (N q d,i)1≤i≤kd2 ) be d2 B-spline bases with degree q. The kernel size is defined in k = (k1, ..., kd2). In SplineCNN, the continuous kernel function gl : [a1, b1]× ...× [ad2 , bd2 ]→ G is defined as:\\ngl(e) = ∑ p∈P wp,l ·Bp(e) (21)\\nwhere P = (Nq1,i)i × ... × (N q d,i)i is the B-spline bases (Piegl & Tiller, 2012) and wp,l is the trainable parameter corresponding to the lth node feature in X, with Bp being the product of the basis functions in P:\\nBp = d∏ i=1 Nqi,pi(ei) (22)\\nwhere e is the pseudo-coordinate in E. Then, given the kernel function g = (g1, ..., gd1) and the node feature X ∈ Gn×d1 , one layer of the convolution at node i in SplineCNN reads (same as Eq. (3)):\\n(x ∗ g)(i) = 1 |N (i)| d1∑ l=1 ∑ j∈N (i) xl(j) · gl(e(i, j)) (23)\\nwhere xl(j) indicates the convolved node feature value of node j at lth dimension. This formulation can be tensorized into Eq. (4) with explicit topology matrix A. In this sense, we can back-propagate the gradient of A. Reader are referred to Fey et al. (2018) for more comprehensive understanding of this method.\\nA.3 DERIVATION OF DLGM-D\\nWe give more details of the optimization on DLGM-D in this section. This part also interprets some basic formulation conversion (e.g. from Eq. (2) to its Bayesian form). First, we assume there is no latent topology A(s) and A(s) at the current stage. In this case, the objective of GM is simply:\\nmax ∏ k Pθ ( Zk|G(s)k ,G (t) k ) (24)\\nwhere Pθ measures the probability of a matching Zk given graph pair G(s)k and G (t) k . If we impose the latent topology A(s) and A(t), as well as some distribution over them, then Eq. (24) can be equivalently expressed as:\\nmax ∏ k Pθ ( Zk|G(s)k ,G (t) k ) = max ∏ k ∫ A (s) k ,A (t) k Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) (25)\\nwhere Pθ ( Zk|G(s)k ,G (t) k ) is the marginal distribution of Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) with respect\\nto Zk, since A (s) k and A (t) k are integrated over some distribution. Herein we can impose another distribution of the topology Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) characterized by parameter φ, then we have:\\nlog ∫ A\\n(s) k ,A (t) k\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = log\\n∫ A\\n(s) k ,A (t) k\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ(A(s)k ,A(t)k |G(s)k ,G(t)k ) Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k )\\n= log E Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ(A (s) k ,A (t) k |G (s) k ,G (t) k ) ≥E\\nQφ(A (s) k ,A (t) k |G (s) k ,G (t) k )\\n[ logPθ(Z,A (s),A(t)|G(s),G(t))− logQφ(A(s),A(t)|G(s),G(t)) ]\\n(26)\\nwhere the final step is derived from Jensen’s inequality. Since optimizating Eq. 25 is difficult, we can alter to maximize the right hand side of inequality of Eq. (26) instead, which is the Evidence Lower Bound (ELBO) (Bishop, 2006). Since two input graphs are handled separately by two identical subroutines (see Fig. 2a), we can then impose the independence of topology A(s)k and A (t) k : Qφ(A (s),A(t)|G(s),G(t)) = Qφ(A(s)|G(s))Qφ(A(t)|G(t)). In this sense, we can utilize the same parameter φ to characterize two identical neural networks (generators) for modeling Qφ.\\nAssuming θ is fixed, ELBO is determined by Qφ. According to Jensen’s inequality, equality of Eq. (26) holds when:\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) Qφ ( A (s) k ,A (t) k |G (s) k ,G (t) k\\n) = c (27) where c 6= 0 is a constant. We then have:∫\\nA (s) k ,A (t) k\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = c ∫ A\\n(s) k ,A (t) k\\nQφ\\n( A\\n(s) k ,A (t) k |G (s) k ,G (t) k\\n) (28)\\nAs Qφ is a distribution, we have:∫ A\\n(s) k ,A (t) k\\nQφ\\n( A\\n(s) k ,A (t) k |G (s) k ,G (t) k\\n) = 1 (29)\\nTherefore, we have: ∫ A\\n(s) k ,A (t) k\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = c (30)\\nWe now have:\\nQφ\\n( A\\n(s) k ,A (t) k |G (s) k ,G (t) k\\n) = Pθ\\n( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) c\\n= Pθ\\n( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) ∫ A\\n(s) k ,A (t) k\\nPθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k ) = Pθ ( Zk,A (s) k ,A (t) k |G (s) k ,G (t) k\\n) Pθ ( Zk|G(s)k ,G (t) k\\n) =Pθ ( A (s) k ,A (t) k |Zk,G (s) k ,G (t) k )\\n(31)\\nEq. (31) shows that, once θ is fixed, maximizing ELBO amounts to finding a distribution Qφ approximating the posterior probability Pθ ( A (s) k ,A (t) k |Zk,G (s) k ,G (t) k ) . This can be done by training the generatorQφ to produce latent topology A given graph pair and the matching Z. This corresponds to the Inference part in Sec. 3.4.2.\\nA.4 ABLATION STUDY\\nIn this part, we evaluate the performance of DLGM-D and DLGM-G by selectively deactivating different loss functions (refer Sec. 3.3 for more details of the functions). We also conduct the test on DLGM-G using different sample size of the generator. This ablation test is conducted on Pascal VOC dataset and average accuracy is reported in Tab. 5.\\nWe first test the performance of both settings of DLGM by selectively activate the designated loss functions. Experimental results are summarized in Tab. 5a. As matching loss LM is essential for GM task, we constantly activate this loss for all settings. We see that the proposed novel losses LC and LL can consistently enhance the matching performance. Besides, DLGM-G indeed delivers better performance than DLGM-D under fair comparison.\\nWe then test the impact of sample size from the generator Qφ under DLGM-G. Experimental results are summarized in Tab. 5b. We see that along with the increasing sample size, the average accuracy ascends. The performance becomes stable when the sample size reaches over 16.\\nTable 5: Ablation test on Pascal VOC dataset. (a) Selectively deactivating loss functions on Pascal VOC. LM , LC and LL are selectively activated in DLGM-D and DLGM-G. “full” indicates all loss functions are activated. Average accuracy (%) is reported. (b) Average matching accuracy under different sampling sizes from the generator Qφ with “full” DLGM-G setting.\\n(a) On losses\\nmethod Ave DLGM-D (LM + LC) 79.8 DLGM-D (LM + LL) 79.5 DLGM-G (LM + LC) 80.9 DLGM-G (LM + LL) 80.4\\nDLGM-D (full) 82.9 DLGM-G (full) 83.8\\n(b) On sample size\\n#Sample Ave 1 82.5 2 83.2 4 83.2 8 83.5\\n16 83.8 32 83.7\\nClass\\naero & bike\\nbird & boat\\nbottle & bus\\ncar & cat\\nchair & cow\\ntable & dog\\nhorse & mbike\\nperson & plant\\nsheep & sofa\\ntrain & tv\\nTable 6: Matching examples of DLGM-G on 20 classes of Pascal VOC. The coloring of graphs and matchings follows the principle of Fig. 1 in the manuscript. Zoom in for better view.\\nA.5 MORE VISUAL EXAMPLES AND ANALYSIS\\nWe show more visual examples of matchings and generated topology using DLGM-G on Pascal VOC in Tab. 6 and Tab. 7, respectively. Each table follows distinct coloring regulation which will be detailed as follows:\\n• Tab. 6. For each class, the left and right images corresponds to Delaunay triangulation. The image in the middle refers to the predicted matching and generated graph topology. Cyan solid and dashed lines correspond to correct and wrong matchings, respectively. Green dashed lines are the ground-truth matchings that are missed by our model. • Tab. 7. In this table, the leftmost and the rightmost columns correspond to original topology\\nconstructed using Delaunay triangulation. The two columns in the middle are the generated topology using our method given Delaunay triangulation as prior. Blue edges are the edges that Delaunay and generated ones have in common. Green edges corresponds to the ones that are in Delaunay but not in generated topology, while red edges are the ones that are generated but not in Delaunay.\\nWe give some analysis for the following questions.\"\n },\n {\n \"heading\": \"In what case a different graph is generated?\",\n \"text\": \"Since there are some generated graphs are identical to Delaunay, this question may naturally arise. We observe that, DLGM tends to produce an identical graph to Delaunay when objects are rarely with distortion and graphs are simple (e.g. tv, bottle and plant in Tab. 6 and last two rows in Tab. 7). However, when Delaunay is not sufficient to reveal the complex geometric relation or objects are with large distortion and feature diversity (e.g. cow and cat in Tab. 6 and person in Tab. 7), DLGM will resort to generating new topology with richer and stronger hint for graph matching. In other words, DLGM somewhat finds a way to identify if current instance pair is difficult or easy to match, and learns an adaptive strategy to handle these two cases.\\nWhy DLGM-G delivers better performance than DLGM-D?\\nIn general, DLGM-D is a deterministic gradient-based method. That is, the solution trajectory of DLGM-D almost follows the gradient direction at each iteration (with some variance from minibatch). Though it is assured to reach a local optima, only following gradient is too greedy since generated graph is coupled with predicted matching. Besides, as the topology is discrete, the optimal continuous solution will have a large objective score gap to its nearest discrete sampled solution once the landspace of the neural network is too sharp. On the other hand, DLGM-G performs discrete sampling under feasible graph distribution at each iteration, which generally but not fully follows the gradient direction. This procedure can thus find better discrete direction with probability, hence better exploring the searching space. This behavior is similar to Reinforcement Learning, but with much higher efficiency. Additionally, EM framework can guarantee the convergence (Bishop, 2006).\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":193,"cells":{"paper_id":{"kind":"string","value":"SP:879ce870f09e422aced7d008abc42fe5a8db29bc"},"summaries":{"kind":"list like","value":["The paper proposes a method for stabilizing the training of GAN as well as overcoming the problem of mode collapse by optimizing several auxiliary models. The first step is to learn a latent space using an autoencoder. Then, this latent space is \"intervened\" by a predefined set of $K$ transformations to generate a set of distributions $p_k$. A classifier is then taught to distinguish between $p_k$. Eventually, the weights of the classifier are shared with those of the discriminator network to produce the desired stabilization/diversification effect. In other words, the authors propose to stabilize GANs by intervening with the discriminator. This is done by sharing its weights with a classifier that trains on a perturbed latent distribution that is somehow related to the original problem via the prior assumption imposed."],"string":"[\n \"The paper proposes a method for stabilizing the training of GAN as well as overcoming the problem of mode collapse by optimizing several auxiliary models. The first step is to learn a latent space using an autoencoder. Then, this latent space is \\\"intervened\\\" by a predefined set of $K$ transformations to generate a set of distributions $p_k$. A classifier is then taught to distinguish between $p_k$. Eventually, the weights of the classifier are shared with those of the discriminator network to produce the desired stabilization/diversification effect. In other words, the authors propose to stabilize GANs by intervening with the discriminator. This is done by sharing its weights with a classifier that trains on a perturbed latent distribution that is somehow related to the original problem via the prior assumption imposed.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"In this paper we propose a novel approach for stabilizing the training process of Generative Adversarial Networks as well as alleviating the mode collapse problem. The main idea is to incorporate a regularization term that we call intervention into the objective. We refer to the resulting generative model as Intervention Generative Adversarial Networks (IVGAN). By perturbing the latent representations of real images obtained from an auxiliary encoder network with Gaussian invariant interventions and penalizing the dissimilarity of the distributions of the resulting generated images, the intervention term provides more informative gradient for the generator, significantly improving training stability and encouraging modecovering behaviour. We demonstrate the performance of our approach via solid theoretical analysis and thorough evaluation on standard real-world datasets as well as the stacked MNIST dataset."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["M Arjovsky","L Bottou"],"title":"Towards principled methods for training generative adversarial networks. arxiv 2017","venue":"arXiv preprint arXiv:1701.04862","year":2017},{"authors":["Martin Arjovsky","Soumith Chintala","Léon Bottou"],"title":"Wasserstein generative adversarial networks","venue":"In International Conference on Machine Learning,","year":2017},{"authors":["Tong Che","Yanran Li","Athul Jacob","Yoshua Bengio","Wenjie Li"],"title":"Mode regularized generative adversarial networks. 2016","venue":null,"year":2016},{"authors":["Adam Coates","Andrew Ng","Honglak Lee"],"title":"An analysis of single-layer networks in unsupervised feature learning","venue":"In Proceedings of the fourteenth international conference on artificial intelligence and statistics,","year":2011},{"authors":["Ian J. Goodfellow","Jean Pouget-Abadie","Mehdi Mirza","Bing Xu","David Warde-Farley","Sherjil Ozair","Aaron C. Courville","Yoshua Bengio"],"title":"Generative adversarial nets","venue":"In Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems","year":2014},{"authors":["Ishaan Gulrajani","Faruk Ahmed","Martin Arjovsky","Vincent Dumoulin","Aaron C Courville"],"title":"Improved training of wasserstein gans","venue":"In Advances in neural information processing systems,","year":2017},{"authors":["Martin Heusel","Hubert Ramsauer","Thomas Unterthiner","Bernhard Nessler","Sepp Hochreiter"],"title":"Gans trained by a two time-scale update rule converge to a local nash equilibrium","venue":"In Advances in neural information processing systems,","year":2017},{"authors":["Simon Jenni","Paolo Favaro"],"title":"On stabilizing generative adversarial training with noise, 2019","venue":null,"year":2019},{"authors":["Hyunjik Kim","Andriy Mnih"],"title":"Disentangling by factorising","venue":"arXiv preprint arXiv:1802.05983,","year":2018},{"authors":["D.P. Kingma","J. Ba"],"title":"Adam: A method for stochastic optimization","venue":"In ICLR,","year":2015},{"authors":["Naveen Kodali","James Hays","Jacob Abernethy","Zsolt Kira"],"title":"On convergence and stability of gans","venue":null,"year":2018},{"authors":["Alex Krizhevsky"],"title":"Learning multiple layers of features from tiny images","venue":null,"year":2009},{"authors":["Anders Boesen Lindbo Larsen","Søren Kaae Sønderby","Ole Winther"],"title":"Autoencoding beyond pixels using a learned similarity","venue":"metric. CoRR,","year":2015},{"authors":["Y. Lecun","L. Bottou","Y. Bengio","P. Haffner"],"title":"Gradient-based learning applied to document recognition","venue":"Proceedings of the IEEE,","year":1998},{"authors":["Zinan Lin","Ashish Khetan","Giulia Fanti","Sewoong Oh"],"title":"Pacgan: The power of two samples in generative adversarial networks","venue":"Advances in Neural Information Processing Systems","year":2018},{"authors":["Xudong Mao","Qing Li","Haoran Xie","Raymond Y.K. Lau","Zhen Wang"],"title":"Multi-class generative adversarial networks with the L2 loss function","venue":"CoRR, abs/1611.04076,","year":2016},{"authors":["Luke Metz","Ben Poole","David Pfau","Jascha Sohl-Dickstein"],"title":"Unrolled generative adversarial networks","venue":"In 5th International Conference on Learning Representations,","year":2017},{"authors":["Alec Radford","Luke Metz","Soumith Chintala"],"title":"Unsupervised representation learning with deep convolutional generative adversarial networks","venue":"arXiv preprint arXiv:1511.06434,","year":2015},{"authors":["Tim Salimans","Ian Goodfellow","Wojciech Zaremba","Vicki Cheung","Alec Radford","Xi Chen"],"title":"Improved techniques for training gans","venue":"In Advances in neural information processing systems,","year":2016},{"authors":["Akash Srivastava","Lazar Valkov","Chris Russell","Michael U Gutmann","Charles Sutton"],"title":"Veegan: Reducing mode collapse in gans using implicit variational learning","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["Ngoc-Trung Tran","Tuan-Anh Bui","Ngai-Man Cheung"],"title":"Dist-gan: An improved gan using distance constraints","venue":"In Proceedings of the European Conference on Computer Vision (ECCV),","year":2018},{"authors":["Fisher Yu","Yinda Zhang","Shuran Song","Ari Seff","Jianxiong Xiao"],"title":"Lsun: Construction of a largescale image dataset using deep learning with humans in the loop","venue":"arXiv preprint arXiv:1506.03365,","year":2015},{"authors":["Zhiming Zhou","Jiadong Liang","Yuxuan Song","Lantao Yu","Hongwei Wang","Weinan Zhang","Yong Yu","Zhihua Zhang"],"title":"Lipschitz generative adversarial nets","venue":null,"year":1902}],"string":"[\n {\n \"authors\": [\n \"M Arjovsky\",\n \"L Bottou\"\n ],\n \"title\": \"Towards principled methods for training generative adversarial networks. arxiv 2017\",\n \"venue\": \"arXiv preprint arXiv:1701.04862\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Martin Arjovsky\",\n \"Soumith Chintala\",\n \"Léon Bottou\"\n ],\n \"title\": \"Wasserstein generative adversarial networks\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Tong Che\",\n \"Yanran Li\",\n \"Athul Jacob\",\n \"Yoshua Bengio\",\n \"Wenjie Li\"\n ],\n \"title\": \"Mode regularized generative adversarial networks. 2016\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Adam Coates\",\n \"Andrew Ng\",\n \"Honglak Lee\"\n ],\n \"title\": \"An analysis of single-layer networks in unsupervised feature learning\",\n \"venue\": \"In Proceedings of the fourteenth international conference on artificial intelligence and statistics,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Ian J. Goodfellow\",\n \"Jean Pouget-Abadie\",\n \"Mehdi Mirza\",\n \"Bing Xu\",\n \"David Warde-Farley\",\n \"Sherjil Ozair\",\n \"Aaron C. Courville\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Generative adversarial nets\",\n \"venue\": \"In Advances in Neural Information Processing Systems 27: Annual Conference on Neural Information Processing Systems\",\n \"year\": 2014\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 Advances in neural information processing systems,\",\n \"year\": 2017\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\": \"In Advances in neural information processing systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Simon Jenni\",\n \"Paolo Favaro\"\n ],\n \"title\": \"On stabilizing generative adversarial training with noise, 2019\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Hyunjik Kim\",\n \"Andriy Mnih\"\n ],\n \"title\": \"Disentangling by factorising\",\n \"venue\": \"arXiv preprint arXiv:1802.05983,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"D.P. Kingma\",\n \"J. Ba\"\n ],\n \"title\": \"Adam: A method for stochastic optimization\",\n \"venue\": \"In ICLR,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Naveen Kodali\",\n \"James Hays\",\n \"Jacob Abernethy\",\n \"Zsolt Kira\"\n ],\n \"title\": \"On convergence and stability of gans\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Alex Krizhevsky\"\n ],\n \"title\": \"Learning multiple layers of features from tiny images\",\n \"venue\": null,\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Anders Boesen Lindbo Larsen\",\n \"Søren Kaae Sønderby\",\n \"Ole Winther\"\n ],\n \"title\": \"Autoencoding beyond pixels using a learned similarity\",\n \"venue\": \"metric. CoRR,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Y. Lecun\",\n \"L. Bottou\",\n \"Y. Bengio\",\n \"P. Haffner\"\n ],\n \"title\": \"Gradient-based learning applied to document recognition\",\n \"venue\": \"Proceedings of the IEEE,\",\n \"year\": 1998\n },\n {\n \"authors\": [\n \"Zinan Lin\",\n \"Ashish Khetan\",\n \"Giulia Fanti\",\n \"Sewoong Oh\"\n ],\n \"title\": \"Pacgan: The power of two samples in generative adversarial networks\",\n \"venue\": \"Advances in Neural Information Processing Systems\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Xudong Mao\",\n \"Qing Li\",\n \"Haoran Xie\",\n \"Raymond Y.K. Lau\",\n \"Zhen Wang\"\n ],\n \"title\": \"Multi-class generative adversarial networks with the L2 loss function\",\n \"venue\": \"CoRR, abs/1611.04076,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Luke Metz\",\n \"Ben Poole\",\n \"David Pfau\",\n \"Jascha Sohl-Dickstein\"\n ],\n \"title\": \"Unrolled generative adversarial networks\",\n \"venue\": \"In 5th International Conference on Learning Representations,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Alec Radford\",\n \"Luke Metz\",\n \"Soumith Chintala\"\n ],\n \"title\": \"Unsupervised representation learning with deep convolutional generative adversarial networks\",\n \"venue\": \"arXiv preprint arXiv:1511.06434,\",\n \"year\": 2015\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 Advances in neural information processing systems,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Akash Srivastava\",\n \"Lazar Valkov\",\n \"Chris Russell\",\n \"Michael U Gutmann\",\n \"Charles Sutton\"\n ],\n \"title\": \"Veegan: Reducing mode collapse in gans using implicit variational learning\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Ngoc-Trung Tran\",\n \"Tuan-Anh Bui\",\n \"Ngai-Man Cheung\"\n ],\n \"title\": \"Dist-gan: An improved gan using distance constraints\",\n \"venue\": \"In Proceedings of the European Conference on Computer Vision (ECCV),\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Fisher Yu\",\n \"Yinda Zhang\",\n \"Shuran Song\",\n \"Ari Seff\",\n \"Jianxiong Xiao\"\n ],\n \"title\": \"Lsun: Construction of a largescale 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 \"Zhiming Zhou\",\n \"Jiadong Liang\",\n \"Yuxuan Song\",\n \"Lantao Yu\",\n \"Hongwei Wang\",\n \"Weinan Zhang\",\n \"Yong Yu\",\n \"Zhihua Zhang\"\n ],\n \"title\": \"Lipschitz generative adversarial nets\",\n \"venue\": null,\n \"year\": 1902\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"As one of the most important advances in generative models in recent years, Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have been attracting great attention in the machine learning community. GANs aim to train a generator network that transforms simple vectors of noise to produce “realistic” samples from the data distribution. In the basic training process of GANs, a discriminator and a target generator are trained in an adversarial manner. The discriminator tries to distinguish the generated fake samples from the real ones, and the generator tries to fool the discriminator into believing the generated samples to be real.\nThough successful, there are two major challenges in training GANs: the instability of the training process and the mode collapse problem. To deal with these problems, one class of approaches focus on designing more informative objective functions (Salimans et al., 2016; Mao et al., 2016; Kodali et al., 2018; Arjovsky & Bottou; Arjovsky et al., 2017; Gulrajani et al., 2017; Zhou et al., 2019). For example, Mao et al. (2016) proposed Least Squares GAN (LSGAN) that uses the least squares loss to penalize the outlier point more harshly. Arjovsky & Bottou discussed the role of the Jensen-Shannon divergence in training GANs and proposed WGAN (Arjovsky et al., 2017) and WGAN-GP (Gulrajani et al., 2017) that use the more informative Wasserstein distance instead. Other approaches enforce proper constraints on latent space representations to better capture the data distribution (Makhzani et al., 2015; Larsen et al., 2015; Che et al., 2016; Tran et al., 2018). A representative work is the Adversarial Autoencoders (AAE) (Makhzani et al., 2015) which uses the discriminator to distinguish the latent representations generated by encoder from Gaussian noise. Larsen et al. (2015) employed image representation in the discriminator as the reconstruction basis of a VAE. Their method turns pixel-wise loss to feature-wise, which can capture the real distribution more simply when some form of invariance is induced. Different from VAE-GAN, Che et al. (2016) regarded the encoder as an auxiliary network, which can promote GANs to pay much attention on missing mode and derive an objective function similar to VAE-GAN. A more detailed discussion of related works can be found in Appendix C.\nIn this paper we propose a novel technique for GANs that improve both the training stability and the quality of generated images. The core of our approach is a regularization term based on the latent representations of real images provided by an encoder network. More specifically, we apply auxiliary intervention operations that preserve the standard Gaussian (e.g., the noise distribution) to these latent representations. The perturbed latent representations are then fed into the generator to produce intervened samples. We then introduce a classifier network to identify the right intervention operations that would have led to these intervened samples. The resulting negative cross-entropy loss\nis added as a regularizer to the objective when training the generator. We call this regularization term the intervention loss and our approach InterVention Generative Adversarial Nets (IVGAN).\nWe show that the intervention loss is equivalent with the JS-divergence among multiple intervened distributions. Furthermore, these intervened distributions interpolate between the original generative distribution of GAN and the data distribution, providing useful information for the generator that is previously unavailable in common GAN models (see a thorough analysis on a toy example in Example 1). We show empirically that our model can be trained efficiently by utilizing the parameter sharing strategy between the discriminator and the classifier. The models trained on the MNIST, CIFAR-10, LSUN and STL-10 datasets successfully generate diverse, visually appealing objects, outperforming state-of-the-art baseline methods such as WGAN-GP and MRGAN in terms of the Frèchet Inception Distance (FID) (proposed in (Heusel et al., 2017)). We also perform a series of experiments on the stacked MNIST dataset and the results show that our proposed method can also effectively alleviate the mode collapse problem. Moreover, an ablation study is conducted, which validates the effectiveness of the proposed intervention loss.\nIn summary, our work offers three major contributions as follows. (i) We propose a novel method that can improve GAN’s training as well as generating performance. (ii) We theoretically analyze our proposed model and give insights on how it makes the gradient of generator more informative and thus stabilizes GAN’s training. (iii) We evaluate the performance of our method on both standard real-world datasets and the stacked MNIST dataset by carefully designed expriments, showing that our approach is able to stabilize GAN’s training and improve the quality and diversity of generated samples as well."},{"heading":"2 PRELIMINARIES","text":"Generative adversarial nets The basic idea of GANs is to utilize a discriminator to continuously push a generator to map Gaussian noise to samples drawn according to an implicit data distribution. The objective function of the vanilla GAN takes the following form:\nmin G max D\n{ V (D,G) , Ex∼pdata log(D(x)) + Ez∼pz log(1−D(G(z))) } , (1)\nwhere pz is a prior distribution (e.g., the standard Gaussian). It can be easily seen that when the discriminator reaches its optimum, that is, D∗(x) = pdata(x)pdata(x)+pG(x) , the objective is equivalent to the Jensen-Shannon (JS) divergence between the generated distribution pG and data distribution pdata:\nJS(pG‖pdata) , 1\n2\n{ KL(pG‖\npG + pdata 2 ) +KL(pdata‖ pG + pdata 2 )\n} .\nMinimizing this JS divergence guarantees that the generated distribution converges to the data distribution given adequate model capacity.\nMulti-distribution JS divergence The JS divergence between two distributions p1 and p2 can be rewritten as\nJS(p1‖p2) = H( p1 + p2 2 )− 1 2 H(p1)− 1 2 H(p2),\nwhere H(p) denotes the entropy of distribution p. We observe that the JS-divergence can be interpreted as the entropy of the mean of the two distribution minus the mean of two distributions’ entropy. So it is immediate to generalize the JS-divergence to the setting of multiple distributions. In particular, we define the JS-divergence of p1, p2, . . . , pn with respect to weights π1, π2, . . . , πn ( ∑ πi = 1 and πi ≥ 0) as\nJSπ1,...,πn(p1, p2, . . . , pn) , H( n∑ i=1 πipi)− n∑ i=1 πiH(pi). (2)\nThe two-distribution case described above is actually a special case of the ‘multi-JS divergence’, where π1 = π2 = 12 . When πi > 0 ∀i, it can be found immediately by Jensen’s inequality that JSπ1,...,πn(p1, p2, . . . , pn) = 0 if and only if p1 = p2 = · · · = pn."},{"heading":"3 METHODOLOGY","text":"Training GAN has been challenging, especially when the generated distribution and the data distribution are far away from each other. In such cases, the discriminator often struggles to provide useful information for the generator, leading to instability and mode collapse problems. The key idea behind our approach is that we construct auxiliary intermediate distributions that interpolate between the generated distribution and the data distribution. To do that, we first introduce an encoder network and combine it with the generator to learn the latent representation of real images within the framework of a standard autoencoder. We then perturb these latent representations with carefully designed intervention operations before feeding them into the generator to create these auxiliary interpolating distributions. A classifier is used to distinguish the intervened samples, which leads to an intervention loss that penalizes the dissimilarity of these intervened distributions. The reconstruction loss and the intervention loss are added as regularization terms to the standard GAN loss for training. We start with an introduction of some notation and definitions. Definition 1 (Intervention). Let O be a transformation on the space of d-dimension random vectors and P be a probability distribution whose support is in Rd. We call O a P-intervention if for any d-dimensional random vector X , X ∼ P⇒ O(X) ∼ P.\nSince the noise distribution in GAN models is usually taken to be standard Gaussian, we use the standard Gaussian distribution as the default choice of P and abbreviate the P-intervention as intervention, unless otherwise claimed. One of the simplest groups of interventions is block substitution. Let Z ∈ Rd be a random vector, k ∈ N and k|d. We slice Z into k blocks so that every block is in R d k . A block substitution intervention Oi is to replace the ith block of Z with Gaussian noise, i = 1, . . . , dk . We will use block substitution interventions in the rest of the paper unless otherwise specified. Note that our theoretical analysis as well as the algorithmic framework do not depend on the specific choice of the intervention group.\nNotation We use E,G,D, f to represent encoder, generator, discriminator and classifier, respectively. Here and later, preal means the distribution of the real data X , and pz is the prior distribution of noise z defined on the latent space (usually is taken to be Gaussian). Let Oi, i = 1, . . . , k denote k different interventions and pi be the distribution of intervened sample Xi created from Oi (namely Xi = G(Oi(E(X)))).\nIntervention loss The intervention loss is the core of our approach. More specifically, given a latent representation z that is generated by an encoder network E, we sample an intervention Oi from a complete group S = {O1, . . . , Ok} and obtain the corresponding intervened latent variable Oi(z) with label ei. These perturbed latent representations are then fed into the generator to produce\nintervened samples. We then introduce an auxiliary classifier network to identify which intervention operations may lead to these intervened samples. The intervention loss LIV (G,E) is simply the resulting negative cross-entropy loss and we add that as a regularizer to the objective function when training the generator. As we can see, the intervention loss is used to penalize the dissimilarity of the distributions of the images generated by different intervention operations. Moreover, it can be noticed that the classifier and the combination of the generator and the encoder are playing a two-player adversarial game and we will train them in an adversarial manner. In particular, we define\nLIV (G,E) = −min f Vclass, where Vclass = Ei∼U([k])Ex′∼pi − eTi log f(x′). (3)\nTheorem 1 (Optimal Classifier). The optimal solution of the classifier is the conditional probability of label y given X ′, where X ′ is the intervened sample generated by the intervention operation sampled from S. And the minimum of the cross entropy loss is equivalent with the negative of the Jensen Shannon divergence among {p1, p2, ..., pk}. That is,\nf∗i (x) = pi(x)∑k j=1 pj(x) and LIV (G,E) = JS(p1, p2, ..., pk) + Const. (4)\nThe proof can be found in Appendix A.1. Clearly, the intervention loss is an approximation of the multi-JS divergence among the intervened distributions {pi : i ∈ [k]}. If the intervention reaches its global minimum, we have p1 = p2 = · · · = pk. And it reaches the maximum log k if and only if the supports of these k distributions do not intersect with each other. This way, the probability that the ‘multi’ JS-divergence has constant value is much smaller, which means the phenomenon of gradient vanishing should be rare in IVGAN. Moreover, as shown in the following example, due to these auxiliary intervened distributions, the intervention is able to provide more informative gradient for the generator that is not previously available in other GAN variants.\nExample 1 (Square fitting). Let X0 be a random vector with distribution U(α), where α = [− 12 , 1 2 ] × [− 1 2 , 1 2 ]. And X1 ∼ U(β), where β = [a− 12 , a+ 1 2 ]× [ 1 2 , 3 2 ] and 0 ≤ a ≤ 1. Assuming we have a perfect discriminator (or classifier), we compute the vanilla GAN loss (i.e. the JS-divergence) and the intervention loss between these two distributions, respectively,\n• JS(X0‖X1) = log 2.\n• In order to compute the intervention loss we need figure out two intervened samples’ distributions evolved from U(α) and U(β). Y1 ∼ U(γ1); γ1 = [− 12 , 1 2 ] × [ 1 2 , 3 2 ] and Y2 ∼\nU(γ2); γ2 = [a − 12 , a + 1 2 ] × [− 1 2 , 1 2 ]. Then the intervention loss is the multi JS-divergence among these four distributions:\nLIV = JS(X0;X1;Y1;Y2) = − ∫ Ac 1 4 log 1 4 dµ− ∫ A 1 2 log 1 2 dµ−H(X0) = log 2 2 [µ(Ac)+µ(A)]\n= log 2\n2 × 2(2− a)−H(X0) = −(log 2)a− Const.\nHere A is the shaded part in Figure 2 and Ac = {α ∪ β ∪ γ1 ∪ γ2} \\A. The most important observation is that the intervention loss is a function of parameter a and the traditional GAN loss is always constant. When we replace the JS with other f -divergence, the metric between U(α) and U(β) would still remain constant. Hence in this situation, we can not get any information from the standard JS for training of the generator but the intervention loss works well.\nAlgorithm 1 Intervention GAN Input learning rate α, regularization parameters λ and µ, dimension d of latent space, number k of blocks in which the hidden space is divided, minibatch size n, Hadamard multiplier ∗\n1: for number of training iterations do 2: Sample minibatch zj , j = 1, ..., n, zj ∼ pz 3: Sample minibatch xj , j = 1, ..., n, xj ∼ preal 4: for number of inner iteration do 5: wj ← E(xj), j = 1, ..., n 6: Sample Gaussian noise 7: Sample ij ∈ [k], j = 1, ..., n 8: x′j ← G(Oij (wj)) 9: Update the parameters of D by:\n10: θD ← θD − α2n∇θDLadv(θD) 11: Update the parameters of f by:\n12: θf ← θf + αn∇θf n∑ j=1 log fij (x ′ j) 13: Calculate LAdv and LIV 14: Update the parameter of G by: 15: θG ← θG + αn∇θG { L̂Adv + λL̂recon + µL̂IV\n} 16: Update the parameter of E by: 17: θE ← θE + αn∇θE { λL̂recon + µL̂IV }\nReconstruction loss In some sense we expect our encoder to be a reverse function of the generator. So it is necessary for the objective function to have a term to push the map composed of the Encoder and the Generator to have the ability to reconstruct the real samples. Not only that, we also hope that the representation can be reconstructed from samples in the pixel space.\nFormally, the reconstruction loss can be defined by the `p-norm (p ≥ 1) between the two samples, or in the from of the Wasserstein distance between samples if images are regarded as a histogram. Here we choose to use the `1-norm as the reconstruction loss:\nLrecon = EX∼preal‖G(E(X))−X‖1 +Ei∼U([k])Ex,z∼preal,pz‖E(G(Oi(z)))−Oi(z)‖1. (5)\nTheorem 2 (Inverse Distribution). Suppose the cumulative distribution function of Oi(z) is qi. For any given positive real number , there exist a δ > 0 such that if Lrecon +LIV ≤ δ , then ∀i, j ∈ [k], sup r ‖qi(r)− qj(r)‖ ≤ .\nThe proof is in A.2.\nAdversarial loss The intervention loss and reconstruction loss can be added as regularization terms to the adversarial loss in many GAN models, e.g., the binary cross entropy loss in vanilla GAN and the least square loss in LSGAN. In the experiments, we use LSGAN (Mao et al., 2016) and DCGAN (Radford et al., 2015) as our base models, and name the resulting IVGAN models IVLSGAN and IVDCGAN respectively.\nNow that we have introduced the essential components in the objective of IVGAN, we can write the loss function of the entire model:\nLmodel = LAdv + λLrecon + µLIV , (6)\nwhere λ and µ are the regularization coefficients for the reconstruction loss and the intervention loss respectively. We summarize the training procedure in Algorithm 1. A diagram of the full workflow of our framework can be found in Figire 3."},{"heading":"4 EXPERIMENTS","text":"In this section we conduct a series of experiments to study IVGAN from multiple aspects. First we evaluate IVGAN’s performance on standard real-world datasets. Then we show IVGAN’s ability to tackle the mode collapse problem on the stacked MNIST dataset. Finally, through an ablation study we investigate the performance of our proposed method under different settings of hyperparameters and demonstrate the effectiveness of the intervention loss.\nWe implement our models using PyTorch (Paszke et al., 2019) with the Adam optimizer (Kingma & Ba, 2015). Network architectures are fairly chosen among the baseline methods and IVGAN (see\nTable B.1 in the appendix for more details). The classifier we use to compute the intervention loss shares the parameters with the discriminator except for the output layer. All input images are resized to have 64× 64 pixels. We use 100-dimensional standard Gaussian distribution as the prior pz . We deploy the instance noise technique as in (Jenni & Favaro, 2019). One may check Appendix B.2 for detailed hyperparameter settings. All experiments are run on one single NVIDIA RTX 2080Ti GPU. Although IVGAN introduces extra computational complexities to the original framework of GANs, the training cost of IVGAN is within an acceptable range1 due to the application of strategies like parameter sharing.\nFigure 4: Random samples of generated images on MNIST, CIFAR-10, LSUN and STL-10.\nReal-world datasets experiments We first test IVGAN on four standard real-world datasets, including CIFAR-10 (Krizhevsky, 2009), MNIST (Lecun et al., 1998), STL-10 (Coates et al., 2011), and a subclass named “church_outdoor\" of the LSUN dataset (Yu et al., 2015), to investigate its training stability and quality of the generated images. We use the Frèchet Inception Distance (FID) (Heusel et al., 2017) to measure the performance of all methods.\nThe FID results are listed in Table 1, and the training curves of the baseline methods and IVGAN on four different datasets are shown in Figure 5. We see that on each datasets, the IVGAN counterparts obtain better FID scores than their corresponding baselines. Moreover, the figure of training curves also suggests the learning processes of IVDCGAN and IVLSGAN are smoother and steadier compared to DCGAN, LSGAN or MRGAN (Che et al., 2016), and converge faster than WGAN-GP. Samples of generated images on all datasets are presented in Figure 4.\nStacked MNIST experiments The original MNIST dataset contains 70K images of 28 × 28 handwritten digits. Following the same approaches in Metz et al. (2017); Srivastava et al. (2017); Lin et al. (2018), we increase the number of modes of the dataset from 10 to 1000 = 10× 10× 10 by\n1Empirically IVGANs are approximately 2 times slower than their corresponding baseline methods.\nstacking three random MNIST images into a 28×28×3 RGB image. The metric we use to evaluate a model’s robustness to mode collapse problem is the number of modes captured by the model, as well as the KL divergence between the generated distribution over modes and the true uniform distribution. The mode of a generated imaged is found from a pre-trained MNIST digit classifier.\nOur results are shown in Table 2. It can be seen that our model works very well to prevent the mode collapse problem. Both IVLSGAN and IVDCGAN are able to reach all 1,000 modes and greatly outperforms early approaches to mitigate mode collapse, such as VEEGAN (Srivastava et al., 2017), and Unrolled GAN (Metz et al., 2017). Moreover, the performance of our model is also comparable to method that is proposed more recently, such as the PacDCGAN (Lin et al., 2018). Figure 6 shows images generated randomly by our model as well as the baseline methods.\nAblation study Our ablation study is conducted on the CIFAR-10 dataset. First, we show the effectiveness of the intervention loss. We consider two cases, IVLSGAN without the intervention loss (µ = 0), and standard IVLSGAN (µ = 0.5). From Figure 7 we can find that the intervention loss makes the training process much smoother and leads to a lower FID score in the end.\nWe also investigate the performance of our model using different number of blocks for the block substitution interventions and different regularization coefficients for the intervention loss. The results are presented in Table 3. It can be noticed that to some extent our models’ performance is not sensitive to the choice of hyperparameters and performs well under several different hyperparameter settings. However, when the number of blocks or the scale of IV loss becomes too large the performance of our model gets worse."},{"heading":"5 CONCLUSION","text":"We have presented a novel model, intervention GAN (IVGAN), to stabilize the training process of GAN and alleviate the mode collapse problem. By introducing auxiliary Gaussian invariant interventions to the latent space of real images and feeding these perturbed latent representations into the generator, we have created intermediate distributions that interpolate between the generated distribution of GAN and the data distribution. The intervention loss based on these auxiliary intervened distributions, together with the reconstruction loss, are added as regularizers to the objective to provide more informative gradients for the generator, significantly improving GAN’s training stability and alleviating the mode collapse problem as well.\nWe have conducted a detailed theoretical analysis of our proposed approach, and illustrated the advantage of the proposed intervention loss on a toy example. Experiments on both real-world datasets and the stacked MNIST dataset demonstrate that, compared to the baseline methods, IVGAN variants are stabler and smoother during training, and are able to generate images of higher quality (achieving state-of-the-art FID scores) and diversity. We believe that our proposed approach can also be applied to other generative models such as Adversarial Autoencoders (Makhzani et al., 2015), which we leave to future work."},{"heading":"A PROOFS","text":""},{"heading":"A.1 PROOF OF THEOREM 1","text":"Proof. The conditional probability ofX ′ given label can be written asP(X ′|ei) = pi(X ′), so further\nP(X ′, ei) = 1 kpi. And we denote the marginal distribution of x as p(x) = 1 k k∑ i=1 pi(x). Cause the activation function at the output layer of the classifier is softmax, we can rewrite the loss function into a more explicit form:\nVclass(f) = Ei∼U [k]Ex′∼pi − eTi log f(x′) = Ei∼U [k]Ex′∼pi − log fi(x)\n= 1\nk ∫ k∑ i=1 −pi(x) log fi(x)dx = ∫ p(x) { − k∑ i=1 p(ei|x) log fi(x) } dx.\nLet gi(x) = fi(x) p(ei|x) , then k∑ i=1 p(ei|x)gi(x) = 1. And notice that k∑ i=1 p(ei|x) = 1. By Jensen’s inequality, we have: k∑ i=1 −p(ei|x) log fi(x) = k∑ i=1 −p(ei|x) log[gi(x)p(ei|x)]\n= k∑ i=1 −p(ei|x) log gi(x) +H(p(·|x)) ≥ log k∑ i=1 p(ei|x)gi(x) +H(pi(·|x))\n= log 1 +H(p(·|x)) = H(p(·|x)). And Vclass(f∗) = ∫ p(x)H(pi(·|x))dx if and only if g∗i (x) = g∗j (x) for any i 6= j, which means\nthat f ∗ i (x) p(ei|x) = r ∀i ∈ [k], where r ∈ R. Notice that k∑ i=1 f∗i (x) = 1, it is not difficult to get that f∗i (x) = p(ei|x). The loss function becomes\n1\nk ∫ k∑ i=1 −pi(x) log p(ei|x)dx = −H(x) + k∑ i=1 1 k H(pi) + log k\n= −JS(p1, p2, ..., pk) + log k\n(7)"},{"heading":"A.2 PROOF OF THEOREM 2","text":"Proof. According to Theorem 1, for a given real number 1, we can find another δ1, when intervention loss is less than δ1, the distance between pi and pj under the measurement of JS-divergence is less than 1. And because JS-divergence and Total Variance distance (TV) are equivalent in the sense of convergence. So we can bound the TV-distance between pi and pj by their JS-divergence. Which means that ∫ |pi − pj |dx ≤ 0 when the intervention loss is less than 1 (we can according to the 0 to finding the appropriate 1). Using this conclusion we can deduce |P (E(G(Oi(z))) ≤ r)− P (E(G(Oj(z))) ≤ r)| ≤ 0, where r is an arbitrary vector inRd. Further, we have:\n|P (Oi(z) ≤ r)− P (Oj(z) ≤ r)| ≤ |P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ > δ)| + |P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ > δ)|+ |P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ ≤ δ) − P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ ≤ δ)|\n(8) We control the three terms on the right side of the inequality sign respectively.\nP (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ > δ)\n≤ P (‖Oi(z)− E(G(Oi(z)))‖ > δ) ≤ E‖Oi(z)− E(G(Oi(z)))‖\nδ\n(9)\nAnd the last term can be bounded by the reconstruction loss. The same trick can be used on P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ > δ). Moreover, we have\nP (E(G(Oi(z))) ≤ r − δ)− P (‖Oi(z)− E(G(Oi(z)))‖ > δ) ≤P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ ≤ δ) ≤ P (E(G(Oi(z))) ≤ r + δ)\n(10)\nNotice that lim δ→0 P (E(G(Oi(z))) ≤ r ± δ) = P (E(G(Oi(z))) ≤ r). Let si(r, δ) = |P (E(G(Oi(z))) ≤ r ± δ)) − P (E(G(Oi(z))) ≤ r)| then the last term of inequalityA.2 can be bounded as: |P (Oi(z) ≤ r;‖Oi(z)− E(G(Oi(z)))‖ ≤ δ)− P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ ≤ δ)|\n≤|P (E(G(Oi(z))) ≤ r)− P (E(G(Oj(z))) ≤ r)|+ P (‖Oi(z)− E(G(Oi(z)))‖ > δ) + si(r, δ) + sj(r, δ)\n(11) Every term on the right hand of the inequality can be controlled close to 0 by the inequalities mentioned above."},{"heading":"B EXPERIMENTAL DETAILS","text":""},{"heading":"B.1 NETWORK ARCHITECTURES","text":""},{"heading":"B.2 HYPERPARAMETER SETTINGS","text":""},{"heading":"C RELATED WORK","text":"In order to address GAN’s unstable training and mode missing problems, many researchers have turned their attention to the latent representations of samples. Makhzani et al. (2015) proposed the Adversarial Autoencoder (AAE). As its name suggests, AAE is essentially a probabilistic autoencoder based on the framework of GANs. Unlike classical GAN models, in the setting of AAE the discriminator’s task is to distinguish the latent representations of real images that are generated by an encoder network from Gaussian noise. And the generator and the encoder are trained to fool the discriminator as well as reconstruct the input image from the encoded representations. However, the generator can only be trained by fitting the reverse of the encoder and cannot get any information from the latent representation.\nThe VAE-GAN (Larsen et al., 2015) combines the objective function from a VAE model with a GAN and utilizes the learned features in the discriminator for better image similarity metrics, which is of great help for the sample visual fidelity. Considering the opposite perspective, Che et al. (2016) claim that the whole learning process of a generative model can be divided into the manifold learning phase and the diffusion learning phase. And the former one is considered to be the source of the mode missing problem. (Che et al., 2016) then proposed Mode Regularized Generative Adversarial\nNets which introduce a reconstruction loss term to the training target of GAN to penalize the missing modes. It is shown that it actually ameliorates GAN’s ’mode missing’-prone weakness to some extent. However, both of them fail to fully excavate the impact of the interaction between VAEs and GANs.\nKim & Mnih (2018) proposed Factor VAE where a regularization term called total correlation penalty is added to the traditional VAE loss. The total correlation is essentially the Kullback-Leibler divergence between the joint distribution p(z1, z2, . . . , zd) and the product of marginal distribution p(zi). Because the closed forms of these two distribution are unavailable, Factor VAE uses adversarial training to approximate the likelihood ratio."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"As one of the most important advances in generative models in recent years, Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have been attracting great attention in the machine learning community. GANs aim to train a generator network that transforms simple vectors of noise to produce “realistic” samples from the data distribution. In the basic training process of GANs, a discriminator and a target generator are trained in an adversarial manner. The discriminator tries to distinguish the generated fake samples from the real ones, and the generator tries to fool the discriminator into believing the generated samples to be real.\\nThough successful, there are two major challenges in training GANs: the instability of the training process and the mode collapse problem. To deal with these problems, one class of approaches focus on designing more informative objective functions (Salimans et al., 2016; Mao et al., 2016; Kodali et al., 2018; Arjovsky & Bottou; Arjovsky et al., 2017; Gulrajani et al., 2017; Zhou et al., 2019). For example, Mao et al. (2016) proposed Least Squares GAN (LSGAN) that uses the least squares loss to penalize the outlier point more harshly. Arjovsky & Bottou discussed the role of the Jensen-Shannon divergence in training GANs and proposed WGAN (Arjovsky et al., 2017) and WGAN-GP (Gulrajani et al., 2017) that use the more informative Wasserstein distance instead. Other approaches enforce proper constraints on latent space representations to better capture the data distribution (Makhzani et al., 2015; Larsen et al., 2015; Che et al., 2016; Tran et al., 2018). A representative work is the Adversarial Autoencoders (AAE) (Makhzani et al., 2015) which uses the discriminator to distinguish the latent representations generated by encoder from Gaussian noise. Larsen et al. (2015) employed image representation in the discriminator as the reconstruction basis of a VAE. Their method turns pixel-wise loss to feature-wise, which can capture the real distribution more simply when some form of invariance is induced. Different from VAE-GAN, Che et al. (2016) regarded the encoder as an auxiliary network, which can promote GANs to pay much attention on missing mode and derive an objective function similar to VAE-GAN. A more detailed discussion of related works can be found in Appendix C.\\nIn this paper we propose a novel technique for GANs that improve both the training stability and the quality of generated images. The core of our approach is a regularization term based on the latent representations of real images provided by an encoder network. More specifically, we apply auxiliary intervention operations that preserve the standard Gaussian (e.g., the noise distribution) to these latent representations. The perturbed latent representations are then fed into the generator to produce intervened samples. We then introduce a classifier network to identify the right intervention operations that would have led to these intervened samples. The resulting negative cross-entropy loss\\nis added as a regularizer to the objective when training the generator. We call this regularization term the intervention loss and our approach InterVention Generative Adversarial Nets (IVGAN).\\nWe show that the intervention loss is equivalent with the JS-divergence among multiple intervened distributions. Furthermore, these intervened distributions interpolate between the original generative distribution of GAN and the data distribution, providing useful information for the generator that is previously unavailable in common GAN models (see a thorough analysis on a toy example in Example 1). We show empirically that our model can be trained efficiently by utilizing the parameter sharing strategy between the discriminator and the classifier. The models trained on the MNIST, CIFAR-10, LSUN and STL-10 datasets successfully generate diverse, visually appealing objects, outperforming state-of-the-art baseline methods such as WGAN-GP and MRGAN in terms of the Frèchet Inception Distance (FID) (proposed in (Heusel et al., 2017)). We also perform a series of experiments on the stacked MNIST dataset and the results show that our proposed method can also effectively alleviate the mode collapse problem. Moreover, an ablation study is conducted, which validates the effectiveness of the proposed intervention loss.\\nIn summary, our work offers three major contributions as follows. (i) We propose a novel method that can improve GAN’s training as well as generating performance. (ii) We theoretically analyze our proposed model and give insights on how it makes the gradient of generator more informative and thus stabilizes GAN’s training. (iii) We evaluate the performance of our method on both standard real-world datasets and the stacked MNIST dataset by carefully designed expriments, showing that our approach is able to stabilize GAN’s training and improve the quality and diversity of generated samples as well.\"\n },\n {\n \"heading\": \"2 PRELIMINARIES\",\n \"text\": \"Generative adversarial nets The basic idea of GANs is to utilize a discriminator to continuously push a generator to map Gaussian noise to samples drawn according to an implicit data distribution. The objective function of the vanilla GAN takes the following form:\\nmin G max D\\n{ V (D,G) , Ex∼pdata log(D(x)) + Ez∼pz log(1−D(G(z))) } , (1)\\nwhere pz is a prior distribution (e.g., the standard Gaussian). It can be easily seen that when the discriminator reaches its optimum, that is, D∗(x) = pdata(x)pdata(x)+pG(x) , the objective is equivalent to the Jensen-Shannon (JS) divergence between the generated distribution pG and data distribution pdata:\\nJS(pG‖pdata) , 1\\n2\\n{ KL(pG‖\\npG + pdata 2 ) +KL(pdata‖ pG + pdata 2 )\\n} .\\nMinimizing this JS divergence guarantees that the generated distribution converges to the data distribution given adequate model capacity.\\nMulti-distribution JS divergence The JS divergence between two distributions p1 and p2 can be rewritten as\\nJS(p1‖p2) = H( p1 + p2 2 )− 1 2 H(p1)− 1 2 H(p2),\\nwhere H(p) denotes the entropy of distribution p. We observe that the JS-divergence can be interpreted as the entropy of the mean of the two distribution minus the mean of two distributions’ entropy. So it is immediate to generalize the JS-divergence to the setting of multiple distributions. In particular, we define the JS-divergence of p1, p2, . . . , pn with respect to weights π1, π2, . . . , πn ( ∑ πi = 1 and πi ≥ 0) as\\nJSπ1,...,πn(p1, p2, . . . , pn) , H( n∑ i=1 πipi)− n∑ i=1 πiH(pi). (2)\\nThe two-distribution case described above is actually a special case of the ‘multi-JS divergence’, where π1 = π2 = 12 . When πi > 0 ∀i, it can be found immediately by Jensen’s inequality that JSπ1,...,πn(p1, p2, . . . , pn) = 0 if and only if p1 = p2 = · · · = pn.\"\n },\n {\n \"heading\": \"3 METHODOLOGY\",\n \"text\": \"Training GAN has been challenging, especially when the generated distribution and the data distribution are far away from each other. In such cases, the discriminator often struggles to provide useful information for the generator, leading to instability and mode collapse problems. The key idea behind our approach is that we construct auxiliary intermediate distributions that interpolate between the generated distribution and the data distribution. To do that, we first introduce an encoder network and combine it with the generator to learn the latent representation of real images within the framework of a standard autoencoder. We then perturb these latent representations with carefully designed intervention operations before feeding them into the generator to create these auxiliary interpolating distributions. A classifier is used to distinguish the intervened samples, which leads to an intervention loss that penalizes the dissimilarity of these intervened distributions. The reconstruction loss and the intervention loss are added as regularization terms to the standard GAN loss for training. We start with an introduction of some notation and definitions. Definition 1 (Intervention). Let O be a transformation on the space of d-dimension random vectors and P be a probability distribution whose support is in Rd. We call O a P-intervention if for any d-dimensional random vector X , X ∼ P⇒ O(X) ∼ P.\\nSince the noise distribution in GAN models is usually taken to be standard Gaussian, we use the standard Gaussian distribution as the default choice of P and abbreviate the P-intervention as intervention, unless otherwise claimed. One of the simplest groups of interventions is block substitution. Let Z ∈ Rd be a random vector, k ∈ N and k|d. We slice Z into k blocks so that every block is in R d k . A block substitution intervention Oi is to replace the ith block of Z with Gaussian noise, i = 1, . . . , dk . We will use block substitution interventions in the rest of the paper unless otherwise specified. Note that our theoretical analysis as well as the algorithmic framework do not depend on the specific choice of the intervention group.\\nNotation We use E,G,D, f to represent encoder, generator, discriminator and classifier, respectively. Here and later, preal means the distribution of the real data X , and pz is the prior distribution of noise z defined on the latent space (usually is taken to be Gaussian). Let Oi, i = 1, . . . , k denote k different interventions and pi be the distribution of intervened sample Xi created from Oi (namely Xi = G(Oi(E(X)))).\\nIntervention loss The intervention loss is the core of our approach. More specifically, given a latent representation z that is generated by an encoder network E, we sample an intervention Oi from a complete group S = {O1, . . . , Ok} and obtain the corresponding intervened latent variable Oi(z) with label ei. These perturbed latent representations are then fed into the generator to produce\\nintervened samples. We then introduce an auxiliary classifier network to identify which intervention operations may lead to these intervened samples. The intervention loss LIV (G,E) is simply the resulting negative cross-entropy loss and we add that as a regularizer to the objective function when training the generator. As we can see, the intervention loss is used to penalize the dissimilarity of the distributions of the images generated by different intervention operations. Moreover, it can be noticed that the classifier and the combination of the generator and the encoder are playing a two-player adversarial game and we will train them in an adversarial manner. In particular, we define\\nLIV (G,E) = −min f Vclass, where Vclass = Ei∼U([k])Ex′∼pi − eTi log f(x′). (3)\\nTheorem 1 (Optimal Classifier). The optimal solution of the classifier is the conditional probability of label y given X ′, where X ′ is the intervened sample generated by the intervention operation sampled from S. And the minimum of the cross entropy loss is equivalent with the negative of the Jensen Shannon divergence among {p1, p2, ..., pk}. That is,\\nf∗i (x) = pi(x)∑k j=1 pj(x) and LIV (G,E) = JS(p1, p2, ..., pk) + Const. (4)\\nThe proof can be found in Appendix A.1. Clearly, the intervention loss is an approximation of the multi-JS divergence among the intervened distributions {pi : i ∈ [k]}. If the intervention reaches its global minimum, we have p1 = p2 = · · · = pk. And it reaches the maximum log k if and only if the supports of these k distributions do not intersect with each other. This way, the probability that the ‘multi’ JS-divergence has constant value is much smaller, which means the phenomenon of gradient vanishing should be rare in IVGAN. Moreover, as shown in the following example, due to these auxiliary intervened distributions, the intervention is able to provide more informative gradient for the generator that is not previously available in other GAN variants.\\nExample 1 (Square fitting). Let X0 be a random vector with distribution U(α), where α = [− 12 , 1 2 ] × [− 1 2 , 1 2 ]. And X1 ∼ U(β), where β = [a− 12 , a+ 1 2 ]× [ 1 2 , 3 2 ] and 0 ≤ a ≤ 1. Assuming we have a perfect discriminator (or classifier), we compute the vanilla GAN loss (i.e. the JS-divergence) and the intervention loss between these two distributions, respectively,\\n• JS(X0‖X1) = log 2.\\n• In order to compute the intervention loss we need figure out two intervened samples’ distributions evolved from U(α) and U(β). Y1 ∼ U(γ1); γ1 = [− 12 , 1 2 ] × [ 1 2 , 3 2 ] and Y2 ∼\\nU(γ2); γ2 = [a − 12 , a + 1 2 ] × [− 1 2 , 1 2 ]. Then the intervention loss is the multi JS-divergence among these four distributions:\\nLIV = JS(X0;X1;Y1;Y2) = − ∫ Ac 1 4 log 1 4 dµ− ∫ A 1 2 log 1 2 dµ−H(X0) = log 2 2 [µ(Ac)+µ(A)]\\n= log 2\\n2 × 2(2− a)−H(X0) = −(log 2)a− Const.\\nHere A is the shaded part in Figure 2 and Ac = {α ∪ β ∪ γ1 ∪ γ2} \\\\A. The most important observation is that the intervention loss is a function of parameter a and the traditional GAN loss is always constant. When we replace the JS with other f -divergence, the metric between U(α) and U(β) would still remain constant. Hence in this situation, we can not get any information from the standard JS for training of the generator but the intervention loss works well.\\nAlgorithm 1 Intervention GAN Input learning rate α, regularization parameters λ and µ, dimension d of latent space, number k of blocks in which the hidden space is divided, minibatch size n, Hadamard multiplier ∗\\n1: for number of training iterations do 2: Sample minibatch zj , j = 1, ..., n, zj ∼ pz 3: Sample minibatch xj , j = 1, ..., n, xj ∼ preal 4: for number of inner iteration do 5: wj ← E(xj), j = 1, ..., n 6: Sample Gaussian noise 7: Sample ij ∈ [k], j = 1, ..., n 8: x′j ← G(Oij (wj)) 9: Update the parameters of D by:\\n10: θD ← θD − α2n∇θDLadv(θD) 11: Update the parameters of f by:\\n12: θf ← θf + αn∇θf n∑ j=1 log fij (x ′ j) 13: Calculate LAdv and LIV 14: Update the parameter of G by: 15: θG ← θG + αn∇θG { L̂Adv + λL̂recon + µL̂IV\\n} 16: Update the parameter of E by: 17: θE ← θE + αn∇θE { λL̂recon + µL̂IV }\\nReconstruction loss In some sense we expect our encoder to be a reverse function of the generator. So it is necessary for the objective function to have a term to push the map composed of the Encoder and the Generator to have the ability to reconstruct the real samples. Not only that, we also hope that the representation can be reconstructed from samples in the pixel space.\\nFormally, the reconstruction loss can be defined by the `p-norm (p ≥ 1) between the two samples, or in the from of the Wasserstein distance between samples if images are regarded as a histogram. Here we choose to use the `1-norm as the reconstruction loss:\\nLrecon = EX∼preal‖G(E(X))−X‖1 +Ei∼U([k])Ex,z∼preal,pz‖E(G(Oi(z)))−Oi(z)‖1. (5)\\nTheorem 2 (Inverse Distribution). Suppose the cumulative distribution function of Oi(z) is qi. For any given positive real number , there exist a δ > 0 such that if Lrecon +LIV ≤ δ , then ∀i, j ∈ [k], sup r ‖qi(r)− qj(r)‖ ≤ .\\nThe proof is in A.2.\\nAdversarial loss The intervention loss and reconstruction loss can be added as regularization terms to the adversarial loss in many GAN models, e.g., the binary cross entropy loss in vanilla GAN and the least square loss in LSGAN. In the experiments, we use LSGAN (Mao et al., 2016) and DCGAN (Radford et al., 2015) as our base models, and name the resulting IVGAN models IVLSGAN and IVDCGAN respectively.\\nNow that we have introduced the essential components in the objective of IVGAN, we can write the loss function of the entire model:\\nLmodel = LAdv + λLrecon + µLIV , (6)\\nwhere λ and µ are the regularization coefficients for the reconstruction loss and the intervention loss respectively. We summarize the training procedure in Algorithm 1. A diagram of the full workflow of our framework can be found in Figire 3.\"\n },\n {\n \"heading\": \"4 EXPERIMENTS\",\n \"text\": \"In this section we conduct a series of experiments to study IVGAN from multiple aspects. First we evaluate IVGAN’s performance on standard real-world datasets. Then we show IVGAN’s ability to tackle the mode collapse problem on the stacked MNIST dataset. Finally, through an ablation study we investigate the performance of our proposed method under different settings of hyperparameters and demonstrate the effectiveness of the intervention loss.\\nWe implement our models using PyTorch (Paszke et al., 2019) with the Adam optimizer (Kingma & Ba, 2015). Network architectures are fairly chosen among the baseline methods and IVGAN (see\\nTable B.1 in the appendix for more details). The classifier we use to compute the intervention loss shares the parameters with the discriminator except for the output layer. All input images are resized to have 64× 64 pixels. We use 100-dimensional standard Gaussian distribution as the prior pz . We deploy the instance noise technique as in (Jenni & Favaro, 2019). One may check Appendix B.2 for detailed hyperparameter settings. All experiments are run on one single NVIDIA RTX 2080Ti GPU. Although IVGAN introduces extra computational complexities to the original framework of GANs, the training cost of IVGAN is within an acceptable range1 due to the application of strategies like parameter sharing.\\nFigure 4: Random samples of generated images on MNIST, CIFAR-10, LSUN and STL-10.\\nReal-world datasets experiments We first test IVGAN on four standard real-world datasets, including CIFAR-10 (Krizhevsky, 2009), MNIST (Lecun et al., 1998), STL-10 (Coates et al., 2011), and a subclass named “church_outdoor\\\" of the LSUN dataset (Yu et al., 2015), to investigate its training stability and quality of the generated images. We use the Frèchet Inception Distance (FID) (Heusel et al., 2017) to measure the performance of all methods.\\nThe FID results are listed in Table 1, and the training curves of the baseline methods and IVGAN on four different datasets are shown in Figure 5. We see that on each datasets, the IVGAN counterparts obtain better FID scores than their corresponding baselines. Moreover, the figure of training curves also suggests the learning processes of IVDCGAN and IVLSGAN are smoother and steadier compared to DCGAN, LSGAN or MRGAN (Che et al., 2016), and converge faster than WGAN-GP. Samples of generated images on all datasets are presented in Figure 4.\\nStacked MNIST experiments The original MNIST dataset contains 70K images of 28 × 28 handwritten digits. Following the same approaches in Metz et al. (2017); Srivastava et al. (2017); Lin et al. (2018), we increase the number of modes of the dataset from 10 to 1000 = 10× 10× 10 by\\n1Empirically IVGANs are approximately 2 times slower than their corresponding baseline methods.\\nstacking three random MNIST images into a 28×28×3 RGB image. The metric we use to evaluate a model’s robustness to mode collapse problem is the number of modes captured by the model, as well as the KL divergence between the generated distribution over modes and the true uniform distribution. The mode of a generated imaged is found from a pre-trained MNIST digit classifier.\\nOur results are shown in Table 2. It can be seen that our model works very well to prevent the mode collapse problem. Both IVLSGAN and IVDCGAN are able to reach all 1,000 modes and greatly outperforms early approaches to mitigate mode collapse, such as VEEGAN (Srivastava et al., 2017), and Unrolled GAN (Metz et al., 2017). Moreover, the performance of our model is also comparable to method that is proposed more recently, such as the PacDCGAN (Lin et al., 2018). Figure 6 shows images generated randomly by our model as well as the baseline methods.\\nAblation study Our ablation study is conducted on the CIFAR-10 dataset. First, we show the effectiveness of the intervention loss. We consider two cases, IVLSGAN without the intervention loss (µ = 0), and standard IVLSGAN (µ = 0.5). From Figure 7 we can find that the intervention loss makes the training process much smoother and leads to a lower FID score in the end.\\nWe also investigate the performance of our model using different number of blocks for the block substitution interventions and different regularization coefficients for the intervention loss. The results are presented in Table 3. It can be noticed that to some extent our models’ performance is not sensitive to the choice of hyperparameters and performs well under several different hyperparameter settings. However, when the number of blocks or the scale of IV loss becomes too large the performance of our model gets worse.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"We have presented a novel model, intervention GAN (IVGAN), to stabilize the training process of GAN and alleviate the mode collapse problem. By introducing auxiliary Gaussian invariant interventions to the latent space of real images and feeding these perturbed latent representations into the generator, we have created intermediate distributions that interpolate between the generated distribution of GAN and the data distribution. The intervention loss based on these auxiliary intervened distributions, together with the reconstruction loss, are added as regularizers to the objective to provide more informative gradients for the generator, significantly improving GAN’s training stability and alleviating the mode collapse problem as well.\\nWe have conducted a detailed theoretical analysis of our proposed approach, and illustrated the advantage of the proposed intervention loss on a toy example. Experiments on both real-world datasets and the stacked MNIST dataset demonstrate that, compared to the baseline methods, IVGAN variants are stabler and smoother during training, and are able to generate images of higher quality (achieving state-of-the-art FID scores) and diversity. We believe that our proposed approach can also be applied to other generative models such as Adversarial Autoencoders (Makhzani et al., 2015), which we leave to future work.\"\n },\n {\n \"heading\": \"A PROOFS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A.1 PROOF OF THEOREM 1\",\n \"text\": \"Proof. The conditional probability ofX ′ given label can be written asP(X ′|ei) = pi(X ′), so further\\nP(X ′, ei) = 1 kpi. And we denote the marginal distribution of x as p(x) = 1 k k∑ i=1 pi(x). Cause the activation function at the output layer of the classifier is softmax, we can rewrite the loss function into a more explicit form:\\nVclass(f) = Ei∼U [k]Ex′∼pi − eTi log f(x′) = Ei∼U [k]Ex′∼pi − log fi(x)\\n= 1\\nk ∫ k∑ i=1 −pi(x) log fi(x)dx = ∫ p(x) { − k∑ i=1 p(ei|x) log fi(x) } dx.\\nLet gi(x) = fi(x) p(ei|x) , then k∑ i=1 p(ei|x)gi(x) = 1. And notice that k∑ i=1 p(ei|x) = 1. By Jensen’s inequality, we have: k∑ i=1 −p(ei|x) log fi(x) = k∑ i=1 −p(ei|x) log[gi(x)p(ei|x)]\\n= k∑ i=1 −p(ei|x) log gi(x) +H(p(·|x)) ≥ log k∑ i=1 p(ei|x)gi(x) +H(pi(·|x))\\n= log 1 +H(p(·|x)) = H(p(·|x)). And Vclass(f∗) = ∫ p(x)H(pi(·|x))dx if and only if g∗i (x) = g∗j (x) for any i 6= j, which means\\nthat f ∗ i (x) p(ei|x) = r ∀i ∈ [k], where r ∈ R. Notice that k∑ i=1 f∗i (x) = 1, it is not difficult to get that f∗i (x) = p(ei|x). The loss function becomes\\n1\\nk ∫ k∑ i=1 −pi(x) log p(ei|x)dx = −H(x) + k∑ i=1 1 k H(pi) + log k\\n= −JS(p1, p2, ..., pk) + log k\\n(7)\"\n },\n {\n \"heading\": \"A.2 PROOF OF THEOREM 2\",\n \"text\": \"Proof. According to Theorem 1, for a given real number 1, we can find another δ1, when intervention loss is less than δ1, the distance between pi and pj under the measurement of JS-divergence is less than 1. And because JS-divergence and Total Variance distance (TV) are equivalent in the sense of convergence. So we can bound the TV-distance between pi and pj by their JS-divergence. Which means that ∫ |pi − pj |dx ≤ 0 when the intervention loss is less than 1 (we can according to the 0 to finding the appropriate 1). Using this conclusion we can deduce |P (E(G(Oi(z))) ≤ r)− P (E(G(Oj(z))) ≤ r)| ≤ 0, where r is an arbitrary vector inRd. Further, we have:\\n|P (Oi(z) ≤ r)− P (Oj(z) ≤ r)| ≤ |P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ > δ)| + |P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ > δ)|+ |P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ ≤ δ) − P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ ≤ δ)|\\n(8) We control the three terms on the right side of the inequality sign respectively.\\nP (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ > δ)\\n≤ P (‖Oi(z)− E(G(Oi(z)))‖ > δ) ≤ E‖Oi(z)− E(G(Oi(z)))‖\\nδ\\n(9)\\nAnd the last term can be bounded by the reconstruction loss. The same trick can be used on P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ > δ). Moreover, we have\\nP (E(G(Oi(z))) ≤ r − δ)− P (‖Oi(z)− E(G(Oi(z)))‖ > δ) ≤P (Oi(z) ≤ r; ‖Oi(z)− E(G(Oi(z)))‖ ≤ δ) ≤ P (E(G(Oi(z))) ≤ r + δ)\\n(10)\\nNotice that lim δ→0 P (E(G(Oi(z))) ≤ r ± δ) = P (E(G(Oi(z))) ≤ r). Let si(r, δ) = |P (E(G(Oi(z))) ≤ r ± δ)) − P (E(G(Oi(z))) ≤ r)| then the last term of inequalityA.2 can be bounded as: |P (Oi(z) ≤ r;‖Oi(z)− E(G(Oi(z)))‖ ≤ δ)− P (Oj(z) ≤ r; ‖Oj(z)− E(G(Oj(z)))‖ ≤ δ)|\\n≤|P (E(G(Oi(z))) ≤ r)− P (E(G(Oj(z))) ≤ r)|+ P (‖Oi(z)− E(G(Oi(z)))‖ > δ) + si(r, δ) + sj(r, δ)\\n(11) Every term on the right hand of the inequality can be controlled close to 0 by the inequalities mentioned above.\"\n },\n {\n \"heading\": \"B EXPERIMENTAL DETAILS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"B.1 NETWORK ARCHITECTURES\",\n \"text\": \"\"\n },\n {\n \"heading\": \"B.2 HYPERPARAMETER SETTINGS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"C RELATED WORK\",\n \"text\": \"In order to address GAN’s unstable training and mode missing problems, many researchers have turned their attention to the latent representations of samples. Makhzani et al. (2015) proposed the Adversarial Autoencoder (AAE). As its name suggests, AAE is essentially a probabilistic autoencoder based on the framework of GANs. Unlike classical GAN models, in the setting of AAE the discriminator’s task is to distinguish the latent representations of real images that are generated by an encoder network from Gaussian noise. And the generator and the encoder are trained to fool the discriminator as well as reconstruct the input image from the encoded representations. However, the generator can only be trained by fitting the reverse of the encoder and cannot get any information from the latent representation.\\nThe VAE-GAN (Larsen et al., 2015) combines the objective function from a VAE model with a GAN and utilizes the learned features in the discriminator for better image similarity metrics, which is of great help for the sample visual fidelity. Considering the opposite perspective, Che et al. (2016) claim that the whole learning process of a generative model can be divided into the manifold learning phase and the diffusion learning phase. And the former one is considered to be the source of the mode missing problem. (Che et al., 2016) then proposed Mode Regularized Generative Adversarial\\nNets which introduce a reconstruction loss term to the training target of GAN to penalize the missing modes. It is shown that it actually ameliorates GAN’s ’mode missing’-prone weakness to some extent. However, both of them fail to fully excavate the impact of the interaction between VAEs and GANs.\\nKim & Mnih (2018) proposed Factor VAE where a regularization term called total correlation penalty is added to the traditional VAE loss. The total correlation is essentially the Kullback-Leibler divergence between the joint distribution p(z1, z2, . . . , zd) and the product of marginal distribution p(zi). Because the closed forms of these two distribution are unavailable, Factor VAE uses adversarial training to approximate the likelihood ratio.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":194,"cells":{"paper_id":{"kind":"string","value":"SP:a9c70bdca13ee3800c633589a6ee028701e5bf51"},"summaries":{"kind":"list like","value":["This work proposed a dimensionality reduction algorithm called Uniform Manifold Approximation with Two-phase Optimization (UMATO), which is an improved version of UMAP (Ref. [3] see below). UMATO has a two-phase optimization approach: global optimization to obtain the overall skeleton of data & local optimization to identify the local structures. "],"string":"[\n \"This work proposed a dimensionality reduction algorithm called Uniform Manifold Approximation with Two-phase Optimization (UMATO), which is an improved version of UMAP (Ref. [3] see below). UMATO has a two-phase optimization approach: global optimization to obtain the overall skeleton of data & local optimization to identify the local structures. \"\n]","isConversational":false},"abstractText":{"kind":"string","value":"We present a dimensionality reduction algorithm called Uniform Manifold Approximation with Two-phase Optimization (UMATO) which produces less biased global structures in the embedding results and is robust over diverse initialization methods than previous methods such as t-SNE and UMAP. We divide the optimization into two phases to alleviate the bias by establishing the global structure early using the representatives of the high-dimensional structures. The phases are 1) global optimization to obtain the overall skeleton of data and 2) local optimization to identify the regional characteristics of local areas. In our experiments with one synthetic and three real-world datasets, UMATO outperformed widely-used baseline algorithms, such as PCA, Isomap, t-SNE, UMAP, topological autoencoders and Anchor t-SNE, in terms of quality metrics and 2D projection results."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Josh Barnes","Piet Hut"],"title":"A hierarchical o (n log n) force-calculation","venue":null,"year":1986},{"authors":["Mikhail Belkin","Partha Niyogi"],"title":"Laplacian eigenmaps and spectral techniques for embedding and clustering","venue":"In Advances in neural information processing systems,","year":2002},{"authors":["Tess Brodie","Elena Brenna","Federica Sallusto"],"title":"Omip-018: Chemokine receptor expression on human t helper cells","venue":"Cytometry Part A,","year":2013},{"authors":["Frédéric Chazal","David Cohen-Steiner","Quentin Mérigot"],"title":"Geometric inference for probability measures","venue":"Foundations of Computational Mathematics,","year":2011},{"authors":["Frédéric Chazal","Brittany Fasy","Fabrizio Lecci","Bertrand Michel","Alessandro Rinaldo","Larry Wasserman"],"title":"Robust topological inference: Distance to a measure and kernel distance","venue":"The Journal of Machine Learning Research,","year":2017},{"authors":["Tarin Clanuwat","Mikel Bober-Irizar","Asanobu Kitamoto","Alex Lamb","Kazuaki Yamamoto","David Ha"],"title":"Deep learning for classical japanese literature, 2018","venue":null,"year":2018},{"authors":["James Cook","Ilya Sutskever","Andriy Mnih","Geoffrey Hinton"],"title":"Visualizing similarity data with a mixture of maps","venue":"In Artificial Intelligence and Statistics,","year":2007},{"authors":["Cong Fu","Yonghui Zhang","Deng Cai","Xiang Ren"],"title":"Atsne: Efficient and robust visualization on gpu through hierarchical optimization","venue":"In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining,","year":2019},{"authors":["Dian Gong","Xuemei Zhao","Gerard Medioni"],"title":"Robust multiple manifolds structure learning","venue":"In Proceedings of the 29th International Conference on Machine Learning,","year":2012},{"authors":["Carmen Bravo González-Blas","Liesbeth Minnoye","Dafni Papasokrati","Sara Aibar","Gert Hulselmans","Valerie Christiaens","Kristofer Davie","Jasper Wouters","Stein Aerts"],"title":"cistopic: cis-regulatory topic modeling on single-cell atac-seq data","venue":"Nature methods,","year":2019},{"authors":["Antonio Gracia","Santiago González","Victor Robles","Ernestina Menasalvas"],"title":"A methodology to compare dimensionality reduction algorithms in terms of loss of quality","venue":"Information Sciences,","year":2014},{"authors":["der Fabisch"],"title":"scikit-optimize/scikit-optimize: v0.5.2, March 2018","venue":"URL https://doi.org/","year":2018},{"authors":["Geoffrey E Hinton","Sam Roweis"],"title":"Stochastic neighbor embedding","venue":"Advances in neural information processing systems,","year":2002},{"authors":["Yann LeCun","Corinna Cortes. MNIST handwritten digit database."],"title":"URL http://yann","venue":"lecun.com/exdb/mnist/.","year":2010},{"authors":["John A Lee","Michel Verleysen"],"title":"Nonlinear dimensionality reduction","venue":"Springer Science & Business Media,","year":2007},{"authors":["John A Lee","Michel Verleysen"],"title":"Quality assessment of dimensionality reduction","venue":"Rank-based criteria. Neurocomputing,","year":2009},{"authors":["George C Linderman","Manas Rachh","Jeremy G Hoskins","Stefan Steinerberger","Yuval Kluger"],"title":"Fast interpolation-based t-sne for improved visualization of single-cell rna-seq data","venue":"Nature methods,","year":2019},{"authors":["Laurens van der Maaten","Geoffrey Hinton"],"title":"Visualizing data using t-sne","venue":"Journal of machine learning research,","year":2008},{"authors":["Leland McInnes","John Healy","James Melville"],"title":"Umap: Uniform manifold approximation and projection for dimension reduction","venue":"arXiv preprint arXiv:1802.03426,","year":2018},{"authors":["Tomas Mikolov","Ilya Sutskever","Kai Chen","Greg S Corrado","Jeff Dean"],"title":"Distributed representations of words and phrases and their compositionality","venue":"In Advances in neural information processing systems,","year":2013},{"authors":["Michael Moor","Max Horn","Bastian Rieck","Karsten Borgwardt"],"title":"Topological autoencoders","venue":"In Proceedings of the 37th International Conference on Machine Learning (ICML), Proceedings of Machine Learning Research. PMLR,","year":2020},{"authors":["Corey J Nolet","Victor Lafargue","Edward Raff","Thejaswi Nanditale","Tim Oates","John Zedlewski","Joshua Patterson"],"title":"Bringing umap closer to the speed of light with gpu acceleration","venue":null,"year":2008},{"authors":["Nicola Pezzotti","Julian Thijssen","Alexander Mordvintsev","Thomas Höllt","Baldur Van Lew","Boudewijn PF Lelieveldt","Elmar Eisemann","Anna Vilanova"],"title":"Gpgpu linear complexity tsne optimization","venue":"IEEE transactions on visualization and computer graphics,","year":2019},{"authors":["Vin D Silva","Joshua B Tenenbaum"],"title":"Global versus local methods in nonlinear dimensionality reduction","venue":"In Advances in neural information processing systems,","year":2003},{"authors":["Josef Spidlen","Karin Breuer","Chad Rosenberg","Nikesh Kotecha","Ryan R Brinkman"],"title":"Flowrepository: A resource of annotated flow cytometry datasets associated with peer-reviewed publications","venue":"Cytometry Part A,","year":2012},{"authors":["Jian Tang","Meng Qu","Mingzhe Wang","Ming Zhang","Jun Yan","Qiaozhu Mei"],"title":"Line: Large-scale information network embedding","venue":"In Proceedings of the 24th international conference on world wide web,","year":2015},{"authors":["Jian Tang","Jingzhou Liu","Ming Zhang","Qiaozhu Mei"],"title":"Visualizing large-scale and highdimensional data","venue":"In Proceedings of the 25th international conference on world wide web,","year":2016},{"authors":["Bosiljka Tasic","Zizhen Yao","Lucas T Graybuck","Kimberly A Smith","Thuc Nghi Nguyen","Darren Bertagnolli","Jeff Goldy","Emma Garren","Michael N Economo","Sarada Viswanathan"],"title":"Shared and distinct transcriptomic cell types across neocortical","venue":"areas. Nature,","year":2018},{"authors":["Joshua B Tenenbaum","Vin De Silva","John C Langford"],"title":"A global geometric framework for nonlinear dimensionality reduction","venue":null,"year":2000},{"authors":["Dmitry Ulyanov"],"title":"Multicore-tsne. https://github.com/DmitryUlyanov/ Multicore-TSNE, 2016","venue":null,"year":2016},{"authors":["Koen Van den Berge","Hector Roux De Bezieux","Kelly Street","Wouter Saelens","Robrecht Cannoodt","Yvan Saeys","Sandrine Dudoit","Lieven Clement"],"title":"Trajectory-based differential expression analysis for single-cell sequencing data","venue":"Nature communications,","year":2020},{"authors":["Laurens Van Der Maaten"],"title":"Accelerating t-sne using tree-based algorithms","venue":"The Journal of Machine Learning Research,","year":2014},{"authors":["Jarkko Venna","Samuel Kaski"],"title":"Neighborhood preservation in nonlinear projection methods: An experimental study","venue":"In International Conference on Artificial Neural Networks,","year":2001},{"authors":["Han Xiao","Kashif Rasul","Roland Vollgraf"],"title":"Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017","venue":null,"year":2017},{"authors":["Lin Yan","Yaodong Zhao","Paul Rosen","Carlos Scheidegger","Bei Wang"],"title":"Homology-preserving dimensionality reduction via manifold landmarking and tearing","venue":"In Visualization in Data Science (VDS),","year":2018},{"authors":["Yuansheng Zhou","Tatyana O Sharpee"],"title":"Using global t-sne to preserve inter-cluster data structure. bioRxiv, 2018","venue":null,"year":2018},{"authors":["Lee","Verleysen"],"title":"The first two metrics are used to test the preservation of global structures and the last two metrics are suggested for the preservation of the local structures. Distance To a Measure considers the dispersion of high- and low-dimensional","venue":null,"year":2007},{"authors":["Moor"],"title":"σ (z) where x is the point in high-dimensional space X and the z is the corresponding projected point in low-dimensional space Z, we can examines the similarity of two datasets. In our experiments, we used the Euclidean distance and the values were normalized between 0 and 1","venue":"The σ ∈ R>0,","year":2020},{"authors":["scikit-optimize (Head"],"title":"2018)) to find the best hyperparameters for topological autoencoders. UMATO has several hyperparameters such as the number of hub points, the number of epochs, and the learning rate for global and local optimization. In our experiments, we configured everything except the number of hub points to the same setting for UMATO. We used 200 hub points for the Spheres dataset and had 300 hubs","venue":null,"year":2018},{"authors":["Tasic"],"title":"Each cell belongs to one of 133 clusters defined by Jaccard–Louvain clustering (for more than 4,000 cells) or a combination of k-means and Ward’s hierarchical clustering. Likewise, each cluster belongs to one of 4 classes: GABAergic (red/purple), Endothelial (brown), Glutamatergic (blue/green), Non-Neuronal (dark green). The embedding result for each method is given in Figure 11. In the case of t-SNE, clusters","venue":null,"year":2018},{"authors":["Van den Berge"],"title":"data analysis where the researchers want to know the distance between samples González-Blas et al","venue":null,"year":2020}],"string":"[\n {\n \"authors\": [\n \"Josh Barnes\",\n \"Piet Hut\"\n ],\n \"title\": \"A hierarchical o (n log n) force-calculation\",\n \"venue\": null,\n \"year\": 1986\n },\n {\n \"authors\": [\n \"Mikhail Belkin\",\n \"Partha Niyogi\"\n ],\n \"title\": \"Laplacian eigenmaps and spectral techniques for embedding and clustering\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2002\n },\n {\n \"authors\": [\n \"Tess Brodie\",\n \"Elena Brenna\",\n \"Federica Sallusto\"\n ],\n \"title\": \"Omip-018: Chemokine receptor expression on human t helper cells\",\n \"venue\": \"Cytometry Part A,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Frédéric Chazal\",\n \"David Cohen-Steiner\",\n \"Quentin Mérigot\"\n ],\n \"title\": \"Geometric inference for probability measures\",\n \"venue\": \"Foundations of Computational Mathematics,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Frédéric Chazal\",\n \"Brittany Fasy\",\n \"Fabrizio Lecci\",\n \"Bertrand Michel\",\n \"Alessandro Rinaldo\",\n \"Larry Wasserman\"\n ],\n \"title\": \"Robust topological inference: Distance to a measure and kernel distance\",\n \"venue\": \"The Journal of Machine Learning Research,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Tarin Clanuwat\",\n \"Mikel Bober-Irizar\",\n \"Asanobu Kitamoto\",\n \"Alex Lamb\",\n \"Kazuaki Yamamoto\",\n \"David Ha\"\n ],\n \"title\": \"Deep learning for classical japanese literature, 2018\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"James Cook\",\n \"Ilya Sutskever\",\n \"Andriy Mnih\",\n \"Geoffrey Hinton\"\n ],\n \"title\": \"Visualizing similarity data with a mixture of maps\",\n \"venue\": \"In Artificial Intelligence and Statistics,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Cong Fu\",\n \"Yonghui Zhang\",\n \"Deng Cai\",\n \"Xiang Ren\"\n ],\n \"title\": \"Atsne: Efficient and robust visualization on gpu through hierarchical optimization\",\n \"venue\": \"In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Dian Gong\",\n \"Xuemei Zhao\",\n \"Gerard Medioni\"\n ],\n \"title\": \"Robust multiple manifolds structure learning\",\n \"venue\": \"In Proceedings of the 29th International Conference on Machine Learning,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"Carmen Bravo González-Blas\",\n \"Liesbeth Minnoye\",\n \"Dafni Papasokrati\",\n \"Sara Aibar\",\n \"Gert Hulselmans\",\n \"Valerie Christiaens\",\n \"Kristofer Davie\",\n \"Jasper Wouters\",\n \"Stein Aerts\"\n ],\n \"title\": \"cistopic: cis-regulatory topic modeling on single-cell atac-seq data\",\n \"venue\": \"Nature methods,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Antonio Gracia\",\n \"Santiago González\",\n \"Victor Robles\",\n \"Ernestina Menasalvas\"\n ],\n \"title\": \"A methodology to compare dimensionality reduction algorithms in terms of loss of quality\",\n \"venue\": \"Information Sciences,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"der Fabisch\"\n ],\n \"title\": \"scikit-optimize/scikit-optimize: v0.5.2, March 2018\",\n \"venue\": \"URL https://doi.org/\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Geoffrey E Hinton\",\n \"Sam Roweis\"\n ],\n \"title\": \"Stochastic neighbor embedding\",\n \"venue\": \"Advances in neural information processing systems,\",\n \"year\": 2002\n },\n {\n \"authors\": [\n \"Yann LeCun\",\n \"Corinna Cortes. MNIST handwritten digit database.\"\n ],\n \"title\": \"URL http://yann\",\n \"venue\": \"lecun.com/exdb/mnist/.\",\n \"year\": 2010\n },\n {\n \"authors\": [\n \"John A Lee\",\n \"Michel Verleysen\"\n ],\n \"title\": \"Nonlinear dimensionality reduction\",\n \"venue\": \"Springer Science & Business Media,\",\n \"year\": 2007\n },\n {\n \"authors\": [\n \"John A Lee\",\n \"Michel Verleysen\"\n ],\n \"title\": \"Quality assessment of dimensionality reduction\",\n \"venue\": \"Rank-based criteria. Neurocomputing,\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"George C Linderman\",\n \"Manas Rachh\",\n \"Jeremy G Hoskins\",\n \"Stefan Steinerberger\",\n \"Yuval Kluger\"\n ],\n \"title\": \"Fast interpolation-based t-sne for improved visualization of single-cell rna-seq data\",\n \"venue\": \"Nature methods,\",\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\": \"Journal of machine learning research,\",\n \"year\": 2008\n },\n {\n \"authors\": [\n \"Leland McInnes\",\n \"John Healy\",\n \"James Melville\"\n ],\n \"title\": \"Umap: Uniform manifold approximation and projection for dimension reduction\",\n \"venue\": \"arXiv preprint arXiv:1802.03426,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Tomas Mikolov\",\n \"Ilya Sutskever\",\n \"Kai Chen\",\n \"Greg S Corrado\",\n \"Jeff Dean\"\n ],\n \"title\": \"Distributed representations of words and phrases and their compositionality\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2013\n },\n {\n \"authors\": [\n \"Michael Moor\",\n \"Max Horn\",\n \"Bastian Rieck\",\n \"Karsten Borgwardt\"\n ],\n \"title\": \"Topological autoencoders\",\n \"venue\": \"In Proceedings of the 37th International Conference on Machine Learning (ICML), Proceedings of Machine Learning Research. PMLR,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Corey J Nolet\",\n \"Victor Lafargue\",\n \"Edward Raff\",\n \"Thejaswi Nanditale\",\n \"Tim Oates\",\n \"John Zedlewski\",\n \"Joshua Patterson\"\n ],\n \"title\": \"Bringing umap closer to the speed of light with gpu acceleration\",\n \"venue\": null,\n \"year\": 2008\n },\n {\n \"authors\": [\n \"Nicola Pezzotti\",\n \"Julian Thijssen\",\n \"Alexander Mordvintsev\",\n \"Thomas Höllt\",\n \"Baldur Van Lew\",\n \"Boudewijn PF Lelieveldt\",\n \"Elmar Eisemann\",\n \"Anna Vilanova\"\n ],\n \"title\": \"Gpgpu linear complexity tsne optimization\",\n \"venue\": \"IEEE transactions on visualization and computer graphics,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Vin D Silva\",\n \"Joshua B Tenenbaum\"\n ],\n \"title\": \"Global versus local methods in nonlinear dimensionality reduction\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2003\n },\n {\n \"authors\": [\n \"Josef Spidlen\",\n \"Karin Breuer\",\n \"Chad Rosenberg\",\n \"Nikesh Kotecha\",\n \"Ryan R Brinkman\"\n ],\n \"title\": \"Flowrepository: A resource of annotated flow cytometry datasets associated with peer-reviewed publications\",\n \"venue\": \"Cytometry Part A,\",\n \"year\": 2012\n },\n {\n \"authors\": [\n \"Jian Tang\",\n \"Meng Qu\",\n \"Mingzhe Wang\",\n \"Ming Zhang\",\n \"Jun Yan\",\n \"Qiaozhu Mei\"\n ],\n \"title\": \"Line: Large-scale information network embedding\",\n \"venue\": \"In Proceedings of the 24th international conference on world wide web,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Jian Tang\",\n \"Jingzhou Liu\",\n \"Ming Zhang\",\n \"Qiaozhu Mei\"\n ],\n \"title\": \"Visualizing large-scale and highdimensional data\",\n \"venue\": \"In Proceedings of the 25th international conference on world wide web,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Bosiljka Tasic\",\n \"Zizhen Yao\",\n \"Lucas T Graybuck\",\n \"Kimberly A Smith\",\n \"Thuc Nghi Nguyen\",\n \"Darren Bertagnolli\",\n \"Jeff Goldy\",\n \"Emma Garren\",\n \"Michael N Economo\",\n \"Sarada Viswanathan\"\n ],\n \"title\": \"Shared and distinct transcriptomic cell types across neocortical\",\n \"venue\": \"areas. Nature,\",\n \"year\": 2018\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 \"Dmitry Ulyanov\"\n ],\n \"title\": \"Multicore-tsne. https://github.com/DmitryUlyanov/ Multicore-TSNE, 2016\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Koen Van den Berge\",\n \"Hector Roux De Bezieux\",\n \"Kelly Street\",\n \"Wouter Saelens\",\n \"Robrecht Cannoodt\",\n \"Yvan Saeys\",\n \"Sandrine Dudoit\",\n \"Lieven Clement\"\n ],\n \"title\": \"Trajectory-based differential expression analysis for single-cell sequencing data\",\n \"venue\": \"Nature communications,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Laurens Van Der Maaten\"\n ],\n \"title\": \"Accelerating t-sne using tree-based algorithms\",\n \"venue\": \"The Journal of Machine Learning Research,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Jarkko Venna\",\n \"Samuel Kaski\"\n ],\n \"title\": \"Neighborhood preservation in nonlinear projection methods: An experimental study\",\n \"venue\": \"In International Conference on Artificial Neural Networks,\",\n \"year\": 2001\n },\n {\n \"authors\": [\n \"Han Xiao\",\n \"Kashif Rasul\",\n \"Roland Vollgraf\"\n ],\n \"title\": \"Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017\",\n \"venue\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Lin Yan\",\n \"Yaodong Zhao\",\n \"Paul Rosen\",\n \"Carlos Scheidegger\",\n \"Bei Wang\"\n ],\n \"title\": \"Homology-preserving dimensionality reduction via manifold landmarking and tearing\",\n \"venue\": \"In Visualization in Data Science (VDS),\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Yuansheng Zhou\",\n \"Tatyana O Sharpee\"\n ],\n \"title\": \"Using global t-sne to preserve inter-cluster data structure. bioRxiv, 2018\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Lee\",\n \"Verleysen\"\n ],\n \"title\": \"The first two metrics are used to test the preservation of global structures and the last two metrics are suggested for the preservation of the local structures. Distance To a Measure considers the dispersion of high- and low-dimensional\",\n \"venue\": null,\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Moor\"\n ],\n \"title\": \"σ (z) where x is the point in high-dimensional space X and the z is the corresponding projected point in low-dimensional space Z, we can examines the similarity of two datasets. In our experiments, we used the Euclidean distance and the values were normalized between 0 and 1\",\n \"venue\": \"The σ ∈ R>0,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"scikit-optimize (Head\"\n ],\n \"title\": \"2018)) to find the best hyperparameters for topological autoencoders. UMATO has several hyperparameters such as the number of hub points, the number of epochs, and the learning rate for global and local optimization. In our experiments, we configured everything except the number of hub points to the same setting for UMATO. We used 200 hub points for the Spheres dataset and had 300 hubs\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Tasic\"\n ],\n \"title\": \"Each cell belongs to one of 133 clusters defined by Jaccard–Louvain clustering (for more than 4,000 cells) or a combination of k-means and Ward’s hierarchical clustering. Likewise, each cluster belongs to one of 4 classes: GABAergic (red/purple), Endothelial (brown), Glutamatergic (blue/green), Non-Neuronal (dark green). The embedding result for each method is given in Figure 11. In the case of t-SNE, clusters\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Van den Berge\"\n ],\n \"title\": \"data analysis where the researchers want to know the distance between samples González-Blas et al\",\n \"venue\": null,\n \"year\": 2020\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"We present a novel dimensionality reduction method, Uniform Manifold Approximation with Twophase Optimization (UMATO) to obtain less biased and robust embedding over diverse initialization methods. One effective way of understanding high-dimensional data in various domains is to reduce its dimensionality and investigate the projection in a lower-dimensional space. The limitation of previous approaches such as t-Stochastic Neighbor Embedding (t-SNE, Maaten & Hinton (2008)) and Uniform Manifold Approximation and Projection (UMAP, McInnes et al. (2018)) is that they are susceptible to different initialization methods, generating considerably different embedding results (Section 5.5).\nt-SNE adopts Kullback-Leibler (KL) divergence as its loss function. The fundamental limitation of the KL divergence is that the penalty for the points that are distant in the original space being close in the projected space is too little (Appendix B). This results in only the local manifolds being captured, while clusters that are far apart change their relative locations from run to run. Meanwhile, UMAP leverages the cross-entropy loss function, which is known to charge a penalty for points that are distant in the original space being close in the projection space and for points that are close in the original space being distant in the projection space (Appendix B). UMAP considers all points in the optimization at once with diverse sampling techniques (i.e., negative sampling and edge sampling). Although the approximation technique in UMAP optimization makes the computation much faster, this raises another problem that the clusters in the embedding become dispersed as the number of epochs increases (Appendix K), which can lead to misinterpretation. UMAP tried to alleviate this by using a fixed number (e.g., 200), which is ad hoc, and by applying a learning rate decay. However, the optimal number of epochs and decay schedule for each initialization method needs to be found in practice.\nTo solve the aforementioned problems, we avoid using approximation during the optimization process, which normally would result in greatly increased computational cost. Instead, we first run optimization only with a small number of points that represent the data (i.e., hub points). Finding the optimal projection for a small number of points using a cross-entropy function is relatively easy and robust, making the additional techniques employed in UMAP unnecessary. Furthermore, it is less sensitive to the initialization method used (Section 5.5). After capturing the overall skeleton of the high-dimensional structure, we gradually append the rest of the points in subsequent phases. Although the same approximation technique as UMAP is used for these points, as we have already embedded the hub points and use them as anchors, the projections become more robust and unbiased. The gradual addition of points can in fact be done in a single phase; we found additional phases\ndo not result in meaningful improvements in the performance but only in the increased computation time (Section 4.5). Therefore, we used only two phases in UMAP: global optimization to capture the global structures (i.e., the pairwise distances in a high-dimensional space) and local optimization to retain the local structures (i.e., the relationships between neighboring points in a high-dimensional space) of the data.\nWe compared UMATO with popular dimensionality reduction techniques including PCA, Isomap (Tenenbaum et al. (2000)), t-SNE, UMAP, topological autoencoders (Moor et al. (2020)) and At-SNE (Fu et al. (2019)). We used one synthetic (101-dimensional spheres) and three realworld (MNIST, Fashion MNIST, and Kuzushiji MNIST) datasets and analyzed the projection results with several quality metrics. In conclusion, UMATO demonstrated better performance than the baseline techniques in all datasets in terms of KLσ with different σ values, meaning that it reasonably preserved the density of data over diverse length scales. Finally, we presented the 2D projections of each dataset, including the replication of an experiment using the synthetic Spheres dataset introduced by Moor et al. (2020) where data points locally constitute multiple small balls globally contained in a larger sphere. Here, we demonstrate that UMATO can better preserve both structures compared to the baseline algorithms (Figure 3)."},{"heading":"2 RELATED WORK","text":"Dimensionality reduction. Most previous dimensionality reduction algorithms focused on preserving the data’s local structures. For example, Maaten & Hinton (2008) proposed t-SNE, focusing on the crowding problem with which the previous attempts (Hinton & Roweis (2002); Cook et al. (2007)) have struggled, to visualize high-dimensional data through projection produced by performing stochastic gradient descent on the KL divergence between two density functions in the original and projection spaces. Van Der Maaten (2014) accelerated t-SNE developing a variant of the Barnes-Hut algorithm (Barnes & Hut (1986)) and reduced the computational complexity from O(N2) into O(N logN). After that, grounded in Riemannian geometry and algebraic topology, McInnes et al. (2018) introduced UMAP as an alternative to t-SNE. Leveraging the cross-entropy function as its loss function, UMAP reduced the computation time by employing negative sampling from Word2Vec (Mikolov et al. (2013)) and edge sampling from LargeVis (Tang et al. (2015; 2016)) (Table 1). Moreover, they showed that UMAP can generate stable projection results compared to t-SNE over repetition.\nOn the other hand, there also exist algorithms that aim to capture the global structures of data. Isomap (Tenenbaum et al. (2000)) was proposed to approximate the geodesic distance of highdimensional data and embed it onto the lower dimension. Global t-SNE (Zhou & Sharpee (2018)) converted the joint probability distribution, P , in the high-dimensional space from Gaussian to Student’s-t distribution, and proposed a variant of KL divergence. By adding it with the original loss function of t-SNE, Global t-SNE assigns a relatively large penalty for a pair of distant data points in high-dimensional space being close in the projection space. Another example is topological autoencoders (Moor et al. (2020)), a deep-learning approach that uses a generative model to make the latent space resemble the high-dimensional space by appending a topological loss to the original reconstruction loss of autoencoders. However, they required a huge amount of time for hyperparameter exploration and training for a dataset, and only focused on the global aspect of data. Unlike other techniques that presented a variation of loss functions in a single pipeline, UMATO is novel as it preserves both structures by dividing the optimization into two phases; this makes it outperform the baselines with respect to quality metrics in our experiments.\nHubs, landmarks, and anchors. Many dimensionality reduction techniques have tried to draw sample points to better model the original space; these points are usually called hubs, landmarks, or anchors. Silva & Tenenbaum (2003) proposed Landmark Isomap, a landmark version of classical multidimensional scaling (MDS) to alleviate its computation cost. Based on the Landmark Isomap, Yan et al. (2018) tried to retain the topological structures (i.e., homology) of high-dimensional data by approximating the geodesic distances of all data points. However, both techniques have the limitation that landmarks were chosen randomly without considering their importance. UMATO uses a k-nearest neighbor graph to extract significant hubs that can represent the overall skeleton of high-dimensional data. The most similar work to ours is At-SNE (Fu et al. (2019)), which optimized the anchor points and all other points with two different loss functions. However, since the anchors wander during the optimization and the KL divergence does not care about distant points, it hardly\ncaptures the global structure. UMATO separates the optimization process into two phases so that the hubs barely moves but guides other points so that the subareas manifest the shape of the highdimensional manifold in the projection. Applying different cross-entropy functions to each phase also helps preserve both structures."},{"heading":"3 UMAP","text":"Since UMATO shares the overall pipeline of UMAP (McInnes et al. (2018)), we briefly introduce UMAP in this section. Although UMAP is grounded in a sophisticated mathematical foundation, its computation can be simply divided into two steps, graph construction and layout optimization, a configuration similar to t-SNE. In this section, we succinctly explain the computation in an abstract manner. For more details about UMAP, please consult the original paper (McInnes et al. (2018)).\nGraph Construction. UMAP starts by generating a weighted k-nearest neighbor graph that represents the distances between data points in the high-dimensional space. Given an input dataset X = {x1, . . . , xn}, the number of neighbors to consider k and a distance metric d : X × X → [0,∞), UMAP first computes Ni, the k-nearest neighbors of xi with respect to d. Then, UMAP computes two parameters, ρi and σi, for each data point xi to identify its local metric space. ρi is a nonzero distance from xi to its nearest neighbor:\nρi = min j∈Ni {d(xi, xj) | d(xi, xj) > 0}. (1)\nUsing binary search, UMAP finds σi that satisfies:∑ j∈Ni exp(−max(0, d(xi, xj)− ρi)/σi) = log2(k). (2)\nNext, UMAP computes:\nvj|i = exp(−max(0, d(xi, xj)− ρi)/σi), (3)\nthe weight of the edge from a point xi to another point xj . To make it symmetric, UMAP computes vij = vj|i + vi|j − vj|i · vi|j , a single edge with combined weight using vj|i and vi|j . Note that vij indicates the similarity between points xi and xj in the original space. Let yi be the projection of xi in a low-dimensional projection space. The similarity between two projected points yi and yj is wij = (1 + a||yi − yj ||2b2 )−1, where a and b are positive constants defined by the user. Setting both a and b to 1 is identical to using Student’s t-distribution to measure the similarity between two points in the projection space as in t-SNE (Maaten & Hinton (2008)).\nLayout Optimization. The goal of layout optimization is to find the yi that minimizes the difference (or loss) between vij and wij . Unlike t-SNE, UMAP employs the cross entropy:\nCUMAP = ∑ i 6=j [vij · log(vij/wij)− (1− vij) · log((1− vij)/(1− wij))], (4)\nbetween vij andwij as the loss function. UMAP initializes yi through spectral embedding (Belkin & Niyogi (2002)) and iteratively optimize its position to minimize CUMAP . Given the output weight wij as 1/(1 + ad2bij ), the attractive gradient is:\nCUMAP yi\n+ = −2abd2(b−1)ij 1 + ad2bij vij(yi − yj), (5)\nand the repulsive gradient is:\nCUMAP yi\n− =\n2b\n( + d2ij)(1 + ad 2b ij )\n(1− vij)(yi − yj), (6)\nwhere is a small value added to prevent division by zero and dij is a Euclidean distance between yi and yj . For efficient optimization, UMAP leverages the negative sampling technique from Word2Vec (Mikolov et al. (2013)). After choosing a target point and its negative samples, the position of the target is updated with the attractive gradient, while the positions of the latter do so with\nthe repulsive gradient. Moreover, UMAP utilizes edge sampling (Tang et al. (2015; 2016)) to accelerate and simplify the optimization process (Table 1). In other words, UMAP randomly samples edges with a probability proportional to their weights, and subsequently treats the selected ones as binary edges. Considering the previous sampling techniques, the modified objective function is:\nO = ∑\n(i,j)∈E\nvij(log(wij) + M∑ k=1 Ejk∼Pn(j)γ log(1− wijk)). (7)\nHere, vij and wij are the similarities in the high and low-dimensional spaces respectively, M is the number of negative samples and Ejk∼Pn(j) indicates that jk is sampled according to a noisy distribution, Pn(j), from Word2Vec (Mikolov et al. (2013))."},{"heading":"4 UMATO","text":"Figure 2 illustrates the computation pipeline of UMATO, which delineates the two-phase optimization (see Figure 9 for a detailed illustration of the overall pipeline). As a novel approach, we split the optimization into global and local so that it could generate a low-dimensional projection keeping both structures well-maintained. We present the pseudocode of UMATO in Appendix A, and made the source codes of it publicly available1.\n4.1 POINTS CLASSIFICATION\nIn the big picture, UMATO follows the pipeline of UMAP. We first find the k-nearest neighbors in the same way as UMAP, by assuming the local connectivity constraint, i.e., no single point is isolated and each point is connected to at least a user-defined number of points. After calculating ρ (Equation 1) and σ (Equation 2) for each point, we obtain the pairwise similarity for every pair of points. Once the k-nearest neighbor indices are established, we unfold it and check the frequency of each point to sort them into descending order so that the index of the popular points come to the front.\nThen, we build a k-nearest neighbor graph by repeating the following steps until no points remain unconnected: 1) choose the most frequent point as a hub among points that are not already connected, 2) retrieve the k-nearest neighbors of the chosen point (i.e., hub), and the points selected from steps 1 and 2 will become a connected component. The gist is that we divide the points into three disjoint sets: hubs, expanded nearest neighbors, and outliers (Figure 1). Thanks to the sorted indices, the most popular point in each iteration—but not too densely located—becomes the hub point. Once the hub points are determined, we recursively seek out their nearest neighbors and again look for the nearest neighbors of those neighbors, until there are no points to be newly appended. In\nother words, we find all connected points that are expanded from the original hub points, which, in turn, is called the expanded nearest neighbors. Any remaining point that is neither a hub point nor a part of any expanded nearest neighbors is classified as an outlier. The main reason to rule out the outliers is, similar to the previous approach (Gong et al. (2012)), to achieve the robustness of the practical manifold learning algorithm. As the characteristics of these classes differ significantly, we take a different approach for each class of points to obtain both structures. That is, we run global optimization for the hub points (Section 4.2), local optimization for the expanded nearest neighbors (Section 4.3), and no optimization for the outliers (Section 4.4). In the next section we explain each in detail."},{"heading":"4.2 GLOBAL OPTIMIZATION","text":"After identifying hub points, we run the global optimization to retrieve the skeletal layout of the data. First, we initialize the positions of hub points using PCA, which makes the optimization process\n1https://www.github.com/anonymous-author/anonymous-repo\nfaster and more stable than using random initial positions. Next, we optimize the positions of hub points by minimizing the cross-entropy function (Equation 4). Let f(X) = {f(xi, xj)|xi, xj ∈ X} and g(Y ) = {g(yi, yj)|yi, yj ∈ Y } be two adjacency matrices in high- and low-dimensional spaces. If Xh represents a set of points selected as hubs in high-dimensional space, and Yh is a set of corresponding points in the projection, we minimize the cross entropy—CE(f(Xh)||g(Yh))— between f(Xh) and g(Yh).\nTable 1: The runtime for each algorithm using MNIST dataset. UMAP and UMATO take much less time than MulticoreTSNE (Ulyanov (2016)) when tested on a Linux server with 40-core Intel Xeon Silver 4210 CPUs. The runtimes are averaged over 10 runs. Isomap (Tenenbaum et al. (2000)) took more than 3 hours to get the embedding result.\nAlgorithm Runtime (s)\nIsomap 3 hours > t-SNE 374.85 ± 11.38 UMAP 26.10 ± 3.97\nUMATO 73.32 ± 8.39\nUMAP computes the cross-entropy between all existing points using two sampling techniques, edge sampling and negative sampling, for speed (Table 1). However, this often ends up capturing only the local properties of data because of the sampling biases and thus it cannot be used for cases that require a comprehensive understanding of the data. On the other hand, in its first phase, UMATO only optimizes for representatives (i.e., the hub points) of data, which takes much less time but can still approximate the manifold effectively."},{"heading":"4.3 LOCAL OPTIMIZATION","text":"In the second phase, UMATO embeds the expanded nearest neighbors to the projection that is computed using only the hub points from the first phase. For each point in the expanded nearest neighbors, we retrieve its nearest m (e.g., 10) hubs in the original high-dimensional space and set its initial position in the projection to the average positions of the hubs in the projection with small random perturbations. We follow a similar optimization process as UMAP in the local optimization with small differences. As explained in Section 3, UMAP first constructs the graph structure; we perform the same task but only with the hubs and expanded nearest neighbors. While doing this, since some points are excluded as outliers, we need to update the k-nearest neighbor indices. This is fast because we recycle the already-built k-nearest neighbor indices by updating the outliers to the new nearest neighbor.\nOnce we compute the similarity between points (Equation 3), to optimize the positions of points, similar to UMAP, we use the cross-entropy loss function with edge sampling and negative sampling (Equation 7). Here, we try to avoid moving the hubs as much as possible since they have already formed the global structure. Thus, we only sample a point p among the expanded nearest neighbors as one end of an edge, while the point q at the other end of the edge can be chosen from all points except outliers. In UMAP implementation, when q pulls in p, p also drags q to facilitate the optimization (Equation 5). When updating the position of q, we only give a penalty to this (e.g., 0.1), if q is a hub point, not letting its position excessively be affected by p. In addition, because the repulsive force can disperse the local attachment, making the point veer off for each epoch and eventually destroying the well-shaped global layout, we multiply a penalty (e.g., 0.1) when calculating the repulsive gradient (Equation 6) for the points selected as negative samples."},{"heading":"4.4 OUTLIERS ARRANGEMENT","text":"Since the isolated points, which we call outliers, mostly have the same distance to all the other data points in high-dimensional space, due to the curse of dimensionality, they both sabotage the global structure we have already made and try to mingle with all other points, thus distorting the overall projection. We do not optimize these points but instead simply append them using the alreadyprojected points (e.g., hubs or expanded nearest neighbors), that belong to each outlier’s connected component of the nearest neighbor graph. That is, if xi ∈ Cn where xi is the target outlier and Cn is the connected component to which xi belongs, we find xj ∈ Cn that has already been projected and is closest to xi. We arrange yi which corresponds to xi in low-dimensional space using the position of yj in the same component offset by a random noise. In this way, we can benefit from the comprehensive composition of the projection that we have already optimized when arranging the outliers. We can ensure that all outliers can find a point as its neighbor since we picked hubs from each connected component of the nearest neighbor graph and thus at least one point is already located and has an optimized position (Section 4.2)."},{"heading":"4.5 MULTI-PHASE OPTIMIZATION","text":"The optimization of UMATO can be easily expanded to multiple phases (e.g., three or more phases). Since we have a recursive procedure to expand the nearest neighbors, we can insert the optimization process each time we expand the neighbors to create a multi-phase algorithm. However, our experiment with three- and four-phase optimization with the Fashion MNIST dataset showed that there is no big difference between two-phase optimization and that with more than two phases. Appendix C contains the quantitative and qualitative results of the experiment for multi-phase optimization."},{"heading":"5 EXPERIMENTS","text":"We conducted experiments to evaluate UMATO’s ability to capture the global and local structures of high-dimensional data. We compared UMATO with six baseline algorithms, PCA, Isomap, t-SNE, UMAP, topological autoencoders, and At-SNE in terms of global (i.e., DTM and KLσ) and local (i.e., trustworthiness, continuity, and MRREs) quality metrics."},{"heading":"5.1 DATASETS","text":"We used four datasets for the experiments: Spheres, MNIST (LeCun & Cortes (2010)), Fashion MNIST (Xiao et al. (2017)), and Kuzushiji MNIST (Clanuwat et al. (2018)). Spheres is a synthetic dataset that has 10,000 rows of 101 dimensions. It has a high-dimensional structure in which ten small spheres are contained in a larger sphere. Specifically, the dataset’s first 5,000 rows are the points sampled from a sphere of radius 25 and 500 points are sampled for each of the ten smaller spheres of radius 5 shifted to a random direction from the origin. This dataset is the one used for the original experiment with topological autoencoders (Moor et al. (2020)). Other datasets are images of digits, fashion items, and Japanese characters, each of which consists of 60,000 784-dimensional (28 × 28) images from 10 classes."},{"heading":"5.2 EXPERIMENTAL SETTING","text":"Evaluation Metrics. To assess how well projections preserve the global structures of highdimensional data, we computed the density estimates (Chazal et al. (2011; 2017)), the so-called Distance To a Measure (DTM), between the original data and the projections. Moor et al. (2020) adopted the Kullback-Leibler divergence between density estimates with different scales (KLσ) to evaluate the global structure preservation. To follow the original experimental setup by Moor et al. (2020), we found the projections with the lowest KL0.1 from all algorithms by adjusting their hyperparameters. Next, to evaluate the local structure preservation of projections, we used the mean relative rank errors (MRREs, Lee & Verleysen (2007)), trustworthiness, and continuity (Venna & Kaski (2001)). All of these local quality metrics estimate how well the nearest neighbors in one space (e.g., high- or low-dimensional space) are preserved in the other space. For more information on the quality metrics, we refer readers to Appendix E.\nBaselines. We set the most widely used dimensionality reduction techniques as our baselines, including PCA, Isomap (Tenenbaum et al. (2000)), t-SNE (Maaten & Hinton (2008)),\nUMAP (McInnes et al. (2018)), and At-SNE (Fu et al. (2019)). In the case of t-SNE, we leveraged Multicore t-SNE (Ulyanov (2016)) for fast computation. To initialize the points’ position, we used PCA for t-SNE, following the recommendation in the previous work (Linderman et al. (2019)), and spectral embedding for UMAP which was set to default. In addition, we compared with topological autoencoders (Moor et al. (2020)) that were developed to capture the global properties of the data using a deep learning-based generative model. Following the convention of visualization in dimensionality reduction, we determined our result projected onto 2D space. We tuned the hyperparameters of each technique to minimize the KL0.1. Appendix F further describes the details of the hyperparameters settings."},{"heading":"5.3 QUANTITATIVE RESULTS","text":"Table 2 displays the experiment results. In most cases, UMATO was the only method that has shown performance both in the global and local quality metrics in most datasets. For local metrics, t-SNE, At-SNE, and UMAP generally had the upper-hand, but UMATO showed comparable MRREX and continuity in Spheres, Fashion MNIST, and Kuzushiji MNIST datasets. Meanwhile, Isomap and topological autoencoders were mostly good at global quality metrics, although UMATO had the lowest (best) KL0.1 and DTM except for the MNIST dataset."},{"heading":"5.4 QUALITATIVE RESULTS","text":"Among the five algorithms, only UMATO could preserve both the global and local structure of the Spheres dataset. If we look at the figure made by UMATO, the outer sphere encircles the inner spheres in a circular form, which is the most intuitive to understand the relationship among different classes and the local linkage in detail. In the results from Isomap, t-SNE, UMAP, and At-SNE,\nthe points representing the surrounding giant sphere mix with those representing the other small inner spheres, thus failing to capture the nested relationships among different classes. Meanwhile, topological autoencoders are able to realize the global relationship between classes in an incomplete manner; the points for the outer sphere are too spread out, thus losing the local characteristics of the class. From this result, we can acknowledge how UMATO can work with high-dimensional data effectively to reveal both global and local structures. 2D visualization results on other datasets (MNIST, Fashion MNIST, Kuzushiji MNIST) can be found in Appendix H. Lastly, we report an additional experiment on the mouse neocortex dataset (Tasic et al. (2018)) in Appendix N which shows the relationship between classes much better than the baseline algorithms like t-SNE and UMAP."},{"heading":"5.5 PROJECTION ROBUSTNESS OVER DIVERSE INITIALIZATION METHODS","text":"We experimented with the robustness of each dimensionality reduction technique with different initialization methods such as PCA, spectral embedding, random position, and class-wise separation. In class-wise separation, we initialized each class with a non-overlapping random position in 2- dimensional space, adding random Gaussian noise. In our results, UMATO embeddings were almost the same on the real-world datasets, while the UMAP and t-SNE results relied highly upon the initialization method. We report this in Table 3 with a quantitative comparison using Procrustes distance. Specifically, given two datasets X = {x1, x2, . . . , xn} and Y = {y′1, y′2, . . . , y′n} where y′i corresponds to xi, the Procrustes distance is defined as\ndP (X,Y ) = √√√√ N∑ i=1 (xi − y′i). (8)\nFor all cases, we ran optimal translation, uniform scaling, and rotation to minimize the Procrustes distance between the two distributions. In the case of the Spheres dataset, as defined in Appendix G, the clusters were equidistant from each other. The embedding results have to be different due to the limitation of the 2-dimensional space since there is no way to express this relationship. However, as we report in Figure 4, the global and local structures of the Spheres data are manifested with UMATO with all different initialization methods."},{"heading":"6 CONCLUSION","text":"We present a two-phase dimensionality reduction algorithm called UMATO that can effectively preserve the global and local properties of high-dimensional data. In our experiments with diverse datasets, we have proven that UMATO can outperform previous widely used baselines (e.g., t-SNE and UMAP) both quantitatively and qualitatively. As future work, we plan to accelerate UMATO, as in previous attempts with other dimensionality reduction techniques (Pezzotti et al. (2019); Nolet et al. (2020)), by implementing it on a heterogeneous system (e.g., GPU) for speedups."},{"heading":"A UMATO ALGORITHM PSEUDOCODE","text":"The pseudocode of UMATO is as below:\nAlgorithm 1 Uniform Manifold Approximation with Two-phase Optimization 1: procedure UMATO(X , kX , d, min dist, nh, eg , el) Input: High-dimensional data X , number of nearest neighbors k, projection dimension d, minimum distance in projection result min dist, number of hub points nh, epochs for global and local optimization eg , el Output: Low-dimensional projection Y 2: Compute k-nearest neighbors of X 3: Obtain sorted list using indices’ frequency of k-nearest neighbors 4: Build k-nearest neighbor graph structure 5: Classify points into hubs, expanded nearest neighbors, and outliers 6: Optimize CE(f(Xh)||g(Yh)) to preserve global configuration (Equation 4) 7: Initialize expanded nearest neighbors using hub locations 8: Update k-nearest neighbors & compute weights (Equation 3) 9: Optimize CE(f(X)||g(Y )) to preserve local configuration (Equation 7) 10: Position outliers 11: return Y 12: end procedure"},{"heading":"B THE MEANING OF USING DIFFERENT LOSS FUNCTIONS IN DIMENSIONALITY REDUCTION","text":"Following the notations from above, we set the similarity between points in high-dimensional space xi and xj as vij and the low-dimensional space yi and yj as wij . Then we can write the KL divergence and cross entropy loss function as:\nKL = ∑ i6=j vij · log(vij/wij), (9)\nCE = ∑ i 6=j vij · log(vij/wij)− ∑ i 6=j (1− vij) · log((1− vij)/(1− wij)). (10)\nIf we use the same probability distributions, vij and wij , the KL divergence and the first term of cross-entropy are exactly the same. If vij and wij are both large (Table 4 a.) or both small (Table 4 d.), this means that the relationship between points in high-dimensional space is well-retained in the projection. Thus, the positions of points in the low-dimensional space do not have to move. As vij and wij are similar, log(vij/wij) becomes zero, producing a small cost in the end.\nHowever, we need to modify the position of wij if there exists a gap between vij and wij . The KL divergence imposes a big penalty when vij is large but wij is small (Table 4 b.). That is, if the neighboring points in high-dimensional space are not captured well in the projection, the KL divergence imposes a high penalty to move the point (vij) into the right position. Thus, we can understand why t-SNE is able to capture the local characteristics of high-dimensional space, but not the global ones. However, the second term of cross-entropy imposes a big penalty when vij is small but wij is large (Table 4 g.). Therefore, it moves points that are close together in the highdimensional space but far apart in the projection."},{"heading":"C MULTI-PHASE OPTIMIZATION","text":"We report the result of multi-phase optimization (e.g., three and four-phase) using the Fashion MNIST dataset both quantitatively (Table 5) and qualitatively (Figure 5). As in Figure 5, we were unable to find any significant differences between the 2D projections, although some outliers were located in different places. Moreover, the quality metrics were almost the same for all three results. The original UMATO was the winner in DTM, KL0.1, KL1, continuity, and MRREZ but came last in other quality metrics. In addition, the gap in metrics between UMATO and the multi-phase optimizations indicated a trivial difference. Thus, we concluded that developing a multi-phase optimization for UMATO does not bring about any notable improvement in the projection result."},{"heading":"D PROJECTION STABILITY","text":"Table 6 denotes the results of our experiment on the projection stability of UMATO and other dimensionality reduction techniques. When the data size grows, we want to sample a portion of it to speed up the visualization. However, the concern is whether the projection run with the sampled indices is consistent with the part of the projection result with indices selected from the full dataset. If the algorithm can generate stable and consistent results, the two projections should contain the least bias possible. To compute the projection stability of dimensionality reduction techniques, we\nused the normalized Procrustes distance (Equation 8) to measure the distance between two comparable distributions. To replicate the experiment by McInnes et al. (2018), we used the same Flow Cytometry dataset (Spidlen et al. (2012); Brodie et al. (2013)), and ran optimal translation, uniform scaling, and rotation to minimize the Procrustes distance between the two distributions. As we can see in Table 6, UMATO outperformed t-SNE and At-SNE for all sub-sample sizes. Moreover, although UMAP is known as stable among existing algorithms, UMATO showed even better (lower) Procrustes distance except for one sub-sample size (60%). From this result, we can acknowledge that UMATO can generate the more stable and consistent results regardless of sub-sample size than many other dimensionality reduction techniques."},{"heading":"E QUALITY METRICS","text":"As UMATO presents a dimensionality reduction technique that can capture both the global and local structures of high-dimensional data, we used several quality metrics to evaluate each aspect respectively. We have referred to some review papers (Gracia et al. (2014); Lee & Verleysen (2009)) for the best use and implementation. Among many quality metrics, we leveraged 1) Distance To a Measure (DTM, Chazal et al. (2011; 2017)), 2) KL divergence between two density functions, 3) trustworthiness and continuity (Venna & Kaski (2001)), and 4) mean relative rank errors (MRREs, Lee & Verleysen (2007)). The first two metrics are used to test the preservation of global structures and the last two metrics are suggested for the preservation of the local structures.\nDistance To a Measure considers the dispersion of high- and low-dimensional data, where it is defined for a given point as fXσ (x) := ∑ y∈X exp (−dist(x, y)2/σ). By summing up the element-\nwise absolute values between two distributions, ∑ x∈X,z∈Z f X σ (x)− fZσ (z) where x is the point in high-dimensional space X and the z is the corresponding projected point in low-dimensional space Z, we can examines the similarity of two datasets. In our experiments, we used the Euclidean distance and the values were normalized between 0 and 1. The σ ∈ R>0, which represents the length scale parameter, was set to 0.1. Moor et al. (2020) proposed the KL divergence of two probability distributions, KLσ := KL(fXσ ||fZσ ), a variation of DTM. Changing σ as a normalizing factor of the distribution, the authors investigated if the algorithms can preserve the global structure of the high-dimensional data. Following the same notion as the experiment in the paper (Moor et al. (2020)), we used three σ values, 1.0, 0.1, and 0.01, to test whether each algorithm can capture the global aspect with respect to diverse density estimates.\nTrustworthiness and continuity measure how much the nearest neighbors are preserved in a space (i.e., high- or low-dimensional space) compared to the other space by analyzing the ranks of knearest neighbors in both spaces. The difference between trustworthiness and continuity comes from which space is held as the base space. Specifically, we first need to find the k-nearest neighbors in both high- and low-dimensional space. Then, we compute the trustworthiness by checking whether the ranks of nearest neighbors in low-dimensional space resemble those of high-dimensional space. If so, we can achieve a high score in trustworthiness. Meanwhile, we achieve a high score in continuity if the ranks of nearest neighbors in high-dimensional space resemble those of low-dimensional space. MRREs take a similar approach to trustworthiness and continuity as it calculates and compares the ranks of the k-nearest neighbors in both spaces, but the normalizing factor is slightly different. Originally, it was better if MRREs had lower values. However, for the ease of comparing\nlocal quality metrics, we defined it as MRREs := 1 −MRREs, so higher MRREs mean that they have better retained the k-nearest neighbors like trustworthiness and continuity."},{"heading":"F HYPERPARAMETER SETTING","text":"As explained in Section 5.2, we generated projections for each dimensionality reduction algorithm that had the lowest KL0.1 measure. To tune each algorithm’s hyperparameters, we employed the grid search for t-SNE, UMAP, and At-SNE. For t-SNE and At-SNE, we changed the perplexity from 5 to 50 with an interval of 5, and the learning rate from 0.1 to 1.0 with a log-uniform scale. In the case of UMAP, we changed the number of nearest neighbors from 5 to 50 with an interval of 5, and the minimum distance between points in the projection from 0.1 to 1.0 with an interval of 0.1. We used the Python library scikit-optimize (Head et al. (2018)) to find the best hyperparameters for topological autoencoders. UMATO has several hyperparameters such as the number of hub points, the number of epochs, and the learning rate for global and local optimization. In our experiments, we configured everything except the number of hub points to the same setting for UMATO. We used 200 hub points for the Spheres dataset and had 300 hubs for others. We used fewer hub points for the Spheres since it has only 10,000 data points in total, while the other datasets have 60,000 data points. We set the number of epochs to 100 for global optimization and to 50 for local optimization. Lastly, the global learning rate was set to 0.0065, and the local learning rate was set to 0.01."},{"heading":"G SYNTHETIC SPHERES DATASET","text":"We leveraged the same Spheres dataset that Moor et al. (2020) used in their experiments of topological antoencoders. The Spheres dataset contains eleven high-dimensional spheres which reside in 101-dimensional space. We first generated ten spheres of radius of 5, and shifted each sphere by adding the same Gaussian noise to a random direction. For this aim, we created d-dimensional Gaussian vectors X ∼ N(0, I(10/ √ d)), where d is 101. As to embed an interesting geometrical structure to the dataset, the ten spheres of relatively small radii of 5 were enclosed by another larger sphere of radius of 25."},{"heading":"H MORE EXPERIMENTS ON SYNTHETIC DATASETS","text":"We leveraged the 3-dimensional S-curve and Swiss roll datasets to test whether UMATO can preserve both the global and local structures of original datasets. As the visualization shows (Figure 6),\nonly PCA and UMATO were able to capture the global and local structures of original datasets. Isomap, t-SNE and UMAP could capture the local manifolds of original datasets, but high-level manifolds of the original datasets were not reflected to the embedding."},{"heading":"I LOCAL QUALITY METRICS WITH DIFFERENT VALUES OF HYPERPARAMETER","text":"We report the result of local quality metrics with diverse hyperparameters (Table 7). We changed the number of nearest neighbors (k) from 5 to 15 with an interval of 5. As we have already reported when the k = 5 (Table 2), below are the cases where k = 10, 15. As we can check from the result, while the values fluctuate a little bit, the ranks are mostly robust over diverse k values.\nJ 2D EMBEDDING RESULTS OF UMATO AND BASELINE ALGORITHMS ON REAL-WORLD DATASETS\nFor the real-world datasets, UMATO showed a similar projection to PCA but with better captured local characteristics. The results from topological autoencoders showed some points detached far apart from their centers, even though the best hyperparameters were used for each. Although AtSNE claimed that it could capture both structures, the results were not significantly different from those of the original t-SNE algorithm when projecting the Spheres and Fashion MNIST datasets."},{"heading":"K EMBEDDING ROBUSTNESS OVER NUMBER OF EPOCHS","text":"We report the experimental result in Figure 8. As we explained, the UMAP embedding results are susceptible to the number of epochs so that the distance between clusters get dispersed. This can induce a misinterpretation that the user considers the distance between clusters as something meaningful. The two-phase optimization of UMATO can solve the problem since the global optimization (first phase) is easy to converge as it runs only with a small portion of points. Therefore, the increasing number of epochs in the global optimization does not harm the final embedding.\nL ILLUSTRATION OF UMATO PIPELINE\nFor the ease of understanding, we provide an illustration of UMATO pipeline in Figure 9. The detailed explanation for UMATO can be found in Section 4."},{"heading":"M EFFECT OF LOCAL LEARNING RATE OF UMATO","text":"By manipulating one of UMATO’s hyperparameters, local learning rate, the user can determine where to focus in the embedding result; to reveal more of the global structures, the user should apply a lower value (e.g., 0.005), while using a higher one (e.g., 0.1) would generate more like a UMAP embedding which prefers to show the local manifolds."},{"heading":"N UMATO ON REAL-WORLD BIOLOGICAL DATASET","text":"To test UMATO on the real-world biological dataset, we took professional advice from an expert who has a Ph.D. in Bioinformatics. We have run UMATO and the baseline algorithms (t-SNE, UMAP) on 23,822 single-cell transcriptomes from two areas at distant poles of the mouse neocortex (Tasic et al. (2018)). Each cell belongs to one of 133 clusters defined by Jaccard–Louvain clustering (for more than 4,000 cells) or a combination of k-means and Ward’s hierarchical clustering. Likewise, each cluster belongs to one of 4 classes: GABAergic (red/purple), Endothelial (brown), Glutamatergic (blue/green), Non-Neuronal (dark green).\nThe embedding result for each method is given in Figure 11. In the case of t-SNE, clusters are well-captured, but the classes are much dispersed, while UMAP adequately separates both classes and clusters. Compared to these baseline algorithms, UMATO is able to capture the relationship between classes much better, retaining some of the local manifolds as well. This means that UMATO focuses more on the manifold at a higher level than the baselines that the hub points worked as the representatives that explain well about the overall dataset. Moreover, there are cases in biological data analysis where the researchers want to know the distance between samples González-Blas et al. (2019); Van den Berge et al. (2020). As the UMAP embedding results are susceptible to the number of epochs, this may cause a negative impact to interpret the results accurately. On the other hand, as UMATO is robust over the number of epochs, we do not have to worry about such biases."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"We present a novel dimensionality reduction method, Uniform Manifold Approximation with Twophase Optimization (UMATO) to obtain less biased and robust embedding over diverse initialization methods. One effective way of understanding high-dimensional data in various domains is to reduce its dimensionality and investigate the projection in a lower-dimensional space. The limitation of previous approaches such as t-Stochastic Neighbor Embedding (t-SNE, Maaten & Hinton (2008)) and Uniform Manifold Approximation and Projection (UMAP, McInnes et al. (2018)) is that they are susceptible to different initialization methods, generating considerably different embedding results (Section 5.5).\\nt-SNE adopts Kullback-Leibler (KL) divergence as its loss function. The fundamental limitation of the KL divergence is that the penalty for the points that are distant in the original space being close in the projected space is too little (Appendix B). This results in only the local manifolds being captured, while clusters that are far apart change their relative locations from run to run. Meanwhile, UMAP leverages the cross-entropy loss function, which is known to charge a penalty for points that are distant in the original space being close in the projection space and for points that are close in the original space being distant in the projection space (Appendix B). UMAP considers all points in the optimization at once with diverse sampling techniques (i.e., negative sampling and edge sampling). Although the approximation technique in UMAP optimization makes the computation much faster, this raises another problem that the clusters in the embedding become dispersed as the number of epochs increases (Appendix K), which can lead to misinterpretation. UMAP tried to alleviate this by using a fixed number (e.g., 200), which is ad hoc, and by applying a learning rate decay. However, the optimal number of epochs and decay schedule for each initialization method needs to be found in practice.\\nTo solve the aforementioned problems, we avoid using approximation during the optimization process, which normally would result in greatly increased computational cost. Instead, we first run optimization only with a small number of points that represent the data (i.e., hub points). Finding the optimal projection for a small number of points using a cross-entropy function is relatively easy and robust, making the additional techniques employed in UMAP unnecessary. Furthermore, it is less sensitive to the initialization method used (Section 5.5). After capturing the overall skeleton of the high-dimensional structure, we gradually append the rest of the points in subsequent phases. Although the same approximation technique as UMAP is used for these points, as we have already embedded the hub points and use them as anchors, the projections become more robust and unbiased. The gradual addition of points can in fact be done in a single phase; we found additional phases\\ndo not result in meaningful improvements in the performance but only in the increased computation time (Section 4.5). Therefore, we used only two phases in UMAP: global optimization to capture the global structures (i.e., the pairwise distances in a high-dimensional space) and local optimization to retain the local structures (i.e., the relationships between neighboring points in a high-dimensional space) of the data.\\nWe compared UMATO with popular dimensionality reduction techniques including PCA, Isomap (Tenenbaum et al. (2000)), t-SNE, UMAP, topological autoencoders (Moor et al. (2020)) and At-SNE (Fu et al. (2019)). We used one synthetic (101-dimensional spheres) and three realworld (MNIST, Fashion MNIST, and Kuzushiji MNIST) datasets and analyzed the projection results with several quality metrics. In conclusion, UMATO demonstrated better performance than the baseline techniques in all datasets in terms of KLσ with different σ values, meaning that it reasonably preserved the density of data over diverse length scales. Finally, we presented the 2D projections of each dataset, including the replication of an experiment using the synthetic Spheres dataset introduced by Moor et al. (2020) where data points locally constitute multiple small balls globally contained in a larger sphere. Here, we demonstrate that UMATO can better preserve both structures compared to the baseline algorithms (Figure 3).\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Dimensionality reduction. Most previous dimensionality reduction algorithms focused on preserving the data’s local structures. For example, Maaten & Hinton (2008) proposed t-SNE, focusing on the crowding problem with which the previous attempts (Hinton & Roweis (2002); Cook et al. (2007)) have struggled, to visualize high-dimensional data through projection produced by performing stochastic gradient descent on the KL divergence between two density functions in the original and projection spaces. Van Der Maaten (2014) accelerated t-SNE developing a variant of the Barnes-Hut algorithm (Barnes & Hut (1986)) and reduced the computational complexity from O(N2) into O(N logN). After that, grounded in Riemannian geometry and algebraic topology, McInnes et al. (2018) introduced UMAP as an alternative to t-SNE. Leveraging the cross-entropy function as its loss function, UMAP reduced the computation time by employing negative sampling from Word2Vec (Mikolov et al. (2013)) and edge sampling from LargeVis (Tang et al. (2015; 2016)) (Table 1). Moreover, they showed that UMAP can generate stable projection results compared to t-SNE over repetition.\\nOn the other hand, there also exist algorithms that aim to capture the global structures of data. Isomap (Tenenbaum et al. (2000)) was proposed to approximate the geodesic distance of highdimensional data and embed it onto the lower dimension. Global t-SNE (Zhou & Sharpee (2018)) converted the joint probability distribution, P , in the high-dimensional space from Gaussian to Student’s-t distribution, and proposed a variant of KL divergence. By adding it with the original loss function of t-SNE, Global t-SNE assigns a relatively large penalty for a pair of distant data points in high-dimensional space being close in the projection space. Another example is topological autoencoders (Moor et al. (2020)), a deep-learning approach that uses a generative model to make the latent space resemble the high-dimensional space by appending a topological loss to the original reconstruction loss of autoencoders. However, they required a huge amount of time for hyperparameter exploration and training for a dataset, and only focused on the global aspect of data. Unlike other techniques that presented a variation of loss functions in a single pipeline, UMATO is novel as it preserves both structures by dividing the optimization into two phases; this makes it outperform the baselines with respect to quality metrics in our experiments.\\nHubs, landmarks, and anchors. Many dimensionality reduction techniques have tried to draw sample points to better model the original space; these points are usually called hubs, landmarks, or anchors. Silva & Tenenbaum (2003) proposed Landmark Isomap, a landmark version of classical multidimensional scaling (MDS) to alleviate its computation cost. Based on the Landmark Isomap, Yan et al. (2018) tried to retain the topological structures (i.e., homology) of high-dimensional data by approximating the geodesic distances of all data points. However, both techniques have the limitation that landmarks were chosen randomly without considering their importance. UMATO uses a k-nearest neighbor graph to extract significant hubs that can represent the overall skeleton of high-dimensional data. The most similar work to ours is At-SNE (Fu et al. (2019)), which optimized the anchor points and all other points with two different loss functions. However, since the anchors wander during the optimization and the KL divergence does not care about distant points, it hardly\\ncaptures the global structure. UMATO separates the optimization process into two phases so that the hubs barely moves but guides other points so that the subareas manifest the shape of the highdimensional manifold in the projection. Applying different cross-entropy functions to each phase also helps preserve both structures.\"\n },\n {\n \"heading\": \"3 UMAP\",\n \"text\": \"Since UMATO shares the overall pipeline of UMAP (McInnes et al. (2018)), we briefly introduce UMAP in this section. Although UMAP is grounded in a sophisticated mathematical foundation, its computation can be simply divided into two steps, graph construction and layout optimization, a configuration similar to t-SNE. In this section, we succinctly explain the computation in an abstract manner. For more details about UMAP, please consult the original paper (McInnes et al. (2018)).\\nGraph Construction. UMAP starts by generating a weighted k-nearest neighbor graph that represents the distances between data points in the high-dimensional space. Given an input dataset X = {x1, . . . , xn}, the number of neighbors to consider k and a distance metric d : X × X → [0,∞), UMAP first computes Ni, the k-nearest neighbors of xi with respect to d. Then, UMAP computes two parameters, ρi and σi, for each data point xi to identify its local metric space. ρi is a nonzero distance from xi to its nearest neighbor:\\nρi = min j∈Ni {d(xi, xj) | d(xi, xj) > 0}. (1)\\nUsing binary search, UMAP finds σi that satisfies:∑ j∈Ni exp(−max(0, d(xi, xj)− ρi)/σi) = log2(k). (2)\\nNext, UMAP computes:\\nvj|i = exp(−max(0, d(xi, xj)− ρi)/σi), (3)\\nthe weight of the edge from a point xi to another point xj . To make it symmetric, UMAP computes vij = vj|i + vi|j − vj|i · vi|j , a single edge with combined weight using vj|i and vi|j . Note that vij indicates the similarity between points xi and xj in the original space. Let yi be the projection of xi in a low-dimensional projection space. The similarity between two projected points yi and yj is wij = (1 + a||yi − yj ||2b2 )−1, where a and b are positive constants defined by the user. Setting both a and b to 1 is identical to using Student’s t-distribution to measure the similarity between two points in the projection space as in t-SNE (Maaten & Hinton (2008)).\\nLayout Optimization. The goal of layout optimization is to find the yi that minimizes the difference (or loss) between vij and wij . Unlike t-SNE, UMAP employs the cross entropy:\\nCUMAP = ∑ i 6=j [vij · log(vij/wij)− (1− vij) · log((1− vij)/(1− wij))], (4)\\nbetween vij andwij as the loss function. UMAP initializes yi through spectral embedding (Belkin & Niyogi (2002)) and iteratively optimize its position to minimize CUMAP . Given the output weight wij as 1/(1 + ad2bij ), the attractive gradient is:\\nCUMAP yi\\n+ = −2abd2(b−1)ij 1 + ad2bij vij(yi − yj), (5)\\nand the repulsive gradient is:\\nCUMAP yi\\n− =\\n2b\\n( + d2ij)(1 + ad 2b ij )\\n(1− vij)(yi − yj), (6)\\nwhere is a small value added to prevent division by zero and dij is a Euclidean distance between yi and yj . For efficient optimization, UMAP leverages the negative sampling technique from Word2Vec (Mikolov et al. (2013)). After choosing a target point and its negative samples, the position of the target is updated with the attractive gradient, while the positions of the latter do so with\\nthe repulsive gradient. Moreover, UMAP utilizes edge sampling (Tang et al. (2015; 2016)) to accelerate and simplify the optimization process (Table 1). In other words, UMAP randomly samples edges with a probability proportional to their weights, and subsequently treats the selected ones as binary edges. Considering the previous sampling techniques, the modified objective function is:\\nO = ∑\\n(i,j)∈E\\nvij(log(wij) + M∑ k=1 Ejk∼Pn(j)γ log(1− wijk)). (7)\\nHere, vij and wij are the similarities in the high and low-dimensional spaces respectively, M is the number of negative samples and Ejk∼Pn(j) indicates that jk is sampled according to a noisy distribution, Pn(j), from Word2Vec (Mikolov et al. (2013)).\"\n },\n {\n \"heading\": \"4 UMATO\",\n \"text\": \"Figure 2 illustrates the computation pipeline of UMATO, which delineates the two-phase optimization (see Figure 9 for a detailed illustration of the overall pipeline). As a novel approach, we split the optimization into global and local so that it could generate a low-dimensional projection keeping both structures well-maintained. We present the pseudocode of UMATO in Appendix A, and made the source codes of it publicly available1.\\n4.1 POINTS CLASSIFICATION\\nIn the big picture, UMATO follows the pipeline of UMAP. We first find the k-nearest neighbors in the same way as UMAP, by assuming the local connectivity constraint, i.e., no single point is isolated and each point is connected to at least a user-defined number of points. After calculating ρ (Equation 1) and σ (Equation 2) for each point, we obtain the pairwise similarity for every pair of points. Once the k-nearest neighbor indices are established, we unfold it and check the frequency of each point to sort them into descending order so that the index of the popular points come to the front.\\nThen, we build a k-nearest neighbor graph by repeating the following steps until no points remain unconnected: 1) choose the most frequent point as a hub among points that are not already connected, 2) retrieve the k-nearest neighbors of the chosen point (i.e., hub), and the points selected from steps 1 and 2 will become a connected component. The gist is that we divide the points into three disjoint sets: hubs, expanded nearest neighbors, and outliers (Figure 1). Thanks to the sorted indices, the most popular point in each iteration—but not too densely located—becomes the hub point. Once the hub points are determined, we recursively seek out their nearest neighbors and again look for the nearest neighbors of those neighbors, until there are no points to be newly appended. In\\nother words, we find all connected points that are expanded from the original hub points, which, in turn, is called the expanded nearest neighbors. Any remaining point that is neither a hub point nor a part of any expanded nearest neighbors is classified as an outlier. The main reason to rule out the outliers is, similar to the previous approach (Gong et al. (2012)), to achieve the robustness of the practical manifold learning algorithm. As the characteristics of these classes differ significantly, we take a different approach for each class of points to obtain both structures. That is, we run global optimization for the hub points (Section 4.2), local optimization for the expanded nearest neighbors (Section 4.3), and no optimization for the outliers (Section 4.4). In the next section we explain each in detail.\"\n },\n {\n \"heading\": \"4.2 GLOBAL OPTIMIZATION\",\n \"text\": \"After identifying hub points, we run the global optimization to retrieve the skeletal layout of the data. First, we initialize the positions of hub points using PCA, which makes the optimization process\\n1https://www.github.com/anonymous-author/anonymous-repo\\nfaster and more stable than using random initial positions. Next, we optimize the positions of hub points by minimizing the cross-entropy function (Equation 4). Let f(X) = {f(xi, xj)|xi, xj ∈ X} and g(Y ) = {g(yi, yj)|yi, yj ∈ Y } be two adjacency matrices in high- and low-dimensional spaces. If Xh represents a set of points selected as hubs in high-dimensional space, and Yh is a set of corresponding points in the projection, we minimize the cross entropy—CE(f(Xh)||g(Yh))— between f(Xh) and g(Yh).\\nTable 1: The runtime for each algorithm using MNIST dataset. UMAP and UMATO take much less time than MulticoreTSNE (Ulyanov (2016)) when tested on a Linux server with 40-core Intel Xeon Silver 4210 CPUs. The runtimes are averaged over 10 runs. Isomap (Tenenbaum et al. (2000)) took more than 3 hours to get the embedding result.\\nAlgorithm Runtime (s)\\nIsomap 3 hours > t-SNE 374.85 ± 11.38 UMAP 26.10 ± 3.97\\nUMATO 73.32 ± 8.39\\nUMAP computes the cross-entropy between all existing points using two sampling techniques, edge sampling and negative sampling, for speed (Table 1). However, this often ends up capturing only the local properties of data because of the sampling biases and thus it cannot be used for cases that require a comprehensive understanding of the data. On the other hand, in its first phase, UMATO only optimizes for representatives (i.e., the hub points) of data, which takes much less time but can still approximate the manifold effectively.\"\n },\n {\n \"heading\": \"4.3 LOCAL OPTIMIZATION\",\n \"text\": \"In the second phase, UMATO embeds the expanded nearest neighbors to the projection that is computed using only the hub points from the first phase. For each point in the expanded nearest neighbors, we retrieve its nearest m (e.g., 10) hubs in the original high-dimensional space and set its initial position in the projection to the average positions of the hubs in the projection with small random perturbations. We follow a similar optimization process as UMAP in the local optimization with small differences. As explained in Section 3, UMAP first constructs the graph structure; we perform the same task but only with the hubs and expanded nearest neighbors. While doing this, since some points are excluded as outliers, we need to update the k-nearest neighbor indices. This is fast because we recycle the already-built k-nearest neighbor indices by updating the outliers to the new nearest neighbor.\\nOnce we compute the similarity between points (Equation 3), to optimize the positions of points, similar to UMAP, we use the cross-entropy loss function with edge sampling and negative sampling (Equation 7). Here, we try to avoid moving the hubs as much as possible since they have already formed the global structure. Thus, we only sample a point p among the expanded nearest neighbors as one end of an edge, while the point q at the other end of the edge can be chosen from all points except outliers. In UMAP implementation, when q pulls in p, p also drags q to facilitate the optimization (Equation 5). When updating the position of q, we only give a penalty to this (e.g., 0.1), if q is a hub point, not letting its position excessively be affected by p. In addition, because the repulsive force can disperse the local attachment, making the point veer off for each epoch and eventually destroying the well-shaped global layout, we multiply a penalty (e.g., 0.1) when calculating the repulsive gradient (Equation 6) for the points selected as negative samples.\"\n },\n {\n \"heading\": \"4.4 OUTLIERS ARRANGEMENT\",\n \"text\": \"Since the isolated points, which we call outliers, mostly have the same distance to all the other data points in high-dimensional space, due to the curse of dimensionality, they both sabotage the global structure we have already made and try to mingle with all other points, thus distorting the overall projection. We do not optimize these points but instead simply append them using the alreadyprojected points (e.g., hubs or expanded nearest neighbors), that belong to each outlier’s connected component of the nearest neighbor graph. That is, if xi ∈ Cn where xi is the target outlier and Cn is the connected component to which xi belongs, we find xj ∈ Cn that has already been projected and is closest to xi. We arrange yi which corresponds to xi in low-dimensional space using the position of yj in the same component offset by a random noise. In this way, we can benefit from the comprehensive composition of the projection that we have already optimized when arranging the outliers. We can ensure that all outliers can find a point as its neighbor since we picked hubs from each connected component of the nearest neighbor graph and thus at least one point is already located and has an optimized position (Section 4.2).\"\n },\n {\n \"heading\": \"4.5 MULTI-PHASE OPTIMIZATION\",\n \"text\": \"The optimization of UMATO can be easily expanded to multiple phases (e.g., three or more phases). Since we have a recursive procedure to expand the nearest neighbors, we can insert the optimization process each time we expand the neighbors to create a multi-phase algorithm. However, our experiment with three- and four-phase optimization with the Fashion MNIST dataset showed that there is no big difference between two-phase optimization and that with more than two phases. Appendix C contains the quantitative and qualitative results of the experiment for multi-phase optimization.\"\n },\n {\n \"heading\": \"5 EXPERIMENTS\",\n \"text\": \"We conducted experiments to evaluate UMATO’s ability to capture the global and local structures of high-dimensional data. We compared UMATO with six baseline algorithms, PCA, Isomap, t-SNE, UMAP, topological autoencoders, and At-SNE in terms of global (i.e., DTM and KLσ) and local (i.e., trustworthiness, continuity, and MRREs) quality metrics.\"\n },\n {\n \"heading\": \"5.1 DATASETS\",\n \"text\": \"We used four datasets for the experiments: Spheres, MNIST (LeCun & Cortes (2010)), Fashion MNIST (Xiao et al. (2017)), and Kuzushiji MNIST (Clanuwat et al. (2018)). Spheres is a synthetic dataset that has 10,000 rows of 101 dimensions. It has a high-dimensional structure in which ten small spheres are contained in a larger sphere. Specifically, the dataset’s first 5,000 rows are the points sampled from a sphere of radius 25 and 500 points are sampled for each of the ten smaller spheres of radius 5 shifted to a random direction from the origin. This dataset is the one used for the original experiment with topological autoencoders (Moor et al. (2020)). Other datasets are images of digits, fashion items, and Japanese characters, each of which consists of 60,000 784-dimensional (28 × 28) images from 10 classes.\"\n },\n {\n \"heading\": \"5.2 EXPERIMENTAL SETTING\",\n \"text\": \"Evaluation Metrics. To assess how well projections preserve the global structures of highdimensional data, we computed the density estimates (Chazal et al. (2011; 2017)), the so-called Distance To a Measure (DTM), between the original data and the projections. Moor et al. (2020) adopted the Kullback-Leibler divergence between density estimates with different scales (KLσ) to evaluate the global structure preservation. To follow the original experimental setup by Moor et al. (2020), we found the projections with the lowest KL0.1 from all algorithms by adjusting their hyperparameters. Next, to evaluate the local structure preservation of projections, we used the mean relative rank errors (MRREs, Lee & Verleysen (2007)), trustworthiness, and continuity (Venna & Kaski (2001)). All of these local quality metrics estimate how well the nearest neighbors in one space (e.g., high- or low-dimensional space) are preserved in the other space. For more information on the quality metrics, we refer readers to Appendix E.\\nBaselines. We set the most widely used dimensionality reduction techniques as our baselines, including PCA, Isomap (Tenenbaum et al. (2000)), t-SNE (Maaten & Hinton (2008)),\\nUMAP (McInnes et al. (2018)), and At-SNE (Fu et al. (2019)). In the case of t-SNE, we leveraged Multicore t-SNE (Ulyanov (2016)) for fast computation. To initialize the points’ position, we used PCA for t-SNE, following the recommendation in the previous work (Linderman et al. (2019)), and spectral embedding for UMAP which was set to default. In addition, we compared with topological autoencoders (Moor et al. (2020)) that were developed to capture the global properties of the data using a deep learning-based generative model. Following the convention of visualization in dimensionality reduction, we determined our result projected onto 2D space. We tuned the hyperparameters of each technique to minimize the KL0.1. Appendix F further describes the details of the hyperparameters settings.\"\n },\n {\n \"heading\": \"5.3 QUANTITATIVE RESULTS\",\n \"text\": \"Table 2 displays the experiment results. In most cases, UMATO was the only method that has shown performance both in the global and local quality metrics in most datasets. For local metrics, t-SNE, At-SNE, and UMAP generally had the upper-hand, but UMATO showed comparable MRREX and continuity in Spheres, Fashion MNIST, and Kuzushiji MNIST datasets. Meanwhile, Isomap and topological autoencoders were mostly good at global quality metrics, although UMATO had the lowest (best) KL0.1 and DTM except for the MNIST dataset.\"\n },\n {\n \"heading\": \"5.4 QUALITATIVE RESULTS\",\n \"text\": \"Among the five algorithms, only UMATO could preserve both the global and local structure of the Spheres dataset. If we look at the figure made by UMATO, the outer sphere encircles the inner spheres in a circular form, which is the most intuitive to understand the relationship among different classes and the local linkage in detail. In the results from Isomap, t-SNE, UMAP, and At-SNE,\\nthe points representing the surrounding giant sphere mix with those representing the other small inner spheres, thus failing to capture the nested relationships among different classes. Meanwhile, topological autoencoders are able to realize the global relationship between classes in an incomplete manner; the points for the outer sphere are too spread out, thus losing the local characteristics of the class. From this result, we can acknowledge how UMATO can work with high-dimensional data effectively to reveal both global and local structures. 2D visualization results on other datasets (MNIST, Fashion MNIST, Kuzushiji MNIST) can be found in Appendix H. Lastly, we report an additional experiment on the mouse neocortex dataset (Tasic et al. (2018)) in Appendix N which shows the relationship between classes much better than the baseline algorithms like t-SNE and UMAP.\"\n },\n {\n \"heading\": \"5.5 PROJECTION ROBUSTNESS OVER DIVERSE INITIALIZATION METHODS\",\n \"text\": \"We experimented with the robustness of each dimensionality reduction technique with different initialization methods such as PCA, spectral embedding, random position, and class-wise separation. In class-wise separation, we initialized each class with a non-overlapping random position in 2- dimensional space, adding random Gaussian noise. In our results, UMATO embeddings were almost the same on the real-world datasets, while the UMAP and t-SNE results relied highly upon the initialization method. We report this in Table 3 with a quantitative comparison using Procrustes distance. Specifically, given two datasets X = {x1, x2, . . . , xn} and Y = {y′1, y′2, . . . , y′n} where y′i corresponds to xi, the Procrustes distance is defined as\\ndP (X,Y ) = √√√√ N∑ i=1 (xi − y′i). (8)\\nFor all cases, we ran optimal translation, uniform scaling, and rotation to minimize the Procrustes distance between the two distributions. In the case of the Spheres dataset, as defined in Appendix G, the clusters were equidistant from each other. The embedding results have to be different due to the limitation of the 2-dimensional space since there is no way to express this relationship. However, as we report in Figure 4, the global and local structures of the Spheres data are manifested with UMATO with all different initialization methods.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"We present a two-phase dimensionality reduction algorithm called UMATO that can effectively preserve the global and local properties of high-dimensional data. In our experiments with diverse datasets, we have proven that UMATO can outperform previous widely used baselines (e.g., t-SNE and UMAP) both quantitatively and qualitatively. As future work, we plan to accelerate UMATO, as in previous attempts with other dimensionality reduction techniques (Pezzotti et al. (2019); Nolet et al. (2020)), by implementing it on a heterogeneous system (e.g., GPU) for speedups.\"\n },\n {\n \"heading\": \"A UMATO ALGORITHM PSEUDOCODE\",\n \"text\": \"The pseudocode of UMATO is as below:\\nAlgorithm 1 Uniform Manifold Approximation with Two-phase Optimization 1: procedure UMATO(X , kX , d, min dist, nh, eg , el) Input: High-dimensional data X , number of nearest neighbors k, projection dimension d, minimum distance in projection result min dist, number of hub points nh, epochs for global and local optimization eg , el Output: Low-dimensional projection Y 2: Compute k-nearest neighbors of X 3: Obtain sorted list using indices’ frequency of k-nearest neighbors 4: Build k-nearest neighbor graph structure 5: Classify points into hubs, expanded nearest neighbors, and outliers 6: Optimize CE(f(Xh)||g(Yh)) to preserve global configuration (Equation 4) 7: Initialize expanded nearest neighbors using hub locations 8: Update k-nearest neighbors & compute weights (Equation 3) 9: Optimize CE(f(X)||g(Y )) to preserve local configuration (Equation 7) 10: Position outliers 11: return Y 12: end procedure\"\n },\n {\n \"heading\": \"B THE MEANING OF USING DIFFERENT LOSS FUNCTIONS IN DIMENSIONALITY REDUCTION\",\n \"text\": \"Following the notations from above, we set the similarity between points in high-dimensional space xi and xj as vij and the low-dimensional space yi and yj as wij . Then we can write the KL divergence and cross entropy loss function as:\\nKL = ∑ i6=j vij · log(vij/wij), (9)\\nCE = ∑ i 6=j vij · log(vij/wij)− ∑ i 6=j (1− vij) · log((1− vij)/(1− wij)). (10)\\nIf we use the same probability distributions, vij and wij , the KL divergence and the first term of cross-entropy are exactly the same. If vij and wij are both large (Table 4 a.) or both small (Table 4 d.), this means that the relationship between points in high-dimensional space is well-retained in the projection. Thus, the positions of points in the low-dimensional space do not have to move. As vij and wij are similar, log(vij/wij) becomes zero, producing a small cost in the end.\\nHowever, we need to modify the position of wij if there exists a gap between vij and wij . The KL divergence imposes a big penalty when vij is large but wij is small (Table 4 b.). That is, if the neighboring points in high-dimensional space are not captured well in the projection, the KL divergence imposes a high penalty to move the point (vij) into the right position. Thus, we can understand why t-SNE is able to capture the local characteristics of high-dimensional space, but not the global ones. However, the second term of cross-entropy imposes a big penalty when vij is small but wij is large (Table 4 g.). Therefore, it moves points that are close together in the highdimensional space but far apart in the projection.\"\n },\n {\n \"heading\": \"C MULTI-PHASE OPTIMIZATION\",\n \"text\": \"We report the result of multi-phase optimization (e.g., three and four-phase) using the Fashion MNIST dataset both quantitatively (Table 5) and qualitatively (Figure 5). As in Figure 5, we were unable to find any significant differences between the 2D projections, although some outliers were located in different places. Moreover, the quality metrics were almost the same for all three results. The original UMATO was the winner in DTM, KL0.1, KL1, continuity, and MRREZ but came last in other quality metrics. In addition, the gap in metrics between UMATO and the multi-phase optimizations indicated a trivial difference. Thus, we concluded that developing a multi-phase optimization for UMATO does not bring about any notable improvement in the projection result.\"\n },\n {\n \"heading\": \"D PROJECTION STABILITY\",\n \"text\": \"Table 6 denotes the results of our experiment on the projection stability of UMATO and other dimensionality reduction techniques. When the data size grows, we want to sample a portion of it to speed up the visualization. However, the concern is whether the projection run with the sampled indices is consistent with the part of the projection result with indices selected from the full dataset. If the algorithm can generate stable and consistent results, the two projections should contain the least bias possible. To compute the projection stability of dimensionality reduction techniques, we\\nused the normalized Procrustes distance (Equation 8) to measure the distance between two comparable distributions. To replicate the experiment by McInnes et al. (2018), we used the same Flow Cytometry dataset (Spidlen et al. (2012); Brodie et al. (2013)), and ran optimal translation, uniform scaling, and rotation to minimize the Procrustes distance between the two distributions. As we can see in Table 6, UMATO outperformed t-SNE and At-SNE for all sub-sample sizes. Moreover, although UMAP is known as stable among existing algorithms, UMATO showed even better (lower) Procrustes distance except for one sub-sample size (60%). From this result, we can acknowledge that UMATO can generate the more stable and consistent results regardless of sub-sample size than many other dimensionality reduction techniques.\"\n },\n {\n \"heading\": \"E QUALITY METRICS\",\n \"text\": \"As UMATO presents a dimensionality reduction technique that can capture both the global and local structures of high-dimensional data, we used several quality metrics to evaluate each aspect respectively. We have referred to some review papers (Gracia et al. (2014); Lee & Verleysen (2009)) for the best use and implementation. Among many quality metrics, we leveraged 1) Distance To a Measure (DTM, Chazal et al. (2011; 2017)), 2) KL divergence between two density functions, 3) trustworthiness and continuity (Venna & Kaski (2001)), and 4) mean relative rank errors (MRREs, Lee & Verleysen (2007)). The first two metrics are used to test the preservation of global structures and the last two metrics are suggested for the preservation of the local structures.\\nDistance To a Measure considers the dispersion of high- and low-dimensional data, where it is defined for a given point as fXσ (x) := ∑ y∈X exp (−dist(x, y)2/σ). By summing up the element-\\nwise absolute values between two distributions, ∑ x∈X,z∈Z f X σ (x)− fZσ (z) where x is the point in high-dimensional space X and the z is the corresponding projected point in low-dimensional space Z, we can examines the similarity of two datasets. In our experiments, we used the Euclidean distance and the values were normalized between 0 and 1. The σ ∈ R>0, which represents the length scale parameter, was set to 0.1. Moor et al. (2020) proposed the KL divergence of two probability distributions, KLσ := KL(fXσ ||fZσ ), a variation of DTM. Changing σ as a normalizing factor of the distribution, the authors investigated if the algorithms can preserve the global structure of the high-dimensional data. Following the same notion as the experiment in the paper (Moor et al. (2020)), we used three σ values, 1.0, 0.1, and 0.01, to test whether each algorithm can capture the global aspect with respect to diverse density estimates.\\nTrustworthiness and continuity measure how much the nearest neighbors are preserved in a space (i.e., high- or low-dimensional space) compared to the other space by analyzing the ranks of knearest neighbors in both spaces. The difference between trustworthiness and continuity comes from which space is held as the base space. Specifically, we first need to find the k-nearest neighbors in both high- and low-dimensional space. Then, we compute the trustworthiness by checking whether the ranks of nearest neighbors in low-dimensional space resemble those of high-dimensional space. If so, we can achieve a high score in trustworthiness. Meanwhile, we achieve a high score in continuity if the ranks of nearest neighbors in high-dimensional space resemble those of low-dimensional space. MRREs take a similar approach to trustworthiness and continuity as it calculates and compares the ranks of the k-nearest neighbors in both spaces, but the normalizing factor is slightly different. Originally, it was better if MRREs had lower values. However, for the ease of comparing\\nlocal quality metrics, we defined it as MRREs := 1 −MRREs, so higher MRREs mean that they have better retained the k-nearest neighbors like trustworthiness and continuity.\"\n },\n {\n \"heading\": \"F HYPERPARAMETER SETTING\",\n \"text\": \"As explained in Section 5.2, we generated projections for each dimensionality reduction algorithm that had the lowest KL0.1 measure. To tune each algorithm’s hyperparameters, we employed the grid search for t-SNE, UMAP, and At-SNE. For t-SNE and At-SNE, we changed the perplexity from 5 to 50 with an interval of 5, and the learning rate from 0.1 to 1.0 with a log-uniform scale. In the case of UMAP, we changed the number of nearest neighbors from 5 to 50 with an interval of 5, and the minimum distance between points in the projection from 0.1 to 1.0 with an interval of 0.1. We used the Python library scikit-optimize (Head et al. (2018)) to find the best hyperparameters for topological autoencoders. UMATO has several hyperparameters such as the number of hub points, the number of epochs, and the learning rate for global and local optimization. In our experiments, we configured everything except the number of hub points to the same setting for UMATO. We used 200 hub points for the Spheres dataset and had 300 hubs for others. We used fewer hub points for the Spheres since it has only 10,000 data points in total, while the other datasets have 60,000 data points. We set the number of epochs to 100 for global optimization and to 50 for local optimization. Lastly, the global learning rate was set to 0.0065, and the local learning rate was set to 0.01.\"\n },\n {\n \"heading\": \"G SYNTHETIC SPHERES DATASET\",\n \"text\": \"We leveraged the same Spheres dataset that Moor et al. (2020) used in their experiments of topological antoencoders. The Spheres dataset contains eleven high-dimensional spheres which reside in 101-dimensional space. We first generated ten spheres of radius of 5, and shifted each sphere by adding the same Gaussian noise to a random direction. For this aim, we created d-dimensional Gaussian vectors X ∼ N(0, I(10/ √ d)), where d is 101. As to embed an interesting geometrical structure to the dataset, the ten spheres of relatively small radii of 5 were enclosed by another larger sphere of radius of 25.\"\n },\n {\n \"heading\": \"H MORE EXPERIMENTS ON SYNTHETIC DATASETS\",\n \"text\": \"We leveraged the 3-dimensional S-curve and Swiss roll datasets to test whether UMATO can preserve both the global and local structures of original datasets. As the visualization shows (Figure 6),\\nonly PCA and UMATO were able to capture the global and local structures of original datasets. Isomap, t-SNE and UMAP could capture the local manifolds of original datasets, but high-level manifolds of the original datasets were not reflected to the embedding.\"\n },\n {\n \"heading\": \"I LOCAL QUALITY METRICS WITH DIFFERENT VALUES OF HYPERPARAMETER\",\n \"text\": \"We report the result of local quality metrics with diverse hyperparameters (Table 7). We changed the number of nearest neighbors (k) from 5 to 15 with an interval of 5. As we have already reported when the k = 5 (Table 2), below are the cases where k = 10, 15. As we can check from the result, while the values fluctuate a little bit, the ranks are mostly robust over diverse k values.\\nJ 2D EMBEDDING RESULTS OF UMATO AND BASELINE ALGORITHMS ON REAL-WORLD DATASETS\\nFor the real-world datasets, UMATO showed a similar projection to PCA but with better captured local characteristics. The results from topological autoencoders showed some points detached far apart from their centers, even though the best hyperparameters were used for each. Although AtSNE claimed that it could capture both structures, the results were not significantly different from those of the original t-SNE algorithm when projecting the Spheres and Fashion MNIST datasets.\"\n },\n {\n \"heading\": \"K EMBEDDING ROBUSTNESS OVER NUMBER OF EPOCHS\",\n \"text\": \"We report the experimental result in Figure 8. As we explained, the UMAP embedding results are susceptible to the number of epochs so that the distance between clusters get dispersed. This can induce a misinterpretation that the user considers the distance between clusters as something meaningful. The two-phase optimization of UMATO can solve the problem since the global optimization (first phase) is easy to converge as it runs only with a small portion of points. Therefore, the increasing number of epochs in the global optimization does not harm the final embedding.\\nL ILLUSTRATION OF UMATO PIPELINE\\nFor the ease of understanding, we provide an illustration of UMATO pipeline in Figure 9. The detailed explanation for UMATO can be found in Section 4.\"\n },\n {\n \"heading\": \"M EFFECT OF LOCAL LEARNING RATE OF UMATO\",\n \"text\": \"By manipulating one of UMATO’s hyperparameters, local learning rate, the user can determine where to focus in the embedding result; to reveal more of the global structures, the user should apply a lower value (e.g., 0.005), while using a higher one (e.g., 0.1) would generate more like a UMAP embedding which prefers to show the local manifolds.\"\n },\n {\n \"heading\": \"N UMATO ON REAL-WORLD BIOLOGICAL DATASET\",\n \"text\": \"To test UMATO on the real-world biological dataset, we took professional advice from an expert who has a Ph.D. in Bioinformatics. We have run UMATO and the baseline algorithms (t-SNE, UMAP) on 23,822 single-cell transcriptomes from two areas at distant poles of the mouse neocortex (Tasic et al. (2018)). Each cell belongs to one of 133 clusters defined by Jaccard–Louvain clustering (for more than 4,000 cells) or a combination of k-means and Ward’s hierarchical clustering. Likewise, each cluster belongs to one of 4 classes: GABAergic (red/purple), Endothelial (brown), Glutamatergic (blue/green), Non-Neuronal (dark green).\\nThe embedding result for each method is given in Figure 11. In the case of t-SNE, clusters are well-captured, but the classes are much dispersed, while UMAP adequately separates both classes and clusters. Compared to these baseline algorithms, UMATO is able to capture the relationship between classes much better, retaining some of the local manifolds as well. This means that UMATO focuses more on the manifold at a higher level than the baselines that the hub points worked as the representatives that explain well about the overall dataset. Moreover, there are cases in biological data analysis where the researchers want to know the distance between samples González-Blas et al. (2019); Van den Berge et al. (2020). As the UMAP embedding results are susceptible to the number of epochs, this may cause a negative impact to interpret the results accurately. On the other hand, as UMATO is robust over the number of epochs, we do not have to worry about such biases.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":195,"cells":{"paper_id":{"kind":"string","value":"SP:fd70696898c5c725ad789565265274a37a6c2ca0"},"summaries":{"kind":"list like","value":["This paper presents a reduction approach to tackle the optimization problem of constrained RL. They propose a Frank-Wolfe type algorithm for the task, which avoids many shortcomings of previous methods, such as the memory complexity. They prove that their algorithm can find an $\\epsilon$-approximate solution with $O(1/\\epsilon)$ invocation. They also show the power of their algorithm with experiments in a grid-world navigation task, though the tasks looks relatively simple."],"string":"[\n \"This paper presents a reduction approach to tackle the optimization problem of constrained RL. They propose a Frank-Wolfe type algorithm for the task, which avoids many shortcomings of previous methods, such as the memory complexity. They prove that their algorithm can find an $\\\\epsilon$-approximate solution with $O(1/\\\\epsilon)$ invocation. They also show the power of their algorithm with experiments in a grid-world navigation task, though the tasks looks relatively simple.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Many applications of reinforcement learning (RL) optimize a long-term reward subject to risk, safety, budget, diversity or other constraints. Though constrained RL problem has been studied to incorporate various constraints, existing methods either tie to specific families of RL algorithms or require storing infinitely many individual policies found by an RL oracle to approach a feasible solution. In this paper, we present a novel reduction approach for constrained RL problem that ensures convergence when using any off-the-shelf RL algorithm to construct an RL oracle yet requires storing at most constantly many policies. The key idea is to reduce the constrained RL problem to a distance minimization problem, and a novel variant of Frank-Wolfe algorithm is proposed for this task. Throughout the learning process, our method maintains at most constantly many individual policies, where the constant is shown to be worst-case optimal to ensure convergence of any RL oracle. Our method comes with rigorous convergence and complexity analysis, and does not introduce any extra hyper-parameter. Experiments on a grid-world navigation task demonstrate the efficiency of our method."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Jacob Abernethy","Peter L Bartlett","Elad Hazan"],"title":"Blackwell approachability and no-regret learning are equivalent","venue":"In Proceedings of the 24th Annual Conference on Learning Theory, pp","year":2011},{"authors":["Jacob D Abernethy","Jun-Kun Wang"],"title":"On frank-wolfe and equilibrium computation","venue":"In Advances in Neural Information Processing Systems,","year":2017},{"authors":["Alekh Agarwal","Alina Beygelzimer","Miroslav Dudı́k","John Langford","Hanna Wallach"],"title":"A reductions approach to fair classification","venue":"arXiv preprint arXiv:1803.02453,","year":2018},{"authors":["Eitan Altman"],"title":"Constrained Markov decision processes, volume 7","venue":"CRC Press,","year":1999},{"authors":["Gabriel Barth-Maron","Matthew W 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Beygelzimer\",\n \"Miroslav Dudı́k\",\n \"John Langford\",\n \"Hanna Wallach\"\n ],\n \"title\": \"A reductions approach to fair classification\",\n \"venue\": \"arXiv preprint arXiv:1803.02453,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Eitan Altman\"\n ],\n \"title\": \"Constrained Markov decision processes, volume 7\",\n \"venue\": \"CRC Press,\",\n \"year\": 1999\n },\n {\n \"authors\": [\n \"Gabriel Barth-Maron\",\n \"Matthew W Hoffman\",\n \"David Budden\",\n \"Will Dabney\",\n \"Dan Horgan\",\n \"Dhruva Tb\",\n \"Alistair Muldal\",\n \"Nicolas Heess\",\n \"Timothy Lillicrap\"\n ],\n \"title\": \"Distributed distributional deterministic policy gradients\",\n \"venue\": \"arXiv preprint arXiv:1804.08617,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Amir Beck\",\n \"Shimrit Shtern\"\n ],\n \"title\": \"Linearly convergent away-step conditional gradient for non-strongly convex functions\",\n \"venue\": \"Mathematical Programming,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Craig Boutilier\",\n \"Tyler Lu\"\n ],\n \"title\": \"Budget allocation using weakly coupled, constrained markov decision processes\",\n \"venue\": null,\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Deeparnab Chakrabarty\",\n \"Prateek Jain\",\n \"Pravesh Kothari\"\n ],\n \"title\": \"Provable submodular minimization using wolfe’s algorithm\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Minmin Chen\",\n \"Alex Beutel\",\n \"Paul Covington\",\n \"Sagar Jain\",\n \"Francois Belletti\",\n \"Ed H Chi\"\n ],\n \"title\": \"Top-k off-policy correction for a reinforce recommender system\",\n \"venue\": \"In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yinlam Chow\",\n \"Mohammad Ghavamzadeh\"\n ],\n \"title\": \"Algorithms for cvar optimization in mdps\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Yinlam Chow\",\n \"Mohammad Ghavamzadeh\",\n \"Lucas Janson\",\n \"Marco Pavone\"\n ],\n \"title\": \"Risk-constrained reinforcement learning with percentile risk criteria\",\n \"venue\": \"The Journal of Machine Learning Research,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yinlam Chow\",\n \"Ofir Nachum\",\n \"Edgar Duenez-Guzman\",\n \"Mohammad Ghavamzadeh\"\n ],\n \"title\": \"A lyapunovbased approach to safe reinforcement learning\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Jesús A De Loera\",\n \"Jamie Haddock\",\n \"Luis Rademacher\"\n ],\n \"title\": \"The minimum euclidean-norm point in a convex polytope: Wolfe’s combinatorial algorithm is exponential\",\n \"venue\": \"In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Miroslav Dudı́k\",\n \"John Langford\",\n \"Lihong Li\"\n ],\n \"title\": \"Doubly robust policy evaluation and learning\",\n \"venue\": \"arXiv preprint arXiv:1103.4601,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Marguerite Frank\",\n \"Philip Wolfe\"\n ],\n \"title\": \"An algorithm for quadratic programming\",\n \"venue\": \"Naval research logistics quarterly,\",\n \"year\": 1956\n },\n {\n \"authors\": [\n \"Yoav Freund\",\n \"Robert E Schapire\"\n ],\n \"title\": \"Adaptive game playing using multiplicative weights\",\n \"venue\": \"Games and Economic Behavior,\",\n \"year\": 1999\n },\n {\n \"authors\": [\n \"Scott Fujimoto\",\n \"Herke Van Hoof\",\n \"David Meger\"\n ],\n \"title\": \"Addressing function approximation error in actor-critic methods\",\n \"venue\": \"arXiv preprint arXiv:1802.09477,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Dan Garber\",\n \"Elad Hazan\"\n ],\n \"title\": \"A linearly convergent conditional gradient algorithm with applications to online and stochastic 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\"year\": 2019\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Contemporary approaches in reinforcement learning (RL) largely focus on optimizing the behavior of an agent against a single reward function. RL algorithms like value function methods (Zou et al., 2019; Zheng et al., 2018) or policy optimization methods (Chen et al., 2019; Zhao et al., 2017) are widely used in real-world tasks. This can be sufficient for simple tasks. However, for complicated applications, designing a reward function that implicitly defines the desired behavior can be challenging. For instance, applications concerning risk (Geibel & Wysotzki, 2005; Chow & Ghavamzadeh, 2014; Chow et al., 2017), safety (Chow et al., 2018) or budget (Boutilier & Lu, 2016; Xiao et al., 2019) are naturally modelled by augmenting the RL problem with orthant constraints. Exploration suggestions, such as to visit all states as evenly as possible, can be modelled by using a vector to measure the behavior of the agent, and to find a policy whose measurement vector lies in a convex set (Miryoosefi et al., 2019).\nTo solve RL problem under constraints, existing methods either ensure convergence only on a specific family of RL algorithms, or treat the underlying RL algorithms as a black box oracle to find individual policy, and look for mixed policy that randomizes among these individual policies. Though the second group of methods has the advantage of working with arbitrary RL algorithms that best suit the underlying problem, existing methods have practically infeasible memory requirement. To get an -approximate solution, they require storing O(1/ ) individual policies, and an exact solution requires storing infinitely many policies. This limits the prevalence of such methods, especially when the individual policy uses deep neural networks.\nIn this paper, we propose a novel reduction approach for the general convex constrained RL (C2RL) problem. Our approach has the advantage of the second group of methods, yet requires storing at most constantly many policies. For a vector-valued Markov Decision Process (MDP) and any given target convex set, our method finds a mixed policy whose measurement vector lies in the target convex set, using any off-the-shelf RL algorithm that optimizes a scalar reward as a RL oracle. To do so, the C2RL problem is reduced to a distance minimization problem between a polytope and a convex set, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this distance minimization problem. To find an -approximate solution in an m-dimensional vector-valued MDP,\nour method only stores at most m + 1 policies, which improves from infinitely many O(1/ ) (Le et al., 2019; Miryoosefi et al., 2019) to a constant. We also show this m + 1 constant is worstcase optimal to ensure convergence of RL algorithms using deterministic policies. Moreover, our method introduces no extra hyper-parameter, which is favorable for practical usage. A preliminary experimental comparison demonstrates the performance of the proposed method and the sparsity of the policy found."},{"heading":"2 RELATED WORK","text":"For high dimensional constrained RL, one line of approaches incorporates the constraint as a penalty signal into the reward function, and makes updates in a multiple time-scale scheme (Tessler et al., 2018; Chow & Ghavamzadeh, 2014). When used with policy gradient or actor-critic algorithms (Sutton & Barto, 2018), this penalty signal guides the policy to converge to a constraint satisfying one (Paternain et al., 2019; Chow et al., 2017). However, the convergence guarantee requires the RL algorithm can find a single policy that satisfies the constraint, hence ruling out methods that search for deterministic policies, such as Deep Q-Networks (DQN) (Mnih et al., 2013), Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015) and their variants (Van Hasselt et al., 2015; Wang et al., 2016; Fujimoto et al., 2018; Barth-Maron et al., 2018).\nAnother line of approaches uses a game-theoretic framework, and does not tie to specific families of RL algorithm. The constrained problem is relaxed to a zero-sum game, whose equilibrium is solved by online learning (Agarwal et al., 2018). The game is played repeatedly, each time any RL algorithm can be used to find a best response policy to play against a no-regret online learner. The mixed policy that uniformly distributed among all played policies can be shown to converge to an optimal policy of the constrained problem (Freund & Schapire, 1999; Abernethy et al., 2011). Taking this approach, Le et al. (2019) uses Lagrangian relaxation to solve the orthant constraint case, and Miryoosefi et al. (2019) uses conic duality to solve the convex constraint case. However, since the convergence is established by the no-regret property, the policy found by these methods requires randomization among policies found during the learning process, which limits their prevalence.\nDifferent from the game-theoretic approaches, we reduce the C2RL to a distance minimization problem and propose a novel variant of Frank-Wolfe (FW) algorithm to solve it. Our result builds on recent finding that the standard FW algorithm emerges as computing the equilibrium of a special convex-convave zero sum game (Abernethy & Wang, 2017). This connects our approach with previous approaches from game-theoretic framework (Agarwal et al., 2018; Le et al., 2019; Miryoosefi et al., 2019). The main advantage of our reduction approach is that the convergence of FW algorithm does not rely on the no-regret property of an online learner. Hence there is no need to introduce extra hyper-parameters, such as learning rate of the online learner, and intuitively, we can eliminate unnecessary policies to achieve better sparsity. To do so, we extend Wolfe’s method for minimum norm point problem (Wolfe, 1976) to solve our distance minimization problem. Throughout the learning process, we maintain an active policy set, and constantly eliminate policies whose measurement vector are affinely dependent of others. Unlike norm function in Wolfe’s method, our objective function is not strongly convex. Hence we cannot achieve the linear convergence of Wolfe’s method as shown in Lacoste-Julien & Jaggi (2015). Instead, we analyze the complexity of our method based on techniques from Chakrabarty et al. (2014). A theoretical comparison between our method and various approaches in constrained RL is provided in Table 1."},{"heading":"3 PRELIMINARIES","text":"A vector-valued Markov decision process can be identified by a tuple {S,A, β, P, c}, where S is a set of states,A is the set of actions and β is the initial state distribution. At the start of each episode, an initial state s0 is drawn following the distribution β. Then, at each step t = 0, 1, . . . , the agent observes a state st ∈ S and makes a decision to take an action at. After at is chosen, at the next observation the state evolves to state st+1 ∈ S with probability P (st+1|st, at). However, instead of a scalar reward, in our setting, the agent receives an m-dimensional vector ct ∈ Rm that may implicitly contain measurements of reward, risk or violation of other constraints. The episode ends after a certain number of steps, called the horizon, or when a terminate state is reached.\nActions are typically selected according to a policy π, where π(s) is a distribution over actions for any s ∈ S . Policies that take a single action for any state are deterministic policies, and can be identified by the mapping π : S 7→ A. The set of all deterministic policies is denoted by Π. For a discount factor γ ∈ [0, 1), the discounted long-term measurement vector of a policy π ∈ Π is defined as\nc(π) := E( T∑ t=0 γtct(st, π(st))), (1)\nwhere the expectation is over trajectories generated by the described random process.\nUnlike unconstrained setting, for a constrained RL problem, it is possible that all feasible policies are non-deterministic (see Appendix D for an example). This limits the usage of RL algorithms that search for deterministic policies in the setting of constrained RL problem.\nOne workaround is to use mixed policies. For a set of policies U , a mixed policy is a distribution over U , and the set of all mixed policies over U is denoted by ∆(U). To execute a mixed policy µ ∈ ∆(U), we first select a policy π ∈ U according to π ∼ µ(π), and then execute π for the entire episode. Altman (1999) shows that any c(·) achievable can be achieved by some mixed deterministic policies µ ∈ ∆(Π). Therefore, though an off-shelves RL algorithm may not converge to any constraint-satisfying policy, it can be used as a subroutine to find individual policies (possibly deterministic), and a randomization among these policies can converge to a feasible policy. The discounted long-term measurement vector of a mixed policy µ ∈ ∆(Π) is defined similarly\nc(µ) := Eπ∼µ(c(π)) = ∑ π∈Π µ(π)c(π). (2)\nFor a mixed policy µ ∈ ∆(U), its active set is defined to be the set of policies with non-zero weights A := {π ∈ U|µ(π) > 0}. The memory requirement of storing µ, is then proportional to the size of its active set. Since a mixed policy can be interpreted as a convex combination of policies in its active set, in the following, the term sparsity of a mixed policy refers to the sparsity of this combination.\nOur learning problem, the convex constrained reinforcement learning (C2RL), is to find a policy whose expected long-term measurement vector lies in a given convex set; i.e., for a given convex target set C ⊂ Rm, our target is to\nfind µ∗ such that c(µ∗) ∈ Ω (C2RL). (3) Any policy µ∗ that satisfies c(µ∗) ∈ Ω is called a feasible policy, and a C2RL problem is feasible if there exists some feasible policies. In the following, we assume the C2RL problem is feasible."},{"heading":"4 APPROACH, ALGORITHM AND ANALYSIS","text":"We now show how the C2RL (3) can be reduced to a distance minimization problem (7) between a polytope and a convex set. A novel variant of Frank-Wolfe-type algorithm is then proposed to solve the distance minimization problem, followed by theoretic analysis about convergence and sparsity of the proposed method."},{"heading":"4.1 REDUCE C2RL TO A DISTANCE MINIMIZATION PROBLEM","text":"Let ||·|| denote the Euclidean norm. For a convex set Ω ∈ Rm, let ProjΩ(x) ∈ arg miny∈Ω ||x−y|| be the projection operator, and dist2(x,Ω) := 12 ||x−ProjΩ(x)|| 2 be half of the squared Euclidean\ndistance function. Then we consider the problem to find a policy whose measurement vector is closest to the target convex set,\narg min µ∈∆(Π)\ndist2(c(µ),Ω). (4)\nA policy µ∗ ∈ ∆(Π) is defined to be an optimal solution if it minimizes (4). Otherwise, the approximation error of µ ∈ ∆(Π) is defined as\nerr(µ) := dist2(c(µ),Ω)− dist2(c(µ∗),Ω) (Approximation Error) (5) where µ∗ is an optimal solution, and a policy is defined to be an -approximate solution if its approximation error is no larger than .\nWhen C2RL (3) is feasible, the equivalence of being optimal to (4) and being feasible to C2RL can be easily established. Since a feasible policy of C2RL problem lies inside Ω, it minimizes the non-negative dist2 function, and hence is optimal to (4). Vice versa, any optimal solution to (4) lies inside Ω and is a feasible solution to C2RL.\nFrom a geometric perspective, let c(Π) := {c(π)|π ∈ Π} be the set of all values achievable by deterministic policies. If the MDP has finite states and actions (though may be extremely large), then Π is finite as well, and hence c(Π) contains finitely many points in Rm. Then the set of values achievable by mixed deterministic policies\nc(∆(Π)) := {c(µ)|µ ∈ ∆(Π)} = { ∑ π µ(π)c(π) | ∑ π µ(π) = 1, µ(π) ≥ 0} ⊂ Rm (6)\nis the convex hull of c(Π); i.e., c(∆(Π)) is a m-dimension polytope whose vertices are c(Π). Therefore finding a policy whose value is closest to the target convex set (4) is equivalent to find a point in the polytope c(∆(Π)) that is closest to the convex set Ω\narg min c(µ)∈c(∆(Π))\ndist2(c(µ),Ω) (Distance minimization problem). (7)\nTo solve this constrained optimization problem, it might be tempting to consider projection methods. However, constructing a projection operator for c(∆(Π)) is non-trivial. For any given measurement vector, it is obscure how to modify a general RL algorithm to update the parameters such that the discounted expected measurement vector is closest to the given value. Therefore, projection-free methods are preferable for this task.\nFrank-Wolfe (FW) algorithm does not require any projection operation, instead it uses a linear minimizer oracle. Intuitively, finding a linear minimizer is similar to the reward maximization process of what a general RL algorithm does. In section 4.3, we formalize this idea. We show that after simple modifications, any RL algorithm that maximizes a scalar reward can be used to construct such a linear minimizer oracle. Before getting into details of the construction process, we discuss FW-type algorithms over polytope and its applications in the distance minimization problem (7)."},{"heading":"4.2 DISTANCE MINIMIZATION BY FRANK-WOLFE-TYPE ALGORITHMS","text":"The Frank-Wolfe algorithm (FW) is a first-order method to minimize a convex function f : P 7→ R over a compact and convex set P , with only access to a linear minimizer oracle. When the feasible set is a polytope P := conv({s1, s2, . . . , sn}) ⊂ Rm defined as the convex hull of finitely many points, FW-type algorithms are discussed by Lacoste-Julien & Jaggi (2015) to optimize\nmin x∈P f(x) using Oracle(v) := arg min s∈{s1,...,sn}\nsTv. (8)\nThe standard FW (Algorithm 2 in Appendix A.1) consists of making repeated calls to the linear minimizer oracle to find an improving point s, followed by a convex averaging step of the current iterate xt−1 and the oracle’s output s.\nIf we have already constructed a RL oracle(λ) that outputs a policy π ∈ arg minπ∈Π λT c(π) together with its measurement vector c(π), then the distance minimizing problem (7) can be solved with standard FW by using\nπ, c(π)← RL oracle(∇dist2(xt−1,Ω)) = RL oracle(xt−1 − ProjΩ(xt−1)) (9)\nAlgorithm 1 Convex Constrained Reinforcement Learning (C2RL) Input. RL Oracle constructed by any RL algorithm, projection operator to target set ProjΩ. Initialize. Random policy π, value x = c(π), active sets Sp := [π],Sc := [x] and weight λ = [1]. Output. Mixed policy µ and its value c(µ) s.t. c(µ) minimizes the distance to the target set Ω.\n1: while true do // Major cycle 2: ω ← ProjΩ(x) 3: π, c(π)← RL Oracle(x− ω) // Potential improving point 4: if (x− ω)T (x− c(π)) ≤ then break 5: if Sc ∪ {c(π)} is affinely independent then Sc ← Sc ∪ {c(π)},Sp ← Sp ∪ {π} 6: while true do // Minor cycle 7: y,α← AffineMinimizer(Sc,ω) // y = arg mins∈aff(Sc) ||s− ω||2 8: if αs > 0 for all s then break // y ∈ conv(Sc) 9: // If y /∈ conv(Sc), then update y to the intersection of conv(Sc) and segment joining x\nand y. Then remove points in Sc unnecessary for describing y. 10: θ ← mini:αi≤0 λiλi−αi // Recall λ satisfies x = ∑ s∈Sc λss"},{"heading":"11: y ← θy + (1− θ)x, λi = θαi + (1− θ)λi","text":"12: Sc ← {c(πi)|c(πi) ∈ Sc and λi > 0},Sp ← {πi|πi ∈ Sp and λi > 0} 13: end while 14: Update µ← ∑ π∈Sp λππ, x← y, λ← α. 15: end while 16: return µ, c(µ)← x\nto find an improving policy and its measurement vector. For ηt := 2t+2 , the convex averaging steps\nµt ← (1− ηt)µt−1 + ηtπ, xt ← (1− ηt)xt−1 + ηtc(π), (10) then maintain the mixed policy, and the corresponding measurement vector, respectively.\nHowever, after T rounds of iteration, the µt found has an active set containing up to T individual polices, and is not sparse enough. If neural networks are used to parameterize the policy, that requires storing T copies of parameters for the individual network, which is unaffordable for largescale usage.\nTo find even more sparse policies, we turn to variants of FW-type algorithms. In particular, Wolfe’s method for minimum norm point in a polytope (Wolfe, 1976; De Loera et al., 2018). In Wolfe’s method (Algorithm 3 in Appendix A.2), the loop in FW is called a major cycle, and the convex averaging step is replaced by a weight optimization process, called minor cycle. Wolfe’s method maintains an active set S, and the current point can be represented by a sparse combination of points in the active set. The minor cycles maintain S to be an affinely independent set such that the affine minimizer is inside St, which Wolfe calls corrals. Recall an affine minimizer is defined as arg mins∈aff(S) ||s||2, where aff(S) := {y|y = ∑ z∈S α T zx, ∑ z∈S αz = 1} is the affine hull formed by S. Since the active set is affinely independent, the number of active atoms is at most m + 1 at any time. Wolfe’s method is shown to strictly decrease the approximation error between two major cycles."},{"heading":"4.3 OUR MAIN ALGORITHM","text":"The main obstacle to apply Wolfe’s method to our distance minimization problem (7) is that the objective function in Wolfe’s method is the norm function. However, in our problem, the objective function is the distance function to a convex set. Unlike the norm function, the distance function to a convex set is not strongly convex and affine minimizer is ill-defined with respect to a convex set. To tackle these problems, we modify the Wolfe’s method. At the core of our new variant of FW algorithm, we add a projection step to Wolfe’s method.\nProjection Step In each major cycle, we minimize the distance to a projected pointω := ProjΩ(x). Intuitively, since the distance to the convex set is upper bounded by the distance to this projected point ω, if the distance to ω converges, so does the distance to the target convex set.\nFormally, for a set of points S ⊂ Rm, and a point x ∈ Rm, we extend the definition of an affine minimizer to define affine minimizer with respect to x as arg mins∈aff(S) ||s − x||2. For x being\nthe affine minimizer of S with respect to ω, the extended affine minimizer property gives\nGiven ω,∀v ∈ aff(S), (v − x)T (x− ω) = 0 (Extended affine minimizer property) (11) Similar to Wolfe’s method, our C2RL method (Algo. 1) contains an outer loop (called major cycle) to find improving policies and their measurement vectors, and an inner loop (called minor cycle) to maintain the affinely independent property of the active set Sc. At the start of each major cycle step, the Sc is an affinely independent set. Then, the RL oracle (defined in (15)) finds a potential improving policy π ∈ U , and its long-term measurement vector c(π). If the c(π) does not get strictly closer to the ω := Proj(x), then we are done, and x is the optimal value. Otherwise, the c(π) is added into the active set, and the minor cycle is run to eliminate policies whose measurement vectors are affinely dependent.\nLine 6 to line 13 contains the minor cycle, which is the same as the original Wolfe’s method (except in line 6, we find affine minimizer with respect to ω). The elimination is executed as a series of affine projections. The minor cycle terminates if active set Sc is affinely independent. Though the interleaving of major and minor cycles oscillate the size of active set Sc, the minor cycles keep |Sc| an affinely independent set, and is terminated whenever Sc contains a single element. Therefore at the start of any major cycle, the size of the active set satisfies |Sc| ∈ [0,m + 1]. More background about the minor cycle in Wolfe’s method is provided in Appendix A.2.\nConstruction of RL Oracle The construction of our RL oracle can use any off-the-shelf RL algorithm that maximizes a scalar reward. For any given λ ∈ Rm, we define any algorithm that finds a policy minimizing the linear function λT c(·) as a RL oracle, that is\nRL oraclep(λ) ∈ arg min π∈Π λT c(π). (12)\nRecall that standard RL algorithm receives a scalar reward after each state transition, instead of the long-term measurement vector c(π) ∈ Rm. We then use the following linear property to reformulate the right hand side of (12) to a standard RL problem\narg min π∈Π λT c(π) = arg min π∈Π λTE( T∑ t=0 γtct) = − arg max π∈Π E( T∑ t=0 γt(−λT ct)). (13)\nThis shows that if we consider the Markov decision process with the same state, action, and transition probability, and construct a scalar reward r := (−λT ct), then any policy that maximizes the expected r is a linear minimizer of (12). Therefore any RL algorithm that best suits the underlying problems can be used to construct a RL oracle.\nCertifying constraint satisfaction amounts to evaluate the measurement vector of the current policy. This is handy in online settings, where simulations can be used to evaluate the measurement vector of the policy directly. Otherwise, in batch settings, various off-policy evaluation methods, such as importance sampling (Precup, 2000; Precup et al., 2001) or doubly robust (Jiang & Li, 2016; Dudı́k et al., 2011), can be used to evaluate the policy.\nRL oraclec(λ) := c(arg min π∈Π λT c(π)) = arg min c(π),π∈Π λT c(π). (14)\nTo simplify notation, we assume a RL Oracle returns a policy as well as its measurement vector\nRL Oracle(λ) := π, c(π) = RL oraclep(λ), RL oraclec(λ) (15)\nFinding Extended Affine Minimizer The process AffineMinimizer(S,x) returns the (y,α) the affine minimizer of S with respect to x where y is the affine minimizer and α := {αs|∀s ∈ Sc} is the set of coefficient expressing y as an affine combination of points in S, that is y =∑\ns∈Sc αss, where αs is the weight associated with s. The process AffineMinimizer(S,x) can be straightforwardly implemented using linear algebra. Wolfe (1976) also provides a more efficient implementation that uses a triangular array representation of the active set."},{"heading":"4.4 CONVERGENCE AND SPARSITY","text":"In this section, we analyze the convergence and complexity of the proposed C2RL method (Algo. 1). We first show that approximation error of C2RL strictly decreases between any two major cycle steps\nand it converges in O(1/t) rate. Then we show our method ensures convergence of arbitrary RL algorithm, including those searching for deterministic policies. Moreover, concerning the memory complexity, we show that maintaining an active policy set ofm+1 is worst case optimal to ensure the convergence of arbitrary RL algorithm. Therefore, the proposed C2RL indeed achieves the optimal sparsity for the found policy, making it favorable for large-scale usage.\nThe main difference between the convergence analysis of C2RL and Wolfe’s method is the addition of the projection step. Intuitively, at each major step, if we are making a significant progress toward the projected point, then the distance to the convex set is decreased by at least the same amount.\nTime Complexity. In our analysis, we consider the approximation error as defined in (5). We use superscript t to denote the variable in t-th major cycle before executing any minor cycle. To simplify notions, we let xt := c(µt) and st := c(πt). When discussing one step with t fixed, let yi denote the affine minimizer found in i-th minor cycle (line 6 of Algo. 1). We first show that the C2RL method strictly reduces approximation error between two calls of the RL oracle.\nTheorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step t, we have err(µt+1) < err(µt). That is, the measurement vector of µt found by the C2RL method gets strictly closer to the convex set Ω after major cycle step.\nThe proof is provided in Appendix B. The idea is to consider the distance between xt and ωt. When the major cycle has no minor cycle, the non-terminal condition and the affine minimizer property implies dist2(xt+1,ωt) < dist2(xt,ωt). Otherwise we show that the first minor cycle strictly reduces the dist2(xt,ωt) by moving along the segment joining x and y, and the subsequent minor cycle cannot increase it. Since ωt ∈ Ω, we conclude err(xt+1) ≤ dist2(xt+1, ωt) < dist2(xt, ωt) = err(xt), and the approximation error strictly decreases.\nGiven the approximation error strictly decreases, Wolfe’s method for minimum norm point can be shown to terminate finitely (Wolfe, 1976). However, this finitely terminating property does not hold for our algorithm. Since a changed ωt may yield a lower distance to the same active set Stc, the active set may stay unchanged across major cycles (see Figure 2 Middle for an example). Therefore we establish the convergence of the C2RL method by the following theorem.\nTheorem 4.2 (Convergence in Approximation Error). For t ≥ 1, the mixed policy µt found by the C2RL method satisfies\nerr(µt) ≤ 16Q2/(t+ 2), (16)\nwhere Q := maxµ∈∆(U) ||c(µ)|| is the maximum norm of a measurement vector.\nThe proof is provided in Appendix C, which relies on the following two lemmas. We briefly discuss the main idea here. Define major cycle steps with at most one minor cycle as ”non-drop step” and major cycle steps with more than one minor cycles as ”drop steps”. We show that in each non-drop step, Algorithm 1 is guaranteed to make enough progress in the following lemma.\nLemma 4.3. For a non-drop step in C2RL method, we have err(µt)−err(µt+1) ≥ err2(µt)/8Q2.\nThough this does not hold for drop steps, we can bound the frequency of drop steps by the following.\nLemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than t/2.\nSince the approximation error strictly decreases (Thm. 4.1), and in more than half of the major cycles steps, the C2RL method makes significantly progress. The Thm. (4.2) can then be proved using an induction argument (Appendix C).\nConvergence with Arbitrary RL Algo. The convergence of the C2RL method when used with RL algorithms that search for deterministic policies, such as DQN, DDPG and variants, is indeed straightforward. In (8), though each time the oracle yields a vertex, the FW-type algorithms indeed optimize over the polytope formed by these vertices. Then since citetaltman1999constrained shows that any c(·) achievable can be achieved by some mixed deterministic policies, we conclude that if the underlying problem is feasible, then our C2RL method is able to find a feasible policy.\nMemory Complexity We then discuss the sparsity of mixed policy for constrained RL problem. We give a constructive proof in Appendix D to show that to ensure convergence for RL algorithms that search for deterministic policies, storing m+ 1 policies is required in the worst case.\nTheorem 4.5 (Memory Complexity Bound). For an constrained RL problem with m-dimensional measurement vector, in the worst case, a mixed policy needs to randomize among m+ 1 individual policies to ensure convergence of RL oracles that search for deterministic policies.\nSince the minor cycles in the C2RL method eliminate policies with affinely dependent measurement vectors, after the termination of minor cycles, the size of the active set is at most m+ 1. That is, the policy found by the C2RL method requires randomization among no more than m + 1 individual policies. Therefore the proposed C2RL indeed achieves the optimal sparsity in the worst case, making it favorable for large-scale usage. Corollary 4.5.1. The C2RL method that randomizes among at most m + 1 policies is worst-case optimal to ensure convergence of any RL oracle."},{"heading":"5 EXPERIMENTS","text":"We evaluate the performance of C2RL in a grid-world navigation task (Fig. 1), and demonstrate its ability to efficiently find sparse policy. In this Risky Mars Rover environment, the agent is required to navigate from the starting point to the goal point, by moving to one of the four neighborhood cells at each step. The episodes terminate when the goal point is reached or after 300 steps. To enforce robustness, we add a risky area to indicate the dangerous states. The agent receives a measurement vector to indicate the steps it takes (0.1 for every step), and whether it stays in the risky area (0.1 for every risky step, and 0 otherwise), with discount factor γ = 0.99. We constrain the agent to reach the goal point with expected cumulative steps measure within 1.1 and the expected cumulative risky steps within 0.05. Note that by design, the shortest path from the starting point to the goal point does not satisfy the constraint. This is common in practice, as robustness typically evolves trade-off between the reward and the constraint satisfaction.\nThe proposed C2RL method is compared with approachability-based policy optimization (ApproPO) (Miryoosefi et al., 2019) and with reward constrained policy optimization (RCPO) (Tessler et al., 2018). ApproPO solves the same convex constrained RL problem by using an RL oracle to play against a no-regret online learner (Hazan et al., 2008; Zinkevich, 2003). Since ApproPO and C2RL both use a RL oracle, ApproPO is a natural baseline to be compared with our method. Besides, we also compare with RCPO, which takes a Lagrangian approach to incorporate the constraints as a penalty signal into the reward. Using an advantage actor critic (A2C) Mnih et al. (2016), RCPO has been shown to converge to a fixed point. For a fair comparison, C2RL and ApproPO uses an A2C agent as the RL oracle, with the same hyperparameter as used in RCPO. The approximation errors are compared after training for the same number of samples.\nNote that the C2RL method does not introduce any extra hyper-parameter. For ApproPO and RCPO, they require extra hyper-parameter for the initialization and learning rate of a variable equivalent to our λ in the outer loop. This is because our approach does not rely on the online learning framework, and therefore there is no need to tune the initialization and learning rate for our λ and ease the usage.\nWe first showcase the consequences of our theoretical results using an optimal RL oracle. For any x ∈ Rm, an optimal policy can be easily found via Dijkstra’s algorithm. If multiple optimal paths exist, one is randomly picked to form a deterministic policy.\nUsing this as an optimal RL oracle, the convergence property of C2RL and ApproPo are compared. Figure 2 Middle shows the value of policies c(µt) found after each call to the oracle. In Figure 2 Right, when approaching the boundary of the feasible set, the iterations of approachability-based methods start to zigzag. Since C2RL contains a minor cycle to re-optimize the weights among the active set, C2RL progresses quickly to reach the exact optimal solution. In Figure 3 Left, the approximation error is shown for 300 calls of the optimal RL oracle.\nWe then compare C2RL, ApproPO and RCPO using the same A2C agent (details of the model structures and hyper-parameters are provided in Appendix E). We run each algorithm for 50 times, and each run for a maximum of 100 thousands of samples. The mean and standard deviation of the results are presented in Figure 3. The original paper of ApproPO suggests using a cache to save memory, and the memory requirement of this variant is also presented. Figure 3 demonstrates that C2RL converges to an optimal policy faster than previous methods, and a sparse combination of individual policies is maintained throughout the iteration process."},{"heading":"6 CONCLUSION","text":"In this paper, we introduce C2RL, an algorithm to solve RL problems under orthant or convex constraints. Our method reduces the constrained RL problem to a distance minimization problem, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this. Our method comes with rigorous theoretical guarantees and does not introduce any extra hyper-parameter. To find an -approximation solution, C2RL takes O(1/ ) calls of any RL oracle and ensures convergence to work with arbitrary RL algorithm. Moreover, C2RL strictly reduces the approximation error between consecutive calls of RL oracle, and form-dimensional constraints, the memory requirement is reduced from storing infinitely many policies (O(1/ )) to storing at most constantly many (m+1) polices. We further show that the constant is worst-case optimal to ensure the convergence for RL algorithms that search for deterministic policies. Experimentally, we demonstrate that the proposed C2RL method finds sparse solution efficiently, and outperforms previous methods."},{"heading":"A MORE ON FRANK-WOLFE-TYPE ALGORITHMS","text":"A.1 STANDARD FRANK-WOLFE ALGORITHM\nAlgorithm 2 Frank-Wolfe algorithm (Frank et al., 1956) Input: obj. f : Y 7→ R, oracle O(·), init. x0 ∈ Y\n1: for t=1, 2, 3 . . . , T do 2: s← Oracle(∇f(xt−1)) = arg mins∈{s1,...,sn} s\nT∇f(xt−1) 3: xt ← (1− ηt)xt−1 + ηts , for ηt := 2t+2 4: end for 5: return xT\nFor a convex function f : X 7→ R the Frank-Wolfe algorithm (FW) solves the constrained optimization problem over a compact and convex set X . The standard FW is known to have a sublinear convergence rate, and various methods are proposed to improve the performance. For example, when the underlying feasible set is a polytope, and the objective function is strongly convex, multiple variants, such as away-step FW (Wolfe, 1970; Jaggi, 2013), pairwise FW (Mitchell et al., 1974), and Wolfe’s method (Wolfe, 1976) are shown to enjoy linear convergence rate. Linear convergence under other conditions is also studied (Beck & Shtern, 2017; Garber & Hazan, 2013a;b).\nA.2 WOLFE’S METHOD FOR MINIMUM NORM POINT\nAlgorithm 3 Wolfe’s Method for Minimum Norm Point Initialize x ∈ P , active set S = [x] and weight λ = [1]. Output: x ∈ P that has the minimum Euclidean norm.\n1: while true do // Major cycle 2: s← Oracle(x) // Potential improving point 3: if ||x||2 ≤ xTs+ then break 4: S ← S ∪ {s} 5: while true do // Minor cycle 6: y,α← AffineMinimizer(S) // y = arg mins∈aff(S) ||s||2 7: if αs > 0 for all s then break // y ∈ conv(S) 8: // If y /∈ conv(S), then update y to the intersection of conv(S) and segment joining x and y. Then remove points in S unnecessary for describing y. 9: θ ← mini:αi≤0 λiλi−αi // Recall λ satisfies x = ∑ s∈S λss"},{"heading":"10: y ← θy + (1− θ)x, λi = θαi + (1− θ)λi","text":"11: S ← {si|si ∈ S and λi > 0} 12: end while 13: Update x = y and λ = α. 14: end while 15: return x\nWolfe’s method is an iterative algorithm for finding the point with minimum Euclidean norm in a polytope, which is defined as the convex hull of a set of finitely many points.\nThe Wolfe’s method consists of a finite number of major cycles, each of which consists of a finite number of minor cycles. At the start of each major cycle, let H(x) := {yTx = xx} be the hyperplane defined by x. If H(x) separates the polytope from the origin, then the major cycle is terminated. Otherwise, it invokes an oracle to find any point on the near side of the hyperplane. The point is then added into the active set S, and starts a minor cycle. In a minor cycle, let y be the point of smallest norm in of the affine hull aff(S). If y is in the relative interior of the convex hull conv(S), then x is updated to y and the minor cycle is terminated. Otherwise, y is updated to the nearest point to y on the line segment conv(S) ∩ [x,y]. Thus y is updated to a boundary point of conv(S), and any point that is not on the face of conv(S) in which y lies is deleted. The minor cycles are executed repeatedly until S becomes a corral, that is, a set\nwhose affine minimizer lies inside its convex hull. Since a set of one point is always a corral, the minor cycles is terminated after a finite number of runs."},{"heading":"B PROOF OF THEOREM 4.1","text":"Theorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step t, we have err(µt+1) < err(µt). That is, the measurement vector of µt found by the C2RL method gets strictly closer to the convex set Ω after major cycle step.\nProof. If the current step is a major cycle with no minor cycle, then xt+1 is the affine minimizer of aff(S ∪ {st}) with respect to ωt. Then the affine minimizer property implies (st − xt+1)(xt+1 − ωt) = 0. Since iteration does not terminate at step t, we have (xt − ωt)T (xt − st) > 0, and therefore xt+1 not equal to xt. Then xt+1 is the unique affine minimizer implies fΩ(xt+1) = minω∈Ω ||xt+1 − ω||2 ≤ ||xt+1 − ωt||2 < ||xt − ωt||2 = fΩ(xt). Otherwise the current step contains one or more minor cycles. In this case, we show that the first minor cycle strictly reduces the approximation error, and the (possibly) following minor cycles cannot increase it. For the first minor cycle, the affine minimizer y0 of aff(S ∪ {st}) with respect to ωt is outside conv(S ∪ {st}). Let z = θy0 + (1 − θ)xt be the intersection of conv(S ∪ {st}) and segment joining x and y. Let V0 := St and Vi denote the active set after the i-th minor cycle. Then since y1 is the affine minimizer of V1 with respect to ωt, we have\n||z − ωt|| = ||θy0 + (1− θ)xt − ωt|| ≤ θ||y0 − ωt||+ (1− θ)||xt − ωt|| < ||xt − ωt||, (17)\nwhere the second step uses the triangle inequality and the last step follows since the segment xty0 intersects the interior of conv(S∪{st}), and the distance to ωt strictly decreases along this segment. Therefore the point z found by first minor cycle satisfies\nfΩ(z) = min ω∈Ω ||z − ω||2 ≤ ||z − ωt||2 < ||xt − ωt|| = fΩ(xt). (18)\nHence h(y1) < h(xt), and the first minor cycle strictly decreases the approximation error. By a similar argument, in subsequent minor cycles the approximation error cannot be increased. However, after the first minor cycle, the iterating point may already at the intersection point and the strict inequality in last step of Eq. 17 need to be replaced by non-strict inequality.\nTherefore any major cycle either finds an improving point and continue, or enters minor cycles where the first minor cycle finds an improving point, and the subsequent minor cycles does not increase the distance. Adding both side of fΩ(xt+1) < fΩ(xt) by fΩ(x∗) and we have the approximation error h(xt+1) < h(xt) strictly decreases."},{"heading":"C PROOF OF THEOREM 4.2","text":"We first prove the Theorem 4.2, using Lemma 4.3 and Lemma 4.4. Then we present the proof of the lemmas.\nTheorem 4.2 (Convergence in Approximation Error). For t ≥ 1, the mixed policy µt found by the C2RL method satisfies\nerr(µt) ≤ 16Q2/(t+ 2). (19)\nwhere Q := maxµ∈∆(U) ||c(µ)|| is the maximum norm of a measurement vector.\nProof. Since Lemma 4.4 shows that drop steps are no more than half of total major cycle steps, and Theorem 4.1 guarantees these drop steps reducing the approximation error, we can safely skip these step, and re-index the step numbers to include non-drop steps only using k.\nFor these non-drop steps, we claim that err(µk) ≤ 8Q2/(k + 1). Using Lemma 4.3, we prove the convergence rate using induction. We first bound the error of any err(µk). For any k ≥ 1\nerr(µk) = dist2(c(µk),Ω)− dist2(c(µ∗),Ω) (20) = 1/2||c(µk)− ProjΩ(c(µ k))||2 − 1/2||c(µ∗)− ProjΩ(c(µ ∗))||2 (21)\n≤ 1/2(||c(µk)||2 + ||ProjΩ(c(µ k))||2 − ||c(µ∗)||2 − ||ProjΩ(c(µ ∗))||2) (22) ≤ ||c(µk)||2 − ||c(µ∗)||2 (23) ≤ ||c(µk)||2 (24) ≤ Q2, (25)\nwhere Eq. 21 uses the definition of our squared Euclidean distance function. Eq. 22 follows from triangle inequality, and Eq. 23 is by the contractive property of the Euclidean distance.\nWhen k = 1, the Eq. 25 established the based case. Now for k ≥ 1, assume that err(µk) ≤ 8Q2/(k + 1) for k ≥ 1, then Lemma 4.3 gives err(µk+1) ≤ err(µk) − err2(µk)/8Q2. Since the quadratic function of the right hand side is monotonically increasing on (−∞, 4Q2], using the inductive hypothesis\nerr(µk+1) ≤ err(µk)− err2(µk)/8Q2 ≤ 8Q2/(k + 1)− 8Q2/(k + 1)2 ≤ Q2/(k + 2) (26)\nSince for t steps of major cycle steps, the number of non-drop steps k > t/2, we conclude that err(µt) ≤ 16Q2/(t+ 2).\nThen we prove the lemmas.\nLemma 4.3. For a non-drop step, we have err(µt)− err(µt+1) ≥ err2(µt)/8Q2.\nProof. The non-drop step contains either no minor cycle or one minor cycle. We first consider the no minor cycle case.\nIf a major cycle contains no minor cycle, then xt+1 is the affine minimizer of the S ∪ {st}.\nerr(µt)− err(µt+1) = dist2(xt,Ω)− dist2(xt+1,Ω) (27) = 1/2(||xt − ωt||2 −min\nω∈Ω ||xt+1 − ω||2) (28)\n≥ 1/2(||xt − ωt||2 − ||xt+1 − ωt||2) (29) = 1/2(||xt − ωt||2 + ||xt+1 − ωt||2 − 2||xt+1 − ωt||2) (30) = 1/2(||xt − ωt||2 + ||xt+1 − ωt||2 − 2(xt − ωt)T (xt+1 − ωt)) (31) = 1/2(||xt − xt+1||2), (32)\nwhere the equation (31) follows from the affine minimizer property Eq. (11). For ||xt − xt+1|| in the last equation, and ∀q ∈ aff(S ∪ {st}), we have\n||xt − xt+1|| ≥ ||xt − xt+1|| ||x t||+ ||q||\n2Q ( Definition of Q) (33)\n≥ ||xt − xt+1|| ||x t − q|| 2Q\n( Triangle inequality) (34)\n≥ 1 2Q (xt − xt+1)(xt − q) ( Cauchy-Schwarz inequality) (35)\n= 1\n2Q (xt − ωt)(xt − q) ( Affine minimizer property). (36)\nThen it suffices to show that (xt − ωt)(xt − q) ≥ err(µt).\nSince Ω is a convex set, the squared Euclidean distance function dist2(x,Ω) is convex for x, which implies\ndist2(xt,Ω) + (q − xt)∇dist2(xt,Ω) ≤ dist2(q,Ω). (37)\nPutting in∇dist2(xt,Ω) = (xt−ProjΩ(xt)) = (xt−ωt), we get (xt−ωt)(xt−q) ≥ err(µt), which together with Eq. 32 and Eq. 36 concludes that for non-drop step with no minor cycles, we have err(µt)− err(µt+1) ≥ err2(µt)/8Q2. For non-drop step with one minor cycle, we use the Theorem 6 of (Chakrabarty et al., 2014). By a linear translation of adding all points with −ωt, it gives\n||xt − ωt||2 − ||xt+1 − ωt||2 ≥ ((xt − ωt)(xt − q))2/8Q2. (38) Then applying the same argument as Eq. 37, and we finished our proof.\nLemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than t/2.\nProof. Recall that at the termination of a minor cycle, the size of the active set |Sc| ∈ [1,m]. Since in each major cycle steps, the size of active set St increases by one, and each drop step reduces the size of St by at least one, the number of drop steps is always less than half of total number of the major cycle steps."},{"heading":"D PROOF OF THEOREM 4.5","text":"Theorem 4.5 (Memory Complexity Bound). For an constrained RL problem with m-dimensional measurement vector, in the worst case, a mixed policy needs to randomize among m+ 1 individual policies to ensure convergence of RL oracles that search for deterministic policies.\nProof. We give a constructive proof. Consider a m-dimensional vector-valued MDP with a single state, m + 1 actions, and c(ai) := ei is the unit vector of i-th dimension for i ∈ [1,m], and c(am+1) := 0, and the episode terminates after 1 steps. The constrained RL problem is to find a policy whose measurement vector lies in the convex set of a single point {1/2m}. By linear programming, it is clear that the only feasible mixed deterministic policy is to select am+1 with 1/2 probability, and the rest m actions with 1/2m probability; i.e. the unique feasible policy to this problem has an active set containing m + 1 deterministic policies. Therefore any method randomize among less than m + 1 individual policies does not ensure convergence when used with RL algorithms searching for deterministic policies."},{"heading":"E ADDITIONAL EXPERIMENT DETAILS","text":"All the methods use the same A2C agent. The input is the one-hot encoded current position index. The A2C is the standard fully connected multi-layer perceptron with ReLU activation function. The actor and critic share the internal representation and have their only final layer. Both actor and critic networks use Adam optimizer with learning rate set to 1e−2. The network is as follows\nActor Critic Input layer One-hot encoded state index (dim=54)\nHidden layer Linear(in=54, out=128, act=”relu”) Output layer Linear(in=128, out=4, act=”relu”) Linear(in=128, out=1, act=”relu”) Output name Action score State value\nFor ApproPO, the constant κ for projection convex set to convex cone is set to be 20. The θ is initialized to 0. Following the original paper.\nFor RCPO, the learning rate of its λ is set to 2.5e−5, and its λ is initialized to 0 and updated by online gradient descent with learning rate set to 1, as used by the original paper.\nThe proposed C2RL introduces no extra hyper-parameters, and has nothing to report."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Contemporary approaches in reinforcement learning (RL) largely focus on optimizing the behavior of an agent against a single reward function. RL algorithms like value function methods (Zou et al., 2019; Zheng et al., 2018) or policy optimization methods (Chen et al., 2019; Zhao et al., 2017) are widely used in real-world tasks. This can be sufficient for simple tasks. However, for complicated applications, designing a reward function that implicitly defines the desired behavior can be challenging. For instance, applications concerning risk (Geibel & Wysotzki, 2005; Chow & Ghavamzadeh, 2014; Chow et al., 2017), safety (Chow et al., 2018) or budget (Boutilier & Lu, 2016; Xiao et al., 2019) are naturally modelled by augmenting the RL problem with orthant constraints. Exploration suggestions, such as to visit all states as evenly as possible, can be modelled by using a vector to measure the behavior of the agent, and to find a policy whose measurement vector lies in a convex set (Miryoosefi et al., 2019).\\nTo solve RL problem under constraints, existing methods either ensure convergence only on a specific family of RL algorithms, or treat the underlying RL algorithms as a black box oracle to find individual policy, and look for mixed policy that randomizes among these individual policies. Though the second group of methods has the advantage of working with arbitrary RL algorithms that best suit the underlying problem, existing methods have practically infeasible memory requirement. To get an -approximate solution, they require storing O(1/ ) individual policies, and an exact solution requires storing infinitely many policies. This limits the prevalence of such methods, especially when the individual policy uses deep neural networks.\\nIn this paper, we propose a novel reduction approach for the general convex constrained RL (C2RL) problem. Our approach has the advantage of the second group of methods, yet requires storing at most constantly many policies. For a vector-valued Markov Decision Process (MDP) and any given target convex set, our method finds a mixed policy whose measurement vector lies in the target convex set, using any off-the-shelf RL algorithm that optimizes a scalar reward as a RL oracle. To do so, the C2RL problem is reduced to a distance minimization problem between a polytope and a convex set, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this distance minimization problem. To find an -approximate solution in an m-dimensional vector-valued MDP,\\nour method only stores at most m + 1 policies, which improves from infinitely many O(1/ ) (Le et al., 2019; Miryoosefi et al., 2019) to a constant. We also show this m + 1 constant is worstcase optimal to ensure convergence of RL algorithms using deterministic policies. Moreover, our method introduces no extra hyper-parameter, which is favorable for practical usage. A preliminary experimental comparison demonstrates the performance of the proposed method and the sparsity of the policy found.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"For high dimensional constrained RL, one line of approaches incorporates the constraint as a penalty signal into the reward function, and makes updates in a multiple time-scale scheme (Tessler et al., 2018; Chow & Ghavamzadeh, 2014). When used with policy gradient or actor-critic algorithms (Sutton & Barto, 2018), this penalty signal guides the policy to converge to a constraint satisfying one (Paternain et al., 2019; Chow et al., 2017). However, the convergence guarantee requires the RL algorithm can find a single policy that satisfies the constraint, hence ruling out methods that search for deterministic policies, such as Deep Q-Networks (DQN) (Mnih et al., 2013), Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015) and their variants (Van Hasselt et al., 2015; Wang et al., 2016; Fujimoto et al., 2018; Barth-Maron et al., 2018).\\nAnother line of approaches uses a game-theoretic framework, and does not tie to specific families of RL algorithm. The constrained problem is relaxed to a zero-sum game, whose equilibrium is solved by online learning (Agarwal et al., 2018). The game is played repeatedly, each time any RL algorithm can be used to find a best response policy to play against a no-regret online learner. The mixed policy that uniformly distributed among all played policies can be shown to converge to an optimal policy of the constrained problem (Freund & Schapire, 1999; Abernethy et al., 2011). Taking this approach, Le et al. (2019) uses Lagrangian relaxation to solve the orthant constraint case, and Miryoosefi et al. (2019) uses conic duality to solve the convex constraint case. However, since the convergence is established by the no-regret property, the policy found by these methods requires randomization among policies found during the learning process, which limits their prevalence.\\nDifferent from the game-theoretic approaches, we reduce the C2RL to a distance minimization problem and propose a novel variant of Frank-Wolfe (FW) algorithm to solve it. Our result builds on recent finding that the standard FW algorithm emerges as computing the equilibrium of a special convex-convave zero sum game (Abernethy & Wang, 2017). This connects our approach with previous approaches from game-theoretic framework (Agarwal et al., 2018; Le et al., 2019; Miryoosefi et al., 2019). The main advantage of our reduction approach is that the convergence of FW algorithm does not rely on the no-regret property of an online learner. Hence there is no need to introduce extra hyper-parameters, such as learning rate of the online learner, and intuitively, we can eliminate unnecessary policies to achieve better sparsity. To do so, we extend Wolfe’s method for minimum norm point problem (Wolfe, 1976) to solve our distance minimization problem. Throughout the learning process, we maintain an active policy set, and constantly eliminate policies whose measurement vector are affinely dependent of others. Unlike norm function in Wolfe’s method, our objective function is not strongly convex. Hence we cannot achieve the linear convergence of Wolfe’s method as shown in Lacoste-Julien & Jaggi (2015). Instead, we analyze the complexity of our method based on techniques from Chakrabarty et al. (2014). A theoretical comparison between our method and various approaches in constrained RL is provided in Table 1.\"\n },\n {\n \"heading\": \"3 PRELIMINARIES\",\n \"text\": \"A vector-valued Markov decision process can be identified by a tuple {S,A, β, P, c}, where S is a set of states,A is the set of actions and β is the initial state distribution. At the start of each episode, an initial state s0 is drawn following the distribution β. Then, at each step t = 0, 1, . . . , the agent observes a state st ∈ S and makes a decision to take an action at. After at is chosen, at the next observation the state evolves to state st+1 ∈ S with probability P (st+1|st, at). However, instead of a scalar reward, in our setting, the agent receives an m-dimensional vector ct ∈ Rm that may implicitly contain measurements of reward, risk or violation of other constraints. The episode ends after a certain number of steps, called the horizon, or when a terminate state is reached.\\nActions are typically selected according to a policy π, where π(s) is a distribution over actions for any s ∈ S . Policies that take a single action for any state are deterministic policies, and can be identified by the mapping π : S 7→ A. The set of all deterministic policies is denoted by Π. For a discount factor γ ∈ [0, 1), the discounted long-term measurement vector of a policy π ∈ Π is defined as\\nc(π) := E( T∑ t=0 γtct(st, π(st))), (1)\\nwhere the expectation is over trajectories generated by the described random process.\\nUnlike unconstrained setting, for a constrained RL problem, it is possible that all feasible policies are non-deterministic (see Appendix D for an example). This limits the usage of RL algorithms that search for deterministic policies in the setting of constrained RL problem.\\nOne workaround is to use mixed policies. For a set of policies U , a mixed policy is a distribution over U , and the set of all mixed policies over U is denoted by ∆(U). To execute a mixed policy µ ∈ ∆(U), we first select a policy π ∈ U according to π ∼ µ(π), and then execute π for the entire episode. Altman (1999) shows that any c(·) achievable can be achieved by some mixed deterministic policies µ ∈ ∆(Π). Therefore, though an off-shelves RL algorithm may not converge to any constraint-satisfying policy, it can be used as a subroutine to find individual policies (possibly deterministic), and a randomization among these policies can converge to a feasible policy. The discounted long-term measurement vector of a mixed policy µ ∈ ∆(Π) is defined similarly\\nc(µ) := Eπ∼µ(c(π)) = ∑ π∈Π µ(π)c(π). (2)\\nFor a mixed policy µ ∈ ∆(U), its active set is defined to be the set of policies with non-zero weights A := {π ∈ U|µ(π) > 0}. The memory requirement of storing µ, is then proportional to the size of its active set. Since a mixed policy can be interpreted as a convex combination of policies in its active set, in the following, the term sparsity of a mixed policy refers to the sparsity of this combination.\\nOur learning problem, the convex constrained reinforcement learning (C2RL), is to find a policy whose expected long-term measurement vector lies in a given convex set; i.e., for a given convex target set C ⊂ Rm, our target is to\\nfind µ∗ such that c(µ∗) ∈ Ω (C2RL). (3) Any policy µ∗ that satisfies c(µ∗) ∈ Ω is called a feasible policy, and a C2RL problem is feasible if there exists some feasible policies. In the following, we assume the C2RL problem is feasible.\"\n },\n {\n \"heading\": \"4 APPROACH, ALGORITHM AND ANALYSIS\",\n \"text\": \"We now show how the C2RL (3) can be reduced to a distance minimization problem (7) between a polytope and a convex set. A novel variant of Frank-Wolfe-type algorithm is then proposed to solve the distance minimization problem, followed by theoretic analysis about convergence and sparsity of the proposed method.\"\n },\n {\n \"heading\": \"4.1 REDUCE C2RL TO A DISTANCE MINIMIZATION PROBLEM\",\n \"text\": \"Let ||·|| denote the Euclidean norm. For a convex set Ω ∈ Rm, let ProjΩ(x) ∈ arg miny∈Ω ||x−y|| be the projection operator, and dist2(x,Ω) := 12 ||x−ProjΩ(x)|| 2 be half of the squared Euclidean\\ndistance function. Then we consider the problem to find a policy whose measurement vector is closest to the target convex set,\\narg min µ∈∆(Π)\\ndist2(c(µ),Ω). (4)\\nA policy µ∗ ∈ ∆(Π) is defined to be an optimal solution if it minimizes (4). Otherwise, the approximation error of µ ∈ ∆(Π) is defined as\\nerr(µ) := dist2(c(µ),Ω)− dist2(c(µ∗),Ω) (Approximation Error) (5) where µ∗ is an optimal solution, and a policy is defined to be an -approximate solution if its approximation error is no larger than .\\nWhen C2RL (3) is feasible, the equivalence of being optimal to (4) and being feasible to C2RL can be easily established. Since a feasible policy of C2RL problem lies inside Ω, it minimizes the non-negative dist2 function, and hence is optimal to (4). Vice versa, any optimal solution to (4) lies inside Ω and is a feasible solution to C2RL.\\nFrom a geometric perspective, let c(Π) := {c(π)|π ∈ Π} be the set of all values achievable by deterministic policies. If the MDP has finite states and actions (though may be extremely large), then Π is finite as well, and hence c(Π) contains finitely many points in Rm. Then the set of values achievable by mixed deterministic policies\\nc(∆(Π)) := {c(µ)|µ ∈ ∆(Π)} = { ∑ π µ(π)c(π) | ∑ π µ(π) = 1, µ(π) ≥ 0} ⊂ Rm (6)\\nis the convex hull of c(Π); i.e., c(∆(Π)) is a m-dimension polytope whose vertices are c(Π). Therefore finding a policy whose value is closest to the target convex set (4) is equivalent to find a point in the polytope c(∆(Π)) that is closest to the convex set Ω\\narg min c(µ)∈c(∆(Π))\\ndist2(c(µ),Ω) (Distance minimization problem). (7)\\nTo solve this constrained optimization problem, it might be tempting to consider projection methods. However, constructing a projection operator for c(∆(Π)) is non-trivial. For any given measurement vector, it is obscure how to modify a general RL algorithm to update the parameters such that the discounted expected measurement vector is closest to the given value. Therefore, projection-free methods are preferable for this task.\\nFrank-Wolfe (FW) algorithm does not require any projection operation, instead it uses a linear minimizer oracle. Intuitively, finding a linear minimizer is similar to the reward maximization process of what a general RL algorithm does. In section 4.3, we formalize this idea. We show that after simple modifications, any RL algorithm that maximizes a scalar reward can be used to construct such a linear minimizer oracle. Before getting into details of the construction process, we discuss FW-type algorithms over polytope and its applications in the distance minimization problem (7).\"\n },\n {\n \"heading\": \"4.2 DISTANCE MINIMIZATION BY FRANK-WOLFE-TYPE ALGORITHMS\",\n \"text\": \"The Frank-Wolfe algorithm (FW) is a first-order method to minimize a convex function f : P 7→ R over a compact and convex set P , with only access to a linear minimizer oracle. When the feasible set is a polytope P := conv({s1, s2, . . . , sn}) ⊂ Rm defined as the convex hull of finitely many points, FW-type algorithms are discussed by Lacoste-Julien & Jaggi (2015) to optimize\\nmin x∈P f(x) using Oracle(v) := arg min s∈{s1,...,sn}\\nsTv. (8)\\nThe standard FW (Algorithm 2 in Appendix A.1) consists of making repeated calls to the linear minimizer oracle to find an improving point s, followed by a convex averaging step of the current iterate xt−1 and the oracle’s output s.\\nIf we have already constructed a RL oracle(λ) that outputs a policy π ∈ arg minπ∈Π λT c(π) together with its measurement vector c(π), then the distance minimizing problem (7) can be solved with standard FW by using\\nπ, c(π)← RL oracle(∇dist2(xt−1,Ω)) = RL oracle(xt−1 − ProjΩ(xt−1)) (9)\\nAlgorithm 1 Convex Constrained Reinforcement Learning (C2RL) Input. RL Oracle constructed by any RL algorithm, projection operator to target set ProjΩ. Initialize. Random policy π, value x = c(π), active sets Sp := [π],Sc := [x] and weight λ = [1]. Output. Mixed policy µ and its value c(µ) s.t. c(µ) minimizes the distance to the target set Ω.\\n1: while true do // Major cycle 2: ω ← ProjΩ(x) 3: π, c(π)← RL Oracle(x− ω) // Potential improving point 4: if (x− ω)T (x− c(π)) ≤ then break 5: if Sc ∪ {c(π)} is affinely independent then Sc ← Sc ∪ {c(π)},Sp ← Sp ∪ {π} 6: while true do // Minor cycle 7: y,α← AffineMinimizer(Sc,ω) // y = arg mins∈aff(Sc) ||s− ω||2 8: if αs > 0 for all s then break // y ∈ conv(Sc) 9: // If y /∈ conv(Sc), then update y to the intersection of conv(Sc) and segment joining x\\nand y. Then remove points in Sc unnecessary for describing y. 10: θ ← mini:αi≤0 λiλi−αi // Recall λ satisfies x = ∑ s∈Sc λss\"\n },\n {\n \"heading\": \"11: y ← θy + (1− θ)x, λi = θαi + (1− θ)λi\",\n \"text\": \"12: Sc ← {c(πi)|c(πi) ∈ Sc and λi > 0},Sp ← {πi|πi ∈ Sp and λi > 0} 13: end while 14: Update µ← ∑ π∈Sp λππ, x← y, λ← α. 15: end while 16: return µ, c(µ)← x\\nto find an improving policy and its measurement vector. For ηt := 2t+2 , the convex averaging steps\\nµt ← (1− ηt)µt−1 + ηtπ, xt ← (1− ηt)xt−1 + ηtc(π), (10) then maintain the mixed policy, and the corresponding measurement vector, respectively.\\nHowever, after T rounds of iteration, the µt found has an active set containing up to T individual polices, and is not sparse enough. If neural networks are used to parameterize the policy, that requires storing T copies of parameters for the individual network, which is unaffordable for largescale usage.\\nTo find even more sparse policies, we turn to variants of FW-type algorithms. In particular, Wolfe’s method for minimum norm point in a polytope (Wolfe, 1976; De Loera et al., 2018). In Wolfe’s method (Algorithm 3 in Appendix A.2), the loop in FW is called a major cycle, and the convex averaging step is replaced by a weight optimization process, called minor cycle. Wolfe’s method maintains an active set S, and the current point can be represented by a sparse combination of points in the active set. The minor cycles maintain S to be an affinely independent set such that the affine minimizer is inside St, which Wolfe calls corrals. Recall an affine minimizer is defined as arg mins∈aff(S) ||s||2, where aff(S) := {y|y = ∑ z∈S α T zx, ∑ z∈S αz = 1} is the affine hull formed by S. Since the active set is affinely independent, the number of active atoms is at most m + 1 at any time. Wolfe’s method is shown to strictly decrease the approximation error between two major cycles.\"\n },\n {\n \"heading\": \"4.3 OUR MAIN ALGORITHM\",\n \"text\": \"The main obstacle to apply Wolfe’s method to our distance minimization problem (7) is that the objective function in Wolfe’s method is the norm function. However, in our problem, the objective function is the distance function to a convex set. Unlike the norm function, the distance function to a convex set is not strongly convex and affine minimizer is ill-defined with respect to a convex set. To tackle these problems, we modify the Wolfe’s method. At the core of our new variant of FW algorithm, we add a projection step to Wolfe’s method.\\nProjection Step In each major cycle, we minimize the distance to a projected pointω := ProjΩ(x). Intuitively, since the distance to the convex set is upper bounded by the distance to this projected point ω, if the distance to ω converges, so does the distance to the target convex set.\\nFormally, for a set of points S ⊂ Rm, and a point x ∈ Rm, we extend the definition of an affine minimizer to define affine minimizer with respect to x as arg mins∈aff(S) ||s − x||2. For x being\\nthe affine minimizer of S with respect to ω, the extended affine minimizer property gives\\nGiven ω,∀v ∈ aff(S), (v − x)T (x− ω) = 0 (Extended affine minimizer property) (11) Similar to Wolfe’s method, our C2RL method (Algo. 1) contains an outer loop (called major cycle) to find improving policies and their measurement vectors, and an inner loop (called minor cycle) to maintain the affinely independent property of the active set Sc. At the start of each major cycle step, the Sc is an affinely independent set. Then, the RL oracle (defined in (15)) finds a potential improving policy π ∈ U , and its long-term measurement vector c(π). If the c(π) does not get strictly closer to the ω := Proj(x), then we are done, and x is the optimal value. Otherwise, the c(π) is added into the active set, and the minor cycle is run to eliminate policies whose measurement vectors are affinely dependent.\\nLine 6 to line 13 contains the minor cycle, which is the same as the original Wolfe’s method (except in line 6, we find affine minimizer with respect to ω). The elimination is executed as a series of affine projections. The minor cycle terminates if active set Sc is affinely independent. Though the interleaving of major and minor cycles oscillate the size of active set Sc, the minor cycles keep |Sc| an affinely independent set, and is terminated whenever Sc contains a single element. Therefore at the start of any major cycle, the size of the active set satisfies |Sc| ∈ [0,m + 1]. More background about the minor cycle in Wolfe’s method is provided in Appendix A.2.\\nConstruction of RL Oracle The construction of our RL oracle can use any off-the-shelf RL algorithm that maximizes a scalar reward. For any given λ ∈ Rm, we define any algorithm that finds a policy minimizing the linear function λT c(·) as a RL oracle, that is\\nRL oraclep(λ) ∈ arg min π∈Π λT c(π). (12)\\nRecall that standard RL algorithm receives a scalar reward after each state transition, instead of the long-term measurement vector c(π) ∈ Rm. We then use the following linear property to reformulate the right hand side of (12) to a standard RL problem\\narg min π∈Π λT c(π) = arg min π∈Π λTE( T∑ t=0 γtct) = − arg max π∈Π E( T∑ t=0 γt(−λT ct)). (13)\\nThis shows that if we consider the Markov decision process with the same state, action, and transition probability, and construct a scalar reward r := (−λT ct), then any policy that maximizes the expected r is a linear minimizer of (12). Therefore any RL algorithm that best suits the underlying problems can be used to construct a RL oracle.\\nCertifying constraint satisfaction amounts to evaluate the measurement vector of the current policy. This is handy in online settings, where simulations can be used to evaluate the measurement vector of the policy directly. Otherwise, in batch settings, various off-policy evaluation methods, such as importance sampling (Precup, 2000; Precup et al., 2001) or doubly robust (Jiang & Li, 2016; Dudı́k et al., 2011), can be used to evaluate the policy.\\nRL oraclec(λ) := c(arg min π∈Π λT c(π)) = arg min c(π),π∈Π λT c(π). (14)\\nTo simplify notation, we assume a RL Oracle returns a policy as well as its measurement vector\\nRL Oracle(λ) := π, c(π) = RL oraclep(λ), RL oraclec(λ) (15)\\nFinding Extended Affine Minimizer The process AffineMinimizer(S,x) returns the (y,α) the affine minimizer of S with respect to x where y is the affine minimizer and α := {αs|∀s ∈ Sc} is the set of coefficient expressing y as an affine combination of points in S, that is y =∑\\ns∈Sc αss, where αs is the weight associated with s. The process AffineMinimizer(S,x) can be straightforwardly implemented using linear algebra. Wolfe (1976) also provides a more efficient implementation that uses a triangular array representation of the active set.\"\n },\n {\n \"heading\": \"4.4 CONVERGENCE AND SPARSITY\",\n \"text\": \"In this section, we analyze the convergence and complexity of the proposed C2RL method (Algo. 1). We first show that approximation error of C2RL strictly decreases between any two major cycle steps\\nand it converges in O(1/t) rate. Then we show our method ensures convergence of arbitrary RL algorithm, including those searching for deterministic policies. Moreover, concerning the memory complexity, we show that maintaining an active policy set ofm+1 is worst case optimal to ensure the convergence of arbitrary RL algorithm. Therefore, the proposed C2RL indeed achieves the optimal sparsity for the found policy, making it favorable for large-scale usage.\\nThe main difference between the convergence analysis of C2RL and Wolfe’s method is the addition of the projection step. Intuitively, at each major step, if we are making a significant progress toward the projected point, then the distance to the convex set is decreased by at least the same amount.\\nTime Complexity. In our analysis, we consider the approximation error as defined in (5). We use superscript t to denote the variable in t-th major cycle before executing any minor cycle. To simplify notions, we let xt := c(µt) and st := c(πt). When discussing one step with t fixed, let yi denote the affine minimizer found in i-th minor cycle (line 6 of Algo. 1). We first show that the C2RL method strictly reduces approximation error between two calls of the RL oracle.\\nTheorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step t, we have err(µt+1) < err(µt). That is, the measurement vector of µt found by the C2RL method gets strictly closer to the convex set Ω after major cycle step.\\nThe proof is provided in Appendix B. The idea is to consider the distance between xt and ωt. When the major cycle has no minor cycle, the non-terminal condition and the affine minimizer property implies dist2(xt+1,ωt) < dist2(xt,ωt). Otherwise we show that the first minor cycle strictly reduces the dist2(xt,ωt) by moving along the segment joining x and y, and the subsequent minor cycle cannot increase it. Since ωt ∈ Ω, we conclude err(xt+1) ≤ dist2(xt+1, ωt) < dist2(xt, ωt) = err(xt), and the approximation error strictly decreases.\\nGiven the approximation error strictly decreases, Wolfe’s method for minimum norm point can be shown to terminate finitely (Wolfe, 1976). However, this finitely terminating property does not hold for our algorithm. Since a changed ωt may yield a lower distance to the same active set Stc, the active set may stay unchanged across major cycles (see Figure 2 Middle for an example). Therefore we establish the convergence of the C2RL method by the following theorem.\\nTheorem 4.2 (Convergence in Approximation Error). For t ≥ 1, the mixed policy µt found by the C2RL method satisfies\\nerr(µt) ≤ 16Q2/(t+ 2), (16)\\nwhere Q := maxµ∈∆(U) ||c(µ)|| is the maximum norm of a measurement vector.\\nThe proof is provided in Appendix C, which relies on the following two lemmas. We briefly discuss the main idea here. Define major cycle steps with at most one minor cycle as ”non-drop step” and major cycle steps with more than one minor cycles as ”drop steps”. We show that in each non-drop step, Algorithm 1 is guaranteed to make enough progress in the following lemma.\\nLemma 4.3. For a non-drop step in C2RL method, we have err(µt)−err(µt+1) ≥ err2(µt)/8Q2.\\nThough this does not hold for drop steps, we can bound the frequency of drop steps by the following.\\nLemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than t/2.\\nSince the approximation error strictly decreases (Thm. 4.1), and in more than half of the major cycles steps, the C2RL method makes significantly progress. The Thm. (4.2) can then be proved using an induction argument (Appendix C).\\nConvergence with Arbitrary RL Algo. The convergence of the C2RL method when used with RL algorithms that search for deterministic policies, such as DQN, DDPG and variants, is indeed straightforward. In (8), though each time the oracle yields a vertex, the FW-type algorithms indeed optimize over the polytope formed by these vertices. Then since citetaltman1999constrained shows that any c(·) achievable can be achieved by some mixed deterministic policies, we conclude that if the underlying problem is feasible, then our C2RL method is able to find a feasible policy.\\nMemory Complexity We then discuss the sparsity of mixed policy for constrained RL problem. We give a constructive proof in Appendix D to show that to ensure convergence for RL algorithms that search for deterministic policies, storing m+ 1 policies is required in the worst case.\\nTheorem 4.5 (Memory Complexity Bound). For an constrained RL problem with m-dimensional measurement vector, in the worst case, a mixed policy needs to randomize among m+ 1 individual policies to ensure convergence of RL oracles that search for deterministic policies.\\nSince the minor cycles in the C2RL method eliminate policies with affinely dependent measurement vectors, after the termination of minor cycles, the size of the active set is at most m+ 1. That is, the policy found by the C2RL method requires randomization among no more than m + 1 individual policies. Therefore the proposed C2RL indeed achieves the optimal sparsity in the worst case, making it favorable for large-scale usage. Corollary 4.5.1. The C2RL method that randomizes among at most m + 1 policies is worst-case optimal to ensure convergence of any RL oracle.\"\n },\n {\n \"heading\": \"5 EXPERIMENTS\",\n \"text\": \"We evaluate the performance of C2RL in a grid-world navigation task (Fig. 1), and demonstrate its ability to efficiently find sparse policy. In this Risky Mars Rover environment, the agent is required to navigate from the starting point to the goal point, by moving to one of the four neighborhood cells at each step. The episodes terminate when the goal point is reached or after 300 steps. To enforce robustness, we add a risky area to indicate the dangerous states. The agent receives a measurement vector to indicate the steps it takes (0.1 for every step), and whether it stays in the risky area (0.1 for every risky step, and 0 otherwise), with discount factor γ = 0.99. We constrain the agent to reach the goal point with expected cumulative steps measure within 1.1 and the expected cumulative risky steps within 0.05. Note that by design, the shortest path from the starting point to the goal point does not satisfy the constraint. This is common in practice, as robustness typically evolves trade-off between the reward and the constraint satisfaction.\\nThe proposed C2RL method is compared with approachability-based policy optimization (ApproPO) (Miryoosefi et al., 2019) and with reward constrained policy optimization (RCPO) (Tessler et al., 2018). ApproPO solves the same convex constrained RL problem by using an RL oracle to play against a no-regret online learner (Hazan et al., 2008; Zinkevich, 2003). Since ApproPO and C2RL both use a RL oracle, ApproPO is a natural baseline to be compared with our method. Besides, we also compare with RCPO, which takes a Lagrangian approach to incorporate the constraints as a penalty signal into the reward. Using an advantage actor critic (A2C) Mnih et al. (2016), RCPO has been shown to converge to a fixed point. For a fair comparison, C2RL and ApproPO uses an A2C agent as the RL oracle, with the same hyperparameter as used in RCPO. The approximation errors are compared after training for the same number of samples.\\nNote that the C2RL method does not introduce any extra hyper-parameter. For ApproPO and RCPO, they require extra hyper-parameter for the initialization and learning rate of a variable equivalent to our λ in the outer loop. This is because our approach does not rely on the online learning framework, and therefore there is no need to tune the initialization and learning rate for our λ and ease the usage.\\nWe first showcase the consequences of our theoretical results using an optimal RL oracle. For any x ∈ Rm, an optimal policy can be easily found via Dijkstra’s algorithm. If multiple optimal paths exist, one is randomly picked to form a deterministic policy.\\nUsing this as an optimal RL oracle, the convergence property of C2RL and ApproPo are compared. Figure 2 Middle shows the value of policies c(µt) found after each call to the oracle. In Figure 2 Right, when approaching the boundary of the feasible set, the iterations of approachability-based methods start to zigzag. Since C2RL contains a minor cycle to re-optimize the weights among the active set, C2RL progresses quickly to reach the exact optimal solution. In Figure 3 Left, the approximation error is shown for 300 calls of the optimal RL oracle.\\nWe then compare C2RL, ApproPO and RCPO using the same A2C agent (details of the model structures and hyper-parameters are provided in Appendix E). We run each algorithm for 50 times, and each run for a maximum of 100 thousands of samples. The mean and standard deviation of the results are presented in Figure 3. The original paper of ApproPO suggests using a cache to save memory, and the memory requirement of this variant is also presented. Figure 3 demonstrates that C2RL converges to an optimal policy faster than previous methods, and a sparse combination of individual policies is maintained throughout the iteration process.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"In this paper, we introduce C2RL, an algorithm to solve RL problems under orthant or convex constraints. Our method reduces the constrained RL problem to a distance minimization problem, and a novel variant of Frank-Wolfe type algorithm is proposed to solve this. Our method comes with rigorous theoretical guarantees and does not introduce any extra hyper-parameter. To find an -approximation solution, C2RL takes O(1/ ) calls of any RL oracle and ensures convergence to work with arbitrary RL algorithm. Moreover, C2RL strictly reduces the approximation error between consecutive calls of RL oracle, and form-dimensional constraints, the memory requirement is reduced from storing infinitely many policies (O(1/ )) to storing at most constantly many (m+1) polices. We further show that the constant is worst-case optimal to ensure the convergence for RL algorithms that search for deterministic policies. Experimentally, we demonstrate that the proposed C2RL method finds sparse solution efficiently, and outperforms previous methods.\"\n },\n {\n \"heading\": \"A MORE ON FRANK-WOLFE-TYPE ALGORITHMS\",\n \"text\": \"A.1 STANDARD FRANK-WOLFE ALGORITHM\\nAlgorithm 2 Frank-Wolfe algorithm (Frank et al., 1956) Input: obj. f : Y 7→ R, oracle O(·), init. x0 ∈ Y\\n1: for t=1, 2, 3 . . . , T do 2: s← Oracle(∇f(xt−1)) = arg mins∈{s1,...,sn} s\\nT∇f(xt−1) 3: xt ← (1− ηt)xt−1 + ηts , for ηt := 2t+2 4: end for 5: return xT\\nFor a convex function f : X 7→ R the Frank-Wolfe algorithm (FW) solves the constrained optimization problem over a compact and convex set X . The standard FW is known to have a sublinear convergence rate, and various methods are proposed to improve the performance. For example, when the underlying feasible set is a polytope, and the objective function is strongly convex, multiple variants, such as away-step FW (Wolfe, 1970; Jaggi, 2013), pairwise FW (Mitchell et al., 1974), and Wolfe’s method (Wolfe, 1976) are shown to enjoy linear convergence rate. Linear convergence under other conditions is also studied (Beck & Shtern, 2017; Garber & Hazan, 2013a;b).\\nA.2 WOLFE’S METHOD FOR MINIMUM NORM POINT\\nAlgorithm 3 Wolfe’s Method for Minimum Norm Point Initialize x ∈ P , active set S = [x] and weight λ = [1]. Output: x ∈ P that has the minimum Euclidean norm.\\n1: while true do // Major cycle 2: s← Oracle(x) // Potential improving point 3: if ||x||2 ≤ xTs+ then break 4: S ← S ∪ {s} 5: while true do // Minor cycle 6: y,α← AffineMinimizer(S) // y = arg mins∈aff(S) ||s||2 7: if αs > 0 for all s then break // y ∈ conv(S) 8: // If y /∈ conv(S), then update y to the intersection of conv(S) and segment joining x and y. Then remove points in S unnecessary for describing y. 9: θ ← mini:αi≤0 λiλi−αi // Recall λ satisfies x = ∑ s∈S λss\"\n },\n {\n \"heading\": \"10: y ← θy + (1− θ)x, λi = θαi + (1− θ)λi\",\n \"text\": \"11: S ← {si|si ∈ S and λi > 0} 12: end while 13: Update x = y and λ = α. 14: end while 15: return x\\nWolfe’s method is an iterative algorithm for finding the point with minimum Euclidean norm in a polytope, which is defined as the convex hull of a set of finitely many points.\\nThe Wolfe’s method consists of a finite number of major cycles, each of which consists of a finite number of minor cycles. At the start of each major cycle, let H(x) := {yTx = xx} be the hyperplane defined by x. If H(x) separates the polytope from the origin, then the major cycle is terminated. Otherwise, it invokes an oracle to find any point on the near side of the hyperplane. The point is then added into the active set S, and starts a minor cycle. In a minor cycle, let y be the point of smallest norm in of the affine hull aff(S). If y is in the relative interior of the convex hull conv(S), then x is updated to y and the minor cycle is terminated. Otherwise, y is updated to the nearest point to y on the line segment conv(S) ∩ [x,y]. Thus y is updated to a boundary point of conv(S), and any point that is not on the face of conv(S) in which y lies is deleted. The minor cycles are executed repeatedly until S becomes a corral, that is, a set\\nwhose affine minimizer lies inside its convex hull. Since a set of one point is always a corral, the minor cycles is terminated after a finite number of runs.\"\n },\n {\n \"heading\": \"B PROOF OF THEOREM 4.1\",\n \"text\": \"Theorem 4.1 (Approximation Error Strictly Decreases). For any non-terminal step t, we have err(µt+1) < err(µt). That is, the measurement vector of µt found by the C2RL method gets strictly closer to the convex set Ω after major cycle step.\\nProof. If the current step is a major cycle with no minor cycle, then xt+1 is the affine minimizer of aff(S ∪ {st}) with respect to ωt. Then the affine minimizer property implies (st − xt+1)(xt+1 − ωt) = 0. Since iteration does not terminate at step t, we have (xt − ωt)T (xt − st) > 0, and therefore xt+1 not equal to xt. Then xt+1 is the unique affine minimizer implies fΩ(xt+1) = minω∈Ω ||xt+1 − ω||2 ≤ ||xt+1 − ωt||2 < ||xt − ωt||2 = fΩ(xt). Otherwise the current step contains one or more minor cycles. In this case, we show that the first minor cycle strictly reduces the approximation error, and the (possibly) following minor cycles cannot increase it. For the first minor cycle, the affine minimizer y0 of aff(S ∪ {st}) with respect to ωt is outside conv(S ∪ {st}). Let z = θy0 + (1 − θ)xt be the intersection of conv(S ∪ {st}) and segment joining x and y. Let V0 := St and Vi denote the active set after the i-th minor cycle. Then since y1 is the affine minimizer of V1 with respect to ωt, we have\\n||z − ωt|| = ||θy0 + (1− θ)xt − ωt|| ≤ θ||y0 − ωt||+ (1− θ)||xt − ωt|| < ||xt − ωt||, (17)\\nwhere the second step uses the triangle inequality and the last step follows since the segment xty0 intersects the interior of conv(S∪{st}), and the distance to ωt strictly decreases along this segment. Therefore the point z found by first minor cycle satisfies\\nfΩ(z) = min ω∈Ω ||z − ω||2 ≤ ||z − ωt||2 < ||xt − ωt|| = fΩ(xt). (18)\\nHence h(y1) < h(xt), and the first minor cycle strictly decreases the approximation error. By a similar argument, in subsequent minor cycles the approximation error cannot be increased. However, after the first minor cycle, the iterating point may already at the intersection point and the strict inequality in last step of Eq. 17 need to be replaced by non-strict inequality.\\nTherefore any major cycle either finds an improving point and continue, or enters minor cycles where the first minor cycle finds an improving point, and the subsequent minor cycles does not increase the distance. Adding both side of fΩ(xt+1) < fΩ(xt) by fΩ(x∗) and we have the approximation error h(xt+1) < h(xt) strictly decreases.\"\n },\n {\n \"heading\": \"C PROOF OF THEOREM 4.2\",\n \"text\": \"We first prove the Theorem 4.2, using Lemma 4.3 and Lemma 4.4. Then we present the proof of the lemmas.\\nTheorem 4.2 (Convergence in Approximation Error). For t ≥ 1, the mixed policy µt found by the C2RL method satisfies\\nerr(µt) ≤ 16Q2/(t+ 2). (19)\\nwhere Q := maxµ∈∆(U) ||c(µ)|| is the maximum norm of a measurement vector.\\nProof. Since Lemma 4.4 shows that drop steps are no more than half of total major cycle steps, and Theorem 4.1 guarantees these drop steps reducing the approximation error, we can safely skip these step, and re-index the step numbers to include non-drop steps only using k.\\nFor these non-drop steps, we claim that err(µk) ≤ 8Q2/(k + 1). Using Lemma 4.3, we prove the convergence rate using induction. We first bound the error of any err(µk). For any k ≥ 1\\nerr(µk) = dist2(c(µk),Ω)− dist2(c(µ∗),Ω) (20) = 1/2||c(µk)− ProjΩ(c(µ k))||2 − 1/2||c(µ∗)− ProjΩ(c(µ ∗))||2 (21)\\n≤ 1/2(||c(µk)||2 + ||ProjΩ(c(µ k))||2 − ||c(µ∗)||2 − ||ProjΩ(c(µ ∗))||2) (22) ≤ ||c(µk)||2 − ||c(µ∗)||2 (23) ≤ ||c(µk)||2 (24) ≤ Q2, (25)\\nwhere Eq. 21 uses the definition of our squared Euclidean distance function. Eq. 22 follows from triangle inequality, and Eq. 23 is by the contractive property of the Euclidean distance.\\nWhen k = 1, the Eq. 25 established the based case. Now for k ≥ 1, assume that err(µk) ≤ 8Q2/(k + 1) for k ≥ 1, then Lemma 4.3 gives err(µk+1) ≤ err(µk) − err2(µk)/8Q2. Since the quadratic function of the right hand side is monotonically increasing on (−∞, 4Q2], using the inductive hypothesis\\nerr(µk+1) ≤ err(µk)− err2(µk)/8Q2 ≤ 8Q2/(k + 1)− 8Q2/(k + 1)2 ≤ Q2/(k + 2) (26)\\nSince for t steps of major cycle steps, the number of non-drop steps k > t/2, we conclude that err(µt) ≤ 16Q2/(t+ 2).\\nThen we prove the lemmas.\\nLemma 4.3. For a non-drop step, we have err(µt)− err(µt+1) ≥ err2(µt)/8Q2.\\nProof. The non-drop step contains either no minor cycle or one minor cycle. We first consider the no minor cycle case.\\nIf a major cycle contains no minor cycle, then xt+1 is the affine minimizer of the S ∪ {st}.\\nerr(µt)− err(µt+1) = dist2(xt,Ω)− dist2(xt+1,Ω) (27) = 1/2(||xt − ωt||2 −min\\nω∈Ω ||xt+1 − ω||2) (28)\\n≥ 1/2(||xt − ωt||2 − ||xt+1 − ωt||2) (29) = 1/2(||xt − ωt||2 + ||xt+1 − ωt||2 − 2||xt+1 − ωt||2) (30) = 1/2(||xt − ωt||2 + ||xt+1 − ωt||2 − 2(xt − ωt)T (xt+1 − ωt)) (31) = 1/2(||xt − xt+1||2), (32)\\nwhere the equation (31) follows from the affine minimizer property Eq. (11). For ||xt − xt+1|| in the last equation, and ∀q ∈ aff(S ∪ {st}), we have\\n||xt − xt+1|| ≥ ||xt − xt+1|| ||x t||+ ||q||\\n2Q ( Definition of Q) (33)\\n≥ ||xt − xt+1|| ||x t − q|| 2Q\\n( Triangle inequality) (34)\\n≥ 1 2Q (xt − xt+1)(xt − q) ( Cauchy-Schwarz inequality) (35)\\n= 1\\n2Q (xt − ωt)(xt − q) ( Affine minimizer property). (36)\\nThen it suffices to show that (xt − ωt)(xt − q) ≥ err(µt).\\nSince Ω is a convex set, the squared Euclidean distance function dist2(x,Ω) is convex for x, which implies\\ndist2(xt,Ω) + (q − xt)∇dist2(xt,Ω) ≤ dist2(q,Ω). (37)\\nPutting in∇dist2(xt,Ω) = (xt−ProjΩ(xt)) = (xt−ωt), we get (xt−ωt)(xt−q) ≥ err(µt), which together with Eq. 32 and Eq. 36 concludes that for non-drop step with no minor cycles, we have err(µt)− err(µt+1) ≥ err2(µt)/8Q2. For non-drop step with one minor cycle, we use the Theorem 6 of (Chakrabarty et al., 2014). By a linear translation of adding all points with −ωt, it gives\\n||xt − ωt||2 − ||xt+1 − ωt||2 ≥ ((xt − ωt)(xt − q))2/8Q2. (38) Then applying the same argument as Eq. 37, and we finished our proof.\\nLemma 4.4. After t major cycle steps of C2RL method, the number of drop steps is less than t/2.\\nProof. Recall that at the termination of a minor cycle, the size of the active set |Sc| ∈ [1,m]. Since in each major cycle steps, the size of active set St increases by one, and each drop step reduces the size of St by at least one, the number of drop steps is always less than half of total number of the major cycle steps.\"\n },\n {\n \"heading\": \"D PROOF OF THEOREM 4.5\",\n \"text\": \"Theorem 4.5 (Memory Complexity Bound). For an constrained RL problem with m-dimensional measurement vector, in the worst case, a mixed policy needs to randomize among m+ 1 individual policies to ensure convergence of RL oracles that search for deterministic policies.\\nProof. We give a constructive proof. Consider a m-dimensional vector-valued MDP with a single state, m + 1 actions, and c(ai) := ei is the unit vector of i-th dimension for i ∈ [1,m], and c(am+1) := 0, and the episode terminates after 1 steps. The constrained RL problem is to find a policy whose measurement vector lies in the convex set of a single point {1/2m}. By linear programming, it is clear that the only feasible mixed deterministic policy is to select am+1 with 1/2 probability, and the rest m actions with 1/2m probability; i.e. the unique feasible policy to this problem has an active set containing m + 1 deterministic policies. Therefore any method randomize among less than m + 1 individual policies does not ensure convergence when used with RL algorithms searching for deterministic policies.\"\n },\n {\n \"heading\": \"E ADDITIONAL EXPERIMENT DETAILS\",\n \"text\": \"All the methods use the same A2C agent. The input is the one-hot encoded current position index. The A2C is the standard fully connected multi-layer perceptron with ReLU activation function. The actor and critic share the internal representation and have their only final layer. Both actor and critic networks use Adam optimizer with learning rate set to 1e−2. The network is as follows\\nActor Critic Input layer One-hot encoded state index (dim=54)\\nHidden layer Linear(in=54, out=128, act=”relu”) Output layer Linear(in=128, out=4, act=”relu”) Linear(in=128, out=1, act=”relu”) Output name Action score State value\\nFor ApproPO, the constant κ for projection convex set to convex cone is set to be 20. The θ is initialized to 0. Following the original paper.\\nFor RCPO, the learning rate of its λ is set to 2.5e−5, and its λ is initialized to 0 and updated by online gradient descent with learning rate set to 1, as used by the original paper.\\nThe proposed C2RL introduces no extra hyper-parameters, and has nothing to report.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":196,"cells":{"paper_id":{"kind":"string","value":"SP:df5fec4899d97f7d5df259a013f467e038895669"},"summaries":{"kind":"list like","value":["The paper proposes a post-hoc uncertainty tuning pipeline for Bayesian neural networks. After getting the point estimate, it adds extra dimensions to the weight matrices and hidden layers, which has no effect on the network output, with the hope that it would influence the variance of the original network weights under the Laplacian approximation. More specifically, it tunes the extra weights by optimizing another objective borrowed from the non-Bayesian robust learning literature, which encourages low uncertainty over real (extra, validation) data, and high uncertainty over manually constructed, out-of-distribution data."],"string":"[\n \"The paper proposes a post-hoc uncertainty tuning pipeline for Bayesian neural networks. After getting the point estimate, it adds extra dimensions to the weight matrices and hidden layers, which has no effect on the network output, with the hope that it would influence the variance of the original network weights under the Laplacian approximation. More specifically, it tunes the extra weights by optimizing another objective borrowed from the non-Bayesian robust learning literature, which encourages low uncertainty over real (extra, validation) data, and high uncertainty over manually constructed, out-of-distribution data.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly “train” the uncertainty in a decoupled way to the prediction itself. To this end we introduce uncertainty units for Laplaceapproximated networks: Hidden units with zero weights that can be added to any pre-trained, point-estimated network. Since these units are inactive, they do not affect the predictions. But their presence changes the geometry (in particular the Hessian) of the loss landscape around the point estimate, thereby affecting the network’s uncertainty estimates under a Laplace approximation. We show that such units can be trained via an uncertainty-aware objective, making the Laplace approximation competitive with more expensive alternative uncertaintyquantification frameworks."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Felix Dangel","Frederik Kunstner","Philipp Hennig"],"title":"BackPACK: Packing more into Backprop","venue":"In ICLR,","year":2020},{"authors":["Yarin Gal","Zoubin Ghahramani"],"title":"Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning","venue":"In ICML,","year":2016},{"authors":["Mark N Gibbs"],"title":"Bayesian gaussian processes for regression and classification","venue":"Ph. D. Thesis, Department of Physics, University of Cambridge,","year":1997},{"authors":["Matthias Hein","Maksym Andriushchenko","Julian Bitterwolf"],"title":"Why ReLU Networks Yield Highconfidence Predictions Far Away from the Training Data and How to Mitigate the Problem","venue":null,"year":2019},{"authors":["Dan Hendrycks","Thomas Dietterich"],"title":"Benchmarking Neural Network Robustness to Common Corruptions and Perturbations","venue":null,"year":2019},{"authors":["Dan Hendrycks","Kevin Gimpel"],"title":"A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks","venue":"In ICLR,","year":2017},{"authors":["Dan Hendrycks","Mantas Mazeika","Thomas Dietterich"],"title":"Deep Anomaly Detection with Outlier Exposure","venue":"In ICLR,","year":2019},{"authors":["Tom Heskes"],"title":"On “Natural” Learning and Pruning in Multilayered Perceptrons","venue":"Neural Computation,","year":2000},{"authors":["Gao Huang","Zhuang Liu","Laurens Van Der Maaten","Kilian Q Weinberger"],"title":"Densely connected convolutional networks","venue":null,"year":2017},{"authors":["Agustinus Kristiadi","Matthias Hein","Philipp Hennig"],"title":"Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks","venue":"In ICML,","year":2020},{"authors":["Balaji Lakshminarayanan","Alexander Pritzel","Charles Blundell"],"title":"Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles","venue":"In NIPS,","year":2017},{"authors":["David JC MacKay"],"title":"The Evidence Framework Applied to Classification Networks","venue":"Neural computation,","year":1992},{"authors":["David JC MacKay"],"title":"A Practical Bayesian Framework For Backpropagation Networks","venue":"Neural computation,","year":1992},{"authors":["Andrey Malinin","Mark Gales"],"title":"Predictive Uncertainty Estimation via Prior Networks","venue":"In NIPS,","year":2018},{"authors":["Andrey Malinin","Mark Gales"],"title":"Reverse KL-Divergence Training of Prior Networks: Improved Uncertainty and Adversarial Robustness","venue":"In NIPS,","year":2019},{"authors":["James Martens","Roger Grosse"],"title":"Optimizing Neural Networks With Kronecker-Factored Approximate Curvature","venue":"In ICML,","year":2015},{"authors":["Alexander Meinke","Matthias Hein"],"title":"Towards Neural Networks that Provably Know when They don’t Know","venue":"In ICLR,","year":2020},{"authors":["Sebastian W Ober","Carl Edward Rasmussen"],"title":"Benchmarking the Neural Linear Model for Regression","venue":"arXiv preprint arXiv:1912.08416,","year":2019},{"authors":["Yaniv Ovadia","Emily Fertig","Jie Ren","Zachary Nado","David Sculley","Sebastian Nowozin","Joshua Dillon","Balaji Lakshminarayanan","Jasper Snoek"],"title":"Can You Trust Your Model’s Uncertainty? Evaluating Predictive Uncertainty under Dataset Shift","venue":"NeurIPS,","year":2019},{"authors":["Hippolyt Ritter","Aleksandar Botev","David Barber"],"title":"Online Structured Laplace Approximations for Overcoming Catastrophic Forgetting","venue":"In NIPS,","year":2018},{"authors":["Hippolyt Ritter","Aleksandar Botev","David Barber"],"title":"A Scalable Laplace Approximation for Neural Networks","venue":"In ICLR,","year":2018}],"string":"[\n {\n \"authors\": [\n \"Felix Dangel\",\n \"Frederik Kunstner\",\n \"Philipp Hennig\"\n ],\n \"title\": \"BackPACK: Packing more into Backprop\",\n \"venue\": \"In ICLR,\",\n \"year\": 2020\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 ICML,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Mark N Gibbs\"\n ],\n \"title\": \"Bayesian gaussian processes for regression and classification\",\n \"venue\": \"Ph. D. Thesis, Department of Physics, University of Cambridge,\",\n \"year\": 1997\n },\n {\n \"authors\": [\n \"Matthias Hein\",\n \"Maksym Andriushchenko\",\n \"Julian Bitterwolf\"\n ],\n \"title\": \"Why ReLU Networks Yield Highconfidence Predictions Far Away from the Training Data and How to Mitigate the Problem\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Dan Hendrycks\",\n \"Thomas Dietterich\"\n ],\n \"title\": \"Benchmarking Neural Network Robustness to Common Corruptions and Perturbations\",\n \"venue\": null,\n \"year\": 2019\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\": \"In ICLR,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Dan Hendrycks\",\n \"Mantas Mazeika\",\n \"Thomas Dietterich\"\n ],\n \"title\": \"Deep Anomaly Detection with Outlier Exposure\",\n \"venue\": \"In ICLR,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Tom Heskes\"\n ],\n \"title\": \"On “Natural” Learning and Pruning in Multilayered Perceptrons\",\n \"venue\": \"Neural Computation,\",\n \"year\": 2000\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\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Agustinus Kristiadi\",\n \"Matthias Hein\",\n \"Philipp Hennig\"\n ],\n \"title\": \"Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks\",\n \"venue\": \"In ICML,\",\n \"year\": 2020\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 NIPS,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"David JC MacKay\"\n ],\n \"title\": \"The Evidence Framework Applied to Classification Networks\",\n \"venue\": \"Neural computation,\",\n \"year\": 1992\n },\n {\n \"authors\": [\n \"David JC MacKay\"\n ],\n \"title\": \"A Practical Bayesian Framework For Backpropagation Networks\",\n \"venue\": \"Neural computation,\",\n \"year\": 1992\n },\n {\n \"authors\": [\n \"Andrey Malinin\",\n \"Mark Gales\"\n ],\n \"title\": \"Predictive Uncertainty Estimation via Prior Networks\",\n \"venue\": \"In NIPS,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Andrey Malinin\",\n \"Mark Gales\"\n ],\n \"title\": \"Reverse KL-Divergence Training of Prior Networks: Improved Uncertainty and Adversarial Robustness\",\n \"venue\": \"In NIPS,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"James Martens\",\n \"Roger Grosse\"\n ],\n \"title\": \"Optimizing Neural Networks With Kronecker-Factored Approximate Curvature\",\n \"venue\": \"In ICML,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Alexander Meinke\",\n \"Matthias Hein\"\n ],\n \"title\": \"Towards Neural Networks that Provably Know when They don’t Know\",\n \"venue\": \"In ICLR,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Sebastian W Ober\",\n \"Carl Edward Rasmussen\"\n ],\n \"title\": \"Benchmarking the Neural Linear Model for Regression\",\n \"venue\": \"arXiv preprint arXiv:1912.08416,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yaniv Ovadia\",\n \"Emily Fertig\",\n \"Jie Ren\",\n \"Zachary Nado\",\n \"David Sculley\",\n \"Sebastian Nowozin\",\n \"Joshua Dillon\",\n \"Balaji Lakshminarayanan\",\n \"Jasper Snoek\"\n ],\n \"title\": \"Can You Trust Your Model’s Uncertainty? Evaluating Predictive Uncertainty under Dataset Shift\",\n \"venue\": \"NeurIPS,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Hippolyt Ritter\",\n \"Aleksandar Botev\",\n \"David Barber\"\n ],\n \"title\": \"Online Structured Laplace Approximations for Overcoming Catastrophic Forgetting\",\n \"venue\": \"In NIPS,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Hippolyt Ritter\",\n \"Aleksandar Botev\",\n \"David Barber\"\n ],\n \"title\": \"A Scalable Laplace Approximation for Neural Networks\",\n \"venue\": \"In ICLR,\",\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":null,"text":"Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly “train” the uncertainty in a decoupled way to the prediction itself. To this end we introduce uncertainty units for Laplaceapproximated networks: Hidden units with zero weights that can be added to any pre-trained, point-estimated network. Since these units are inactive, they do not affect the predictions. But their presence changes the geometry (in particular the Hessian) of the loss landscape around the point estimate, thereby affecting the network’s uncertainty estimates under a Laplace approximation. We show that such units can be trained via an uncertainty-aware objective, making the Laplace approximation competitive with more expensive alternative uncertaintyquantification frameworks."},{"heading":"1 INTRODUCTION","text":"The point estimates of neural networks (NNs)—constructed as maximum a posteriori (MAP) estimates via (regularized) empirical risk minimization—empirically achieve high predictive performance. However, they tend to underestimate the uncertainty of their predictions, leading to an overconfidence problem (Hein et al., 2019), which could be disastrous in safety-critical applications such as autonomous driving. Bayesian inference offers a principled path to overcome this issue. The goal is to turn a “vanilla” NN into a Bayesian neural network (BNN), where the posterior distribution over the network’s weights are inferred via Bayes’ rule and subsequently taken into account when making predictions. Since the cost of exact posterior inference in a BNN is often prohibitive, approximate Bayesian methods are employed instead.\nLaplace approximations (LAs) are classic methods for such a purpose (MacKay, 1992b). The key idea is to obtain an approximate posterior by “surrounding” a MAP estimate of a network with a Gaussian, based on the loss landscape’s geometry around it. A standard practice in LAs is to tune a single hyperparameter—the prior precision—which is inflexible (Ritter et al., 2018b; Kristiadi et al., 2020). Here, we aim at improving the flexibility of uncertainty tuning in LAs. To this end, we introduce Learnable Uncertainty under Laplace Approximations (LULA) units, which are hidden units associated with a zeroed weight. They can be added to the hidden layers of any MAP-trained network. Because they are inactive, such units do not affect the prediction of the underlying network. However, they can still contribute to the Hessian of the loss with respect to the parameters, and hence induce additional structures to the posterior covariance under a LA. LULA units can be trained via an uncertainty-aware objective (Hendrycks et al., 2019; Hein et al., 2019, etc.), such that they improve the predictive uncertainty-quantification (UQ) performance of the Laplace-approximated BNN. Figure 1 demonstrates trained LULA units in action: They improve the UQ performance of a standard LA, while keeping the MAP predictions in both regression and classification tasks.\nIn summary, we (i) introduce LULA units: inactive hidden units for uncertainty tuning of a LA, (ii) bring a robust training technique from non-Bayesian literature for training these units, and (iii) show empirically that LULA-augmented Laplace-approximated BNNs can yield better UQ performance compared to both previous tuning techniques and contemporary, more expensive baselines."},{"heading":"2 BACKGROUND","text":""},{"heading":"2.1 BAYESIAN NEURAL NETWORKS","text":"Let f : Rn × Rd → Rk defined by (x, θ) 7→ f(x; θ) be an L-layer neural network. Here, θ is the concatenation of all the parameters of f . Suppose that the size of each layer of f is given by the sequence of (nl ∈ Z>0)Ll=1. Then, for each l = 1, . . . , L, the l-th layer of f is defined by\na(l) := W (l)h(l−1) + b(l), with h(l) := { ϕ(a(l)) if l < L a(l) if l = L ,\n(1)\nwhere W (l) ∈ Rnl×nl−1 and b(l) ∈ Rnl are the weight matrix and bias vector of the layer, and ϕ is a component-wise activation function. We call the vector h(l) ∈ Rnl the l-th hidden units of f . Note that by convention, we consider n0 := n and nL := k, while h(0) := x and h(L) := f(x; θ).\nFrom the Bayesian perspective, the ubiquitous training formalism of neural networks amounts to MAP estimation: The empirical risk and the regularizer are interpretable as the negative loglikelihood under an i.i.d. dataset D := {xi, yi}mi=1 and the negative log-prior, respectively. That is, the loss function is interpreted as\nL(θ) := − m∑ i=1 log p(yi | f(xi; θ))− log p(θ) = − log p(θ | D) . (2)\nIn this view, the de facto weight decay regularizer amounts to a zero-mean isotropic Gaussian prior p(θ) = N (θ | 0, λ−1I) with a scalar precision hyperparameter λ. Meanwhile, the usual softmax and quadratic output losses correspond to the Categorical and Gaussian distributions over yi in the case of classification and regression, respectively.\nMAP-trained neural networks have been shown to be overconfident (Hein et al., 2019) and BNNs can mitigate this issue (Kristiadi et al., 2020). They quantify epistemic uncertainty by inferring the full posterior distribution of the parameters θ (instead of just a single point estimate in MAP training). Given that p(θ | D) is the posterior, then the prediction for any test point x ∈ Rn is obtained via marginalization\np(y | x,D) = ∫ p(y | f(x; θ)) p(θ | D) dθ , (3)\nwhich captures the uncertainty encoded in the posterior."},{"heading":"2.2 LAPLACE APPROXIMATIONS","text":"In deep learning, since the exact Bayesian posterior is intractable, approximate Bayesian inference methods are used. An important family of such methods is formed by LAs. Let θMAP be the minimizer of (2), which corresponds to a mode of the posterior distribution. A LA locally approximates the posterior using a Gaussian\np(θ | D) ≈ N (θ | θMAP, Σ) := N (θ | θMAP, (∇2L|θMAP)−1) . Thus, LAs construct an approximate Gaussian posterior around θMAP, whose precision equals the Hessian of the loss at θMAP—the “curvature” of the loss landscape at θMAP. While the covariance of a LA is tied to the weight decay of the loss, a common practice in LAs is to tune the prior precision under some objective, in a post-hoc manner. In other words, the MAP estimation and the covariance inference are thought as separate, independent processes. For example, given a fixed MAP estimate, one can maximize the log-likelihood of a LA w.r.t. the prior precision to obtain the covariance. This hyperparameter tuning can thus be thought as an uncertainty tuning.\nA recent example of LAs is the Kronecker-factored Laplace (KFL, Ritter et al., 2018b). The key idea is to approximate the Hessian matrix with the layer-wise Kronecker factorization scheme proposed by Heskes (2000); Martens & Grosse (2015). That is, for each layer l = 1, . . . , L, KFL assumes that the Hessian corresponding to the l-th weight matrix W (l) ∈ Rnl×nl−1 can be written as the Kronecker product G(l) ⊗ A(l) for some G(l) ∈ Rnl×nl and A(l) ∈ Rnl−1×nl−1 . This assumption brings the inversion cost of the Hessian down to Θ(n3l +n 3 l−1), instead of the usual Θ(n 3 l n 3 l−1) cost. The approximate Hessian can easily be computed via tools such as BackPACK (Dangel et al., 2020).\nEven with a closed-form Laplace-approximated posterior, the predictive distribution (3) in general does not have an analytic solution since f is nonlinear. Instead, one can employ Monte-Carlo (MC) integration by sampling from the Gaussian:\np(y | x,D) ≈ 1 S S∑ s=1 p(y | f(x; θs)) ; θs ∼ N (θ | θMAP, Σ) ,\nfor S number of samples. In the case of binary classification with f : Rn × Rd → R, one can use the following well-known approximation, due to MacKay (1992a):\np(y = 1 | x,D) ≈ σ ( f(x; θMAP)√ 1 + π/8 v(x) ) , (4)\nwhere σ is the logistic-sigmoid function and v(x) is the marginal variance of the network output f(x), which is often approximated via a linearization of the network around the MAP estimate:\nv(x) ≈ (∇θf(x; θ)|θMAP) > Σ (∇θf(x; θ)|θMAP) . (5)\n(This approximation has also been generalized to multi-class classifications by Gibbs (1997).) In particular, as v(x) increases, the predictive probability of y = 1 goes to 0.5 and therefore the uncertainty increases. This relationship has also been shown empirically in multi-class classifications with MC-integration (Kristiadi et al., 2020)."},{"heading":"3 LULA UNITS","text":"The problem with the standard uncertainty tuning in LAs is that the only degree-of-freedom available for performing the optimization is the scalar prior precision and therefore inflexible.1 We shall address this by introducing “uncertainty units”, which can be added on top of the hidden units of any MAP-trained network (Section 3.1) and can be trained via an uncertainty-aware loss (Section 3.2)."},{"heading":"3.1 CONSTRUCTION","text":"Let f : Rn × Rd → Rk be a MAP-trained L-layer neural network with parameters θMAP = {W (l)MAP, b (l) MAP}Ll=1. The premise of our method is simple: At each hidden layer l = 1, . . . , L − 1,\n1While one can also use a non-scalar prior precision, it appears to be uncommon in deep learning. In any case, such a element-wise weight-cost would interact with the training procedure.\nMAP. The\nadditional units are represented by the additional block at the bottom of each layer. Dashed lines correspond to the free parameters Ŵ (1), . . . , Ŵ (L−1), while dotted lines to the zero weights.\nsuppose we add ml ∈ Z≥0 additional hidden units, under the original activation function, to h(l). As a consequence, we need to augment each of the weight matrices to accommodate them.\nConsider the following construction: for each layer l = 1, . . . , L − 1 of the network, we expand W (l) and b(l) to obtain the block matrix and vector\nW̃ (l) :=\n( W\n(l) MAP 0\nŴ (l) 1 Ŵ (l) 2\n) ∈ R(nl+ml)×(nl−1+ml−1) ; b̃(l) := ( b (l) MAP\nb̂(l)\n) ∈ Rnl+ml , (6)\nrespectively, with m0 = 0 since we do not add additional units to the input. For l = L, we define\nW̃ (L) := (W (L) MAP, 0) ∈ R k×(nL−1+mL−1); b̃(L) := b (L) MAP ∈ R k ,\nso that the output dimensionality is unchanged. For brevity, we denote Ŵ (l) := (Ŵ (l)1 , Ŵ (l) 2 ). Refer to Figure 2 for an illustration and Algorithm 2 in Appendix B for a step-by-step summary. Taken together, we denote the resulting augmented network as f̃ and the resulting parameter vector as θ̃MAP ∈ Rd̃, where d̃ it the resulting number of parameters. Note that we can easily extend this construction to convolutional nets by expanding the “channel” of a hidden layer.2\nLet us inspect the implication of this construction. Here for each l = 1, . . . , L − 1, since they are zero, the upper-right quadrant of W̃ (l) deactivates the ml−1 additional hidden units in the previous layer, thus they do not contribute to the original hidden units in the l-th layer. Meanwhile, the submatrix Ŵ (l) and the sub-vector b̂(l) contain parameters for the additional ml hidden units in the l-th layer. We are free to choose the the values of these parameters since the following proposition guarantees that they will not change the output of the network (the proof is in Appendix A).\nProposition 1. Let f : Rn × Rd → Rk be a MAP-trained L-layer network parametrized by θMAP. Suppose f̃ : Rn × Rd̃ → R and θ̃MAP ∈ Rd̃ are obtained via the previous construction. For any input x ∈ Rn, we have f̃(x; θ̃MAP) = f(x; θMAP).\nSo far, it looks like all our changes to the network are inconsequential. However, they do affect the curvature of the landscape of L,3 and thus the uncertainty arising in a LA. Let θ̃ be a random variable in Rd̃ and θ̃MAP be an instance of it. Suppose we have a Laplace-approximated posterior p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃) over θ̃, where the covariance Σ̃ is the inverse Hessian of the negative log-posterior w.r.t. the augmented parameters at θ̃MAP. Then, Σ̃ contains additional dimensions (and thus in general, additional structured, non-zero uncertainty) absent in the original network, which depend on the values of the free parameters {Ŵ (l), b̂(l)}L−1l=1 .\n2E.g. if the hidden units are a 3D array of (channel × height × width), then we expand the first dimension. 3More formally: The principal curvatures of the graph of L, seen as a d-dimensional submanifold of Rd+1.\nThe implication of the previous finding can be seen clearly in real-valued networks with diagonal LA posteriors. The following proposition shows that, under such a network and posterior, the construction above will affect the output uncertainty of the original network f (the proof is in Appendix A).\nProposition 2. Suppose f : Rn × Rd → R is a real-valued network and f̃ is as constructed above. Suppose further that diagonal Laplace-approximated posteriors N (θ | θMAP,diag(σ)), N (θ̃ | θ̃,diag(σ̃)) are employed. Using the linearization (5), for any input x ∈ Rn, the variance over the output f̃(x; θ̃) is at least that of f(x; θ).\nIn summary, the construction along with Propositions 1 and 2 imply that the additional hidden units we have added to the original network are uncertainty units under LAs, i.e. hidden units that only contribute to the Laplace-approximated uncertainty and not the predictions. This property gives rise to the name Learnable Uncertainty under Laplace Approximations (LULA) units."},{"heading":"3.2 TRAINING","text":"We have seen that by adding LULA units to a network, we obtain additional free parameters that only affect uncertainty under a LA. These parameters are thus useful for uncertainty calibration. Our goal is therefore to train them to induce low uncertainty over the data (inliers) and high uncertainty on outliers—the so-called out-of-distribution (OOD) data. Specifically, this can be done by minimizing the output variance over inliers while maximizing it over outliers. Note that using variance makes sense in both the regression and classification cases: In the former, this objective directly maintains narrow error bars near the data while widen those far-away from them—cf. Figure 1 (c, top). Meanwhile, in classifications, variances over function outputs directly impact predictive confidences, as we have noted in the discussion of (4)—higher variance implies lower confidence.\nThus, following the contemporary technique from non-Bayesian robust learning literature (Hendrycks et al., 2019; Hein et al., 2019, etc.), we construct the following loss. Let f : Rn×Rd → Rk be an L-layer neural network with a MAP-trained parameters θMAP and let f̃ : Rn × Rd̃ → Rk along with θ̃MAP be obtained by adding LULA units. Denoting the dataset sampled i.i.d. from the data distribution as Din and that from some outlier distribution as Dout, we define\nLLULA(θ̃MAP) := 1 |Din| ∑ xin∈Din ν(f̃(xin); θ̃MAP)− 1 |Dout| ∑ xout∈Dout ν(f̃(xout); θ̃MAP) , (7)\nwhere ν(f̃(x); θ̃MAP) is the total variance over the k components of the network output f̃1(x; θ̃), . . . , f̃k(x; θ̃) under the Laplace-approximated posterior p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)), which can be approximated via an S-samples MC-integral\nν(f̃(x);θ̃MAP) := k∑ i=1 varp(θ̃|D) f̃i(x; θ̃)\n≈ k∑ i=1\n( 1\nS S∑ s=1 f̃i(x; θ̃s) 2\n) − ( 1\nS S∑ s=1 f̃i(x; θ̃s)\n)2 ; with θ̃s ∼ p(θ̃ | D).\n(8)\nHere, for clarity, we have shown explicitly the dependency of Σ̃ on θ̃MAP. Note that we can simply set Din to be the training set of a dataset. Furthermore, throughout this paper, we use the simple OOD dataset proposed by Hein et al. (2019) which is constructed via permutation, blurring, and contrast rescaling of the in-distribution datasetDin. As we shall show in Section 5, this artificial, uninformative OOD dataset is sufficient for obtaining good results across benchmark problems. More complex dataset as Dout might improve LULA’s performance further but is not strictly necessary. Since our aim is to solely improve the uncertainty, we must maintain the structure of all weights and biases in θ̃MAP, in accordance to (6). This can simply be enforced via gradient masking: For all l = 1, . . . , L− 1, set the gradients of the blocks of W̃ (l) and b̃(l) not corresponding to Ŵ (l) and b̂(l), respectively, to zero. Furthermore, since the covariance matrix Σ̃(θ̃) of the Laplace-approximated posterior is a function of θ̃, it needs to be updated at every iteration during the optimization ofLLULA.\nAlgorithm 1 Training LULA units. Input:\nMAP-trained network f . Dataset D, OOD dataset Dout. Learning rate α. Number of epochs E. 1: Construct f̃ from f by following Section 3.1. 2: for i = 1, . . . , E do 3: p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)) . Obtain a Laplace-approximated posterior of f̃ 4: Compute LLULA(θ̃MAP) via (7) using p(θ̃ | D), D, and Dout 5: g = ∇LLULA(θ̃MAP) 6: ĝ = mask gradient(g) . Zero out the derivatives not corresponding to θ̂ 7: θ̃MAP = θ̃MAP − αĝ 8: end for 9: p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)) . Obtain the final Laplace approximation\n10: return f̃ and p(θ̃ | D)\nThe cost scales in the network’s depth and can thus be expensive. Inspired by recent findings that last-layer Bayesian methods are competitive to all-layer alternatives (Ober & Rasmussen, 2019), we thus consider a last-layer LA (Kristiadi et al., 2020) as a proxy: We apply a LA only at the last hidden layer and assume that the first L− 2 layers are learnable feature extractor. We use a diagonal last-layer Fisher matrix to approximate the last-layer Hessian. Note that backpropagation through this matrix does not pose a difficulty since modern deep learning libraries such as PyTorch and TensorFlow supports “double backprop” (backpropagation through a gradient) efficiently. Finally, the loss LLULA can be minimized using standard gradient-based optimizers. Refer to Algorithm 1 for a summary.\nLast but not least, the intuition of LULA training is as follows. By adding LULA units, we obtain an augmented version of the network’s loss landscape. The goal of LULA training is then to exploit the weight-space symmetry (i.e. different parameters but induce the same output) arisen from the construction as shown by Proposition 1 and pick one of these parameters that is symmetric to the original parameter but has “better” curvatures. Here, we define a “good curvature” in terms of the above objective. These curvatures, then, when used in a Laplace approximation, could yield better uncertainty estimates compared to the standard non-LULA-augmented Laplace approximations."},{"heading":"4 RELATED WORK","text":"While traditionally hyperparameter optimization in a LA requires re-training of the network—under the evidence framework (MacKay, 1992b) or empirical Bayes (Bernardo & Smith, 2009), tuning it in a post-hoc manner has increasingly becomes a common practice. Ritter et al. (2018a;b) tune the prior precision of a LA by maximizing the predictive log-likelihood. Kristiadi et al. (2020) extend this procedure by also using OOD data to better calibrate the uncertainty. However, they are limited in terms of flexibility since the prior precision of the LAs constitutes to just a single parameter. LULA can be seen as an extension of these approaches with greater flexibility and is complementary to them since LULA is independent to the prior precision.\nConfidence calibration via OOD data has achieved state-of-the-art performance in non-Bayesian outlier detection. Hendrycks et al. (2019); Hein et al. (2019); Meinke & Hein (2020) use OOD data to regularize the standard maximum-likelihood training. Malinin & Gales (2018; 2019) use OOD data to train probabilistic models based on the Dirichlet distribution. All these methods are non-Bayesian and non-post-hoc. Our work is thus orthogonal since we aim at improving a class of Bayesian models in a post-hoc manner. LULA can be seen as bringing the best of both worlds: Bayesian uncertainty that is tuned via the state-of-the-art non-Bayesian technique."},{"heading":"5 EXPERIMENTS","text":"In this section, we focus on classification using the standard OOD benchmark problems. Supplementary results on regression are in Section C.2."},{"heading":"5.1 IMAGE CLASSIFICATIONS","text":"Here, we aim to show that LULA units and the proposed training procedure are (i) a significantly better method for tuning the uncertainty of a LA than previous methods and (ii) able to make a vanilla LA better than non-post-hoc (thus more expensive) UQ methods. For the purpose of (i), we compare\nLULA-augmented KFL (KFL+LULA) against the vanilla KFL with exact prior precision (KFL), KFL where the prior precision is found by maximizing validation log-likelihood (KFL+LL; Ritter et al., 2018b), and KFL with OOD training (KFL+OOD; Kristiadi et al., 2020). Moreover, for the purpose of (ii), we also compare our method with the following baselines, which have been found to yield strong results in UQ (Ovadia et al., 2019): Monte-Carlo dropout (MCD; Gal & Ghahramani, 2016) and deep ensemble (DE; Lakshminarayanan et al., 2017).\nWe use 5- and 8-layer CNNs for MNIST and CIFAR-10, SVHN, CIFAR-100, respectively. These networks achieve around 99%, 90%, and 50% accuracies for MNIST, CIFAR-10, and CIFAR-100, respectively. For MC-integration during the predictions, we use 10 posterior samples. We quantify the results using the standard metrics: the mean-maximum-confidence (MMC) and area-under-ROC (AUR) metrics (Hendrycks & Gimpel, 2017). All results reported are averages over ten prediction runs. Finally, we use standard test OOD datasets along with the “asymptotic” dataset introduced by Kristiadi et al. (2020) where random uniform noise images are scaled with a large number (5000).\nFor simplicity, we add LULA units and apply only at the last layer of the network. For each indistribution dataset, the training of the free parameters of LULA units is performed for a single epoch using Adam over the respective validation dataset. The number of these units is obtained via a grid-search over the set {64, 128, 256, 512}, balancing both the in- and out-distribution confidences (see Appendix B for the details).\nOOD Detection The results for MMC and AUR are shown in Table 1 and Table 2, respectively. First, we would like to direct the reader’s attention to the last four columns of the table. We can see that in general KFL+LULA performs the best among all LA tuning methods. These results validate the effectiveness of the additional flexibility given by LULA units and the proposed training procedure. Indeed, without this additional flexibility, OOD training on just the prior precision becomes less effective, as shown by the results of KFL+OOD. Finally, as we can see in the results on the Asymptotic OOD dataset, LULA makes KFL significantly better at mitigating overconfidence far away from the training data. Now, compared to the contemporary baselines (MCD, DE), we can see that the vanilla KFL yields somewhat worse results. Augmenting the base KFL with LULA makes it competitive to MCD and DE in general. Keep in mind that both KFL and LULA are post-hoc methods.\nDataset Shift We use the corrupted CIFAR-10 dataset (Hendrycks & Dietterich, 2019) for measuring the robustness of LULA-augmented LA to dataset shifts, following Ovadia et al. (2019). Note that dataset shift is a slightly different concept to OOD data: it concerns about small perturbations of the true data, while OOD data are data that do not come from the true data distribution. Intuitively, humans perceive that the data under a dataset shift lies near the true data while OOD data are farther away. We present the results in Table 3. Focusing first on the last four columns in the table, we see that LULA yields the best results compared other tuning methods for KFL. Furthermore, we see that KFL+LULA outperforms DE, which has been shown by Ovadia et al. (2019) to give the state-of-the-art results in terms of robustness to dataset shifts. Finally, while MCD achieve the best results in this experiment, considering its performance in the previous OOD experiment, we draw a conclusion that KFL+LULA provides a more consistent performance over different tasks.\nComparison with DPN Finally we compare KFL+LULA with the (non-Bayesian) Dirichlet prior network (DPN, Malinin & Gales, 2018) in the Rotated-MNIST benchmark (Ovadia et al., 2019)\n(Figure 3). We found that LULA makes the performance of KFL competitive to DPN. We stress that KFL and LULA are post-hoc methods, while DPN requires training from scratch."},{"heading":"5.2 COST ANALYSIS","text":"The cost of constructing a LULA network is negligible even for our deepest network: on both the 5- and 8-layer CNNs, the wall-clock time required (with a standard consumer GPU) to add additional LULA units is on average 0.01 seconds (over ten trials). For training, using the last-layer LA as a proxy of the true LA posterior, it took on average 7.25 seconds and 35 seconds for MNIST and CIFAR-10, SVHN, CIFAR-100, respectively. This tuning cost is cheap relative to the training time of the base network, which ranges between several minutes to more than an hour. We refer the reader to Table 9 (Appendix C) for the detail. All in all, LULA is not only effective, but also cost-efficient."},{"heading":"6 CONCLUSION","text":"We have proposed LULA units: hidden units that can be added to any pre-trained MAP network for the purpose of exclusively tuning the uncertainty of a Laplace approximation without affecting its predictive performance. They can be trained via an objective that depends on both inlier and outlier datasets to minimize (resp. maximize) the network’s output variance, bringing the state-ofthe-art technique from non-Bayesian robust learning literature to the Bayesian world. Even with very simple outlier dataset for training, we show in extensive experiments that LULA units provide more effective post-hoc uncertainty tuning for Laplace approximations and make their performance competitive to more expensive baselines which require re-training of the whole network."},{"heading":"APPENDIX A PROOFS","text":"Proposition 1. Let f : Rn × Rd → Rk be a MAP-trained L-layer network parametrized by θMAP. Suppose f̃ : Rn × Rd̃ → R and θ̃MAP ∈ Rd̃ are obtained via the previous construction. For any input x ∈ Rn, we have f̃(x; θ̃MAP) = f(x; θMAP).\nProof. Let x ∈ Rn be arbitrary. For each layer l = 1, . . . , L we denote the hidden units and preactivations of f̃ as h̃(l) and ã(l), respectively. We need to show that the output of f̃ , i.e. the last pre-activations ã(L), is equal to the last pre-activations a(L) of f .\nFor the first layer, we have that\nã(1) = W̃ (1)x+ b̃(1) =\n( W (1)\nŴ (1)\n) x+ ( b(1)\nb̂(1)\n) = ( W (1)x+ b(1)\nŴ (1)x+ b̂(1)\n) =: ( a(1)\nâ(1)\n) .\nFor every layer l = 1, . . . , L− 1, we denote the hidden units as the block vector\nh̃(l) = ( ϕ(a(l)) ϕ(â(l)) ) = ( h(l) ĥ(l) ) .\nNow, for the intermediate layer l = 2, . . . , L− 1, we observe that\nã(l) = W̃ (l)h̃(l−1) + b̃(l) =\n( W (l) 0\nŴ (l) 1 Ŵ (l) 2\n)( h(l−1)\nĥ(l−1)\n) + ( b(l)\nb̂(l) ) = ( W (l)h(l−1) + 0 + b(l)\nŴ (l) 1 h (l−1) + Ŵ (l) 2 ĥ (l−1) + b̂(l)\n) =: ( a(l)\nâ(l)\n) .\nFinally, for the last-layer, we get\nã(L) = W̃ (L)x+ b̃(L) = ( W (L) 0 )(h(L−1) ĥ(L−1) ) + b(L) = W (L)h(L−1) + 0 + b(L) = a(L) .\nThis ends the proof.\nProposition 2. Suppose f : Rn × Rd → R is a real-valued network and f̃ is as constructed above. Suppose further that diagonal Laplace-approximated posteriors N (θ | θMAP,diag(σ)), N (θ̃ | θ̃,diag(σ̃)) are employed. Using the linearization (5), for any input x ∈ Rn, the variance over the output f̃(x; θ̃) is at least that of f(x; θ).\nProof. W.l.o.g. we arrange the parameters θ̃ := (θ>, θ̂>)> where θ̂ ∈ Rd̃−d contains the weights corresponding to the the additional LULA units. If g(x) is the gradient of the output f(x; θ) w.r.t. θ, then the gradient of f̃(x; θ̃) w.r.t. θ̃, say g̃(x), can be written as the concatenation (g(x)>, ĝ(x)>)>\nwhere ĝ(x) is the corresponding gradient of θ̂. Furthermore, diag(σ̃) has diagonal elements( σ11, . . . , σdd, σ̂11, . . . , σ̂d̃−d,d̃−d )> =: (σ>, σ̂>)> .\nHence we have\nṽ(x) = g̃(x)>diag(σ̃)g̃(x)\n= g(x)>diag(σ)g(x)︸ ︷︷ ︸ =v(x) +ĝ(x)>diag(σ̂)ĝ(x)\n≥ v(x) ,\nsince diag(σ̂) is positive-definite.\nAlgorithm 2 Adding LULA units. Input:\nL-layer net with a MAP estimate θMAP = (W (l) MAP, b (l) MAP) L l=1. Sequence of non-negative integers\n(ml) L l=1.\n1: for l = 1, . . . , L− 1 do 2: vec Ŵ (l) ∼ p(vec Ŵ (l)) . Draw from a prior 3: b̂(l) ∼ p(̂b(l)) . Draw from a prior\n4: W̃ (l)MAP = W (l)MAP 0 Ŵ\n(l) 1 Ŵ (l) 2 . The zero submatrix 0 is of size nl ×ml−1 5: b̃(l)MAP := b(l)MAP b̂(l)\n 6: end for\n7: W̃ (L)MAP = (W (L) MAP, 0) . The zero submatrix is of size k ×mL−1 8: b̃(L)MAP = b (L) MAP 9: θ̃MAP = (W̃ (l) MAP, b̃ (l) MAP) L l=1\n10: return θ̃MAP"},{"heading":"APPENDIX B IMPLEMENTATION","text":"We summarize the augmentation of a network with LULA units in Algorithm 2. Note that the priors of the free parameters Ŵ (l), b̂(l) (lines 2 and 3) can be chosen as independent Gaussians—this reflects the standard procedure for initializing NNs’ parameters.\nWe train LULA units for a single epoch (since for each dataset, we have a large amount of training points) with learning rate 0.01. For each dataset, the number of the additional last-layer units mL is obtained via a grid search over the set {64, 128, 256, 512} =: ML, minimizing the absolute distance to the optimal MMC, i.e. 1 and 1/k for the in- and out-distribution validation set, respectively:\nmL = arg min m∈ML\n|1−MMCin(m)|+ |1/k −MMCout(m)| , (9)\nwhere MMCin and MMCout are the validation in- and out-distribution MMC of the Laplaceapproximated, trained-LULA, respectively."},{"heading":"APPENDIX C ADDITIONAL EXPERIMENT RESULTS","text":"C.1 TOY DATASET\nIn practice, one can simply set {Ŵ (l), b̂(l)}L−1l=1 randomly given a prior, e.g. the standard Gaussian (see also Algorithm 2). To validate this practice, we show the vanilla Laplace, untrained LULA,\nand fine-tuned LULA’s confidence in Figure 4. Even when set randomly from the standard Gaussian prior, LULA weights provide a significant improvement over the vanilla Laplace. Moreover, training them yields even better predictive confidence estimates. In particular, far from the data, the confidence becomes even lower, while still maintaining high confidence in the data regions.\nC.2 UCI REGRESSION\nTo validate the performance of LULA in regressions, we employ a subset of the UCI regression benchmark datasets. Following previous works, the network architecture used here is a singlehidden-layer ReLU network with 50 hidden units. The data are standardized to have zero mean and unit variance. We use 50 LULA units and optimize them for 40 epochs using OOD data sampled uniformly from [−10, 10]n. For MCD, KFL, and KFL+LULA, each prediction is done via MCintegration with 100 samples. For the evaluation of each dataset, we use a 60-20-20 train-validationtest split. We repeat each train-test process 10 times and take the average.\nIn Table 5 we report the average predictive standard deviation for each dataset. Note that this metric is the direct generalization of the 1D error bar in Figure 1 (top) to multi dimension. The outliers are sampled uniformly from [−10, 10]n. Note that since the inlier data are centered around the origin and have unit variance, they lie approximately in a Euclidean ball with radius 2. Therefore, these outliers are very far away from them. Thus, naturally, high uncertainty values over these outliers are desirable. Uncertainties over the test sets are generally low for all methods, although KFL+LULA has slightly higher uncertainties compared to the base KFL. However, KFL+LULA yield much higher uncertainties over outliers across all datasets, significantly more than the baselines. Moreover, in Table 4, we show that KFL+LULA maintains the predictive performance of the base KFL. Altogether, they imply that KFL+LULA can detect outliers better than other methods without costing the predictive performance.\nC.3 IMAGE CLASSIFICATION\nIn Table 6 we present the sensitivity analysis of confidences under a Laplace approximation w.r.t. the number of additional LULA units. Generally, we found that small number of additional LULA units,\ne.g. 32 and 64, is optimal. It is clear that increasing the number of LULA units decreases both the in- and out-distribution confidences. In the case of larger networks, we found that larger values (e.g. 512 in CIFAR-10) make the Hessian badly conditioned, resulting in numerical instability during its inversion. One might be able to alleviate this issue by additionally tuning the prior precision hyperparameter of the Laplace approximation (as in (Ritter et al., 2018b; Kristiadi et al., 2020)), which corresponds to varying the strength of the diagonal correction of the Hessian. However, we emphasize that even with small amounts of additional LULA units, we can already improve vanilla Laplace approximations significantly, as shown in the main text (Section 5.1).\nWe present the predictive performances of all methods in Table 7. LULA achieves similar accuracies to the base MAP and KFL baselines. Differences in their exact values likely due to various approximations used (e.g. MC-integral). In the case of CIFAR-100, we found that MAP underperforms compared to MCD and DE. This might be because of overfitting, since only weight decay is used for regularization, in contrast to MCD where dropout is used on top of weight decay. Due to MAP’s underperformance, LULA also underperform. However, we stress that whenever the base MAP model performs well, by construction LULA will also perform well.\nAs a supplement we show the performance of KFL+LULA against DPN in OOD detection on MNIST (Table 8). We found that KFL+LULA performance on OOD detection are competitive or better than DPN.\nC.4 COMPUTATIONAL COST\nTo supplement the cost analysis in the main text, we show the wall-clock times required for the construction and training of LULA units in Table 9.\nC.5 DEEPER NETWORKS\nWe also asses the performance of LULA in larger network. We use a 20-layer CNN on CIFAR10, SVHN, and CIFAR-100. Both the KFL and LULA are applied only at the last-layer of the network. The results, in terms of MMC, expected calibration error (ECE), and AUR, are presented in Table 10 and Table 11. We observe that LULA is the best method for uncertainty tuning in LA: It makes KFL better calibrated in both in- and out-distribution settings. Moreover, the LULA-imbued KFL is competitive to DE, which has been shown by Ovadia et al. (2019) to be the best Bayesian method for uncertainty quantification. Note that, KFL+LULA is a post-hoc method and thus can be applied to any pre-trained network. In contrast, DE requires training multiple networks (usually 5) from scratch which can be very expensive.\nWe additionally show the performance of LULA when applied on top of a KFL-approximated DenseNet-121 (Huang et al., 2017) in Tables 12 and 13. LULA generally outperforms previous uncertainty tuning methods for LA and is competitive to DE. However, we observe in SVHN that LULA do not improve KFL significantly. This issue is due to the usage the Smooth Noise dataset, which has already assigned low confidence in this case, for training LULA. Thus, we re-train LULA with the Uniform Noise dataset and present the result in Table 14. We show that using this dataset, we obtain better OOD calibration performance, outperforming DE."}],"string":"[\n {\n \"heading\": null,\n \"text\": \"Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly “train” the uncertainty in a decoupled way to the prediction itself. To this end we introduce uncertainty units for Laplaceapproximated networks: Hidden units with zero weights that can be added to any pre-trained, point-estimated network. Since these units are inactive, they do not affect the predictions. But their presence changes the geometry (in particular the Hessian) of the loss landscape around the point estimate, thereby affecting the network’s uncertainty estimates under a Laplace approximation. We show that such units can be trained via an uncertainty-aware objective, making the Laplace approximation competitive with more expensive alternative uncertaintyquantification frameworks.\"\n },\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"The point estimates of neural networks (NNs)—constructed as maximum a posteriori (MAP) estimates via (regularized) empirical risk minimization—empirically achieve high predictive performance. However, they tend to underestimate the uncertainty of their predictions, leading to an overconfidence problem (Hein et al., 2019), which could be disastrous in safety-critical applications such as autonomous driving. Bayesian inference offers a principled path to overcome this issue. The goal is to turn a “vanilla” NN into a Bayesian neural network (BNN), where the posterior distribution over the network’s weights are inferred via Bayes’ rule and subsequently taken into account when making predictions. Since the cost of exact posterior inference in a BNN is often prohibitive, approximate Bayesian methods are employed instead.\\nLaplace approximations (LAs) are classic methods for such a purpose (MacKay, 1992b). The key idea is to obtain an approximate posterior by “surrounding” a MAP estimate of a network with a Gaussian, based on the loss landscape’s geometry around it. A standard practice in LAs is to tune a single hyperparameter—the prior precision—which is inflexible (Ritter et al., 2018b; Kristiadi et al., 2020). Here, we aim at improving the flexibility of uncertainty tuning in LAs. To this end, we introduce Learnable Uncertainty under Laplace Approximations (LULA) units, which are hidden units associated with a zeroed weight. They can be added to the hidden layers of any MAP-trained network. Because they are inactive, such units do not affect the prediction of the underlying network. However, they can still contribute to the Hessian of the loss with respect to the parameters, and hence induce additional structures to the posterior covariance under a LA. LULA units can be trained via an uncertainty-aware objective (Hendrycks et al., 2019; Hein et al., 2019, etc.), such that they improve the predictive uncertainty-quantification (UQ) performance of the Laplace-approximated BNN. Figure 1 demonstrates trained LULA units in action: They improve the UQ performance of a standard LA, while keeping the MAP predictions in both regression and classification tasks.\\nIn summary, we (i) introduce LULA units: inactive hidden units for uncertainty tuning of a LA, (ii) bring a robust training technique from non-Bayesian literature for training these units, and (iii) show empirically that LULA-augmented Laplace-approximated BNNs can yield better UQ performance compared to both previous tuning techniques and contemporary, more expensive baselines.\"\n },\n {\n \"heading\": \"2 BACKGROUND\",\n \"text\": \"\"\n },\n {\n \"heading\": \"2.1 BAYESIAN NEURAL NETWORKS\",\n \"text\": \"Let f : Rn × Rd → Rk defined by (x, θ) 7→ f(x; θ) be an L-layer neural network. Here, θ is the concatenation of all the parameters of f . Suppose that the size of each layer of f is given by the sequence of (nl ∈ Z>0)Ll=1. Then, for each l = 1, . . . , L, the l-th layer of f is defined by\\na(l) := W (l)h(l−1) + b(l), with h(l) := { ϕ(a(l)) if l < L a(l) if l = L ,\\n(1)\\nwhere W (l) ∈ Rnl×nl−1 and b(l) ∈ Rnl are the weight matrix and bias vector of the layer, and ϕ is a component-wise activation function. We call the vector h(l) ∈ Rnl the l-th hidden units of f . Note that by convention, we consider n0 := n and nL := k, while h(0) := x and h(L) := f(x; θ).\\nFrom the Bayesian perspective, the ubiquitous training formalism of neural networks amounts to MAP estimation: The empirical risk and the regularizer are interpretable as the negative loglikelihood under an i.i.d. dataset D := {xi, yi}mi=1 and the negative log-prior, respectively. That is, the loss function is interpreted as\\nL(θ) := − m∑ i=1 log p(yi | f(xi; θ))− log p(θ) = − log p(θ | D) . (2)\\nIn this view, the de facto weight decay regularizer amounts to a zero-mean isotropic Gaussian prior p(θ) = N (θ | 0, λ−1I) with a scalar precision hyperparameter λ. Meanwhile, the usual softmax and quadratic output losses correspond to the Categorical and Gaussian distributions over yi in the case of classification and regression, respectively.\\nMAP-trained neural networks have been shown to be overconfident (Hein et al., 2019) and BNNs can mitigate this issue (Kristiadi et al., 2020). They quantify epistemic uncertainty by inferring the full posterior distribution of the parameters θ (instead of just a single point estimate in MAP training). Given that p(θ | D) is the posterior, then the prediction for any test point x ∈ Rn is obtained via marginalization\\np(y | x,D) = ∫ p(y | f(x; θ)) p(θ | D) dθ , (3)\\nwhich captures the uncertainty encoded in the posterior.\"\n },\n {\n \"heading\": \"2.2 LAPLACE APPROXIMATIONS\",\n \"text\": \"In deep learning, since the exact Bayesian posterior is intractable, approximate Bayesian inference methods are used. An important family of such methods is formed by LAs. Let θMAP be the minimizer of (2), which corresponds to a mode of the posterior distribution. A LA locally approximates the posterior using a Gaussian\\np(θ | D) ≈ N (θ | θMAP, Σ) := N (θ | θMAP, (∇2L|θMAP)−1) . Thus, LAs construct an approximate Gaussian posterior around θMAP, whose precision equals the Hessian of the loss at θMAP—the “curvature” of the loss landscape at θMAP. While the covariance of a LA is tied to the weight decay of the loss, a common practice in LAs is to tune the prior precision under some objective, in a post-hoc manner. In other words, the MAP estimation and the covariance inference are thought as separate, independent processes. For example, given a fixed MAP estimate, one can maximize the log-likelihood of a LA w.r.t. the prior precision to obtain the covariance. This hyperparameter tuning can thus be thought as an uncertainty tuning.\\nA recent example of LAs is the Kronecker-factored Laplace (KFL, Ritter et al., 2018b). The key idea is to approximate the Hessian matrix with the layer-wise Kronecker factorization scheme proposed by Heskes (2000); Martens & Grosse (2015). That is, for each layer l = 1, . . . , L, KFL assumes that the Hessian corresponding to the l-th weight matrix W (l) ∈ Rnl×nl−1 can be written as the Kronecker product G(l) ⊗ A(l) for some G(l) ∈ Rnl×nl and A(l) ∈ Rnl−1×nl−1 . This assumption brings the inversion cost of the Hessian down to Θ(n3l +n 3 l−1), instead of the usual Θ(n 3 l n 3 l−1) cost. The approximate Hessian can easily be computed via tools such as BackPACK (Dangel et al., 2020).\\nEven with a closed-form Laplace-approximated posterior, the predictive distribution (3) in general does not have an analytic solution since f is nonlinear. Instead, one can employ Monte-Carlo (MC) integration by sampling from the Gaussian:\\np(y | x,D) ≈ 1 S S∑ s=1 p(y | f(x; θs)) ; θs ∼ N (θ | θMAP, Σ) ,\\nfor S number of samples. In the case of binary classification with f : Rn × Rd → R, one can use the following well-known approximation, due to MacKay (1992a):\\np(y = 1 | x,D) ≈ σ ( f(x; θMAP)√ 1 + π/8 v(x) ) , (4)\\nwhere σ is the logistic-sigmoid function and v(x) is the marginal variance of the network output f(x), which is often approximated via a linearization of the network around the MAP estimate:\\nv(x) ≈ (∇θf(x; θ)|θMAP) > Σ (∇θf(x; θ)|θMAP) . (5)\\n(This approximation has also been generalized to multi-class classifications by Gibbs (1997).) In particular, as v(x) increases, the predictive probability of y = 1 goes to 0.5 and therefore the uncertainty increases. This relationship has also been shown empirically in multi-class classifications with MC-integration (Kristiadi et al., 2020).\"\n },\n {\n \"heading\": \"3 LULA UNITS\",\n \"text\": \"The problem with the standard uncertainty tuning in LAs is that the only degree-of-freedom available for performing the optimization is the scalar prior precision and therefore inflexible.1 We shall address this by introducing “uncertainty units”, which can be added on top of the hidden units of any MAP-trained network (Section 3.1) and can be trained via an uncertainty-aware loss (Section 3.2).\"\n },\n {\n \"heading\": \"3.1 CONSTRUCTION\",\n \"text\": \"Let f : Rn × Rd → Rk be a MAP-trained L-layer neural network with parameters θMAP = {W (l)MAP, b (l) MAP}Ll=1. The premise of our method is simple: At each hidden layer l = 1, . . . , L − 1,\\n1While one can also use a non-scalar prior precision, it appears to be uncommon in deep learning. In any case, such a element-wise weight-cost would interact with the training procedure.\\nMAP. The\\nadditional units are represented by the additional block at the bottom of each layer. Dashed lines correspond to the free parameters Ŵ (1), . . . , Ŵ (L−1), while dotted lines to the zero weights.\\nsuppose we add ml ∈ Z≥0 additional hidden units, under the original activation function, to h(l). As a consequence, we need to augment each of the weight matrices to accommodate them.\\nConsider the following construction: for each layer l = 1, . . . , L − 1 of the network, we expand W (l) and b(l) to obtain the block matrix and vector\\nW̃ (l) :=\\n( W\\n(l) MAP 0\\nŴ (l) 1 Ŵ (l) 2\\n) ∈ R(nl+ml)×(nl−1+ml−1) ; b̃(l) := ( b (l) MAP\\nb̂(l)\\n) ∈ Rnl+ml , (6)\\nrespectively, with m0 = 0 since we do not add additional units to the input. For l = L, we define\\nW̃ (L) := (W (L) MAP, 0) ∈ R k×(nL−1+mL−1); b̃(L) := b (L) MAP ∈ R k ,\\nso that the output dimensionality is unchanged. For brevity, we denote Ŵ (l) := (Ŵ (l)1 , Ŵ (l) 2 ). Refer to Figure 2 for an illustration and Algorithm 2 in Appendix B for a step-by-step summary. Taken together, we denote the resulting augmented network as f̃ and the resulting parameter vector as θ̃MAP ∈ Rd̃, where d̃ it the resulting number of parameters. Note that we can easily extend this construction to convolutional nets by expanding the “channel” of a hidden layer.2\\nLet us inspect the implication of this construction. Here for each l = 1, . . . , L − 1, since they are zero, the upper-right quadrant of W̃ (l) deactivates the ml−1 additional hidden units in the previous layer, thus they do not contribute to the original hidden units in the l-th layer. Meanwhile, the submatrix Ŵ (l) and the sub-vector b̂(l) contain parameters for the additional ml hidden units in the l-th layer. We are free to choose the the values of these parameters since the following proposition guarantees that they will not change the output of the network (the proof is in Appendix A).\\nProposition 1. Let f : Rn × Rd → Rk be a MAP-trained L-layer network parametrized by θMAP. Suppose f̃ : Rn × Rd̃ → R and θ̃MAP ∈ Rd̃ are obtained via the previous construction. For any input x ∈ Rn, we have f̃(x; θ̃MAP) = f(x; θMAP).\\nSo far, it looks like all our changes to the network are inconsequential. However, they do affect the curvature of the landscape of L,3 and thus the uncertainty arising in a LA. Let θ̃ be a random variable in Rd̃ and θ̃MAP be an instance of it. Suppose we have a Laplace-approximated posterior p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃) over θ̃, where the covariance Σ̃ is the inverse Hessian of the negative log-posterior w.r.t. the augmented parameters at θ̃MAP. Then, Σ̃ contains additional dimensions (and thus in general, additional structured, non-zero uncertainty) absent in the original network, which depend on the values of the free parameters {Ŵ (l), b̂(l)}L−1l=1 .\\n2E.g. if the hidden units are a 3D array of (channel × height × width), then we expand the first dimension. 3More formally: The principal curvatures of the graph of L, seen as a d-dimensional submanifold of Rd+1.\\nThe implication of the previous finding can be seen clearly in real-valued networks with diagonal LA posteriors. The following proposition shows that, under such a network and posterior, the construction above will affect the output uncertainty of the original network f (the proof is in Appendix A).\\nProposition 2. Suppose f : Rn × Rd → R is a real-valued network and f̃ is as constructed above. Suppose further that diagonal Laplace-approximated posteriors N (θ | θMAP,diag(σ)), N (θ̃ | θ̃,diag(σ̃)) are employed. Using the linearization (5), for any input x ∈ Rn, the variance over the output f̃(x; θ̃) is at least that of f(x; θ).\\nIn summary, the construction along with Propositions 1 and 2 imply that the additional hidden units we have added to the original network are uncertainty units under LAs, i.e. hidden units that only contribute to the Laplace-approximated uncertainty and not the predictions. This property gives rise to the name Learnable Uncertainty under Laplace Approximations (LULA) units.\"\n },\n {\n \"heading\": \"3.2 TRAINING\",\n \"text\": \"We have seen that by adding LULA units to a network, we obtain additional free parameters that only affect uncertainty under a LA. These parameters are thus useful for uncertainty calibration. Our goal is therefore to train them to induce low uncertainty over the data (inliers) and high uncertainty on outliers—the so-called out-of-distribution (OOD) data. Specifically, this can be done by minimizing the output variance over inliers while maximizing it over outliers. Note that using variance makes sense in both the regression and classification cases: In the former, this objective directly maintains narrow error bars near the data while widen those far-away from them—cf. Figure 1 (c, top). Meanwhile, in classifications, variances over function outputs directly impact predictive confidences, as we have noted in the discussion of (4)—higher variance implies lower confidence.\\nThus, following the contemporary technique from non-Bayesian robust learning literature (Hendrycks et al., 2019; Hein et al., 2019, etc.), we construct the following loss. Let f : Rn×Rd → Rk be an L-layer neural network with a MAP-trained parameters θMAP and let f̃ : Rn × Rd̃ → Rk along with θ̃MAP be obtained by adding LULA units. Denoting the dataset sampled i.i.d. from the data distribution as Din and that from some outlier distribution as Dout, we define\\nLLULA(θ̃MAP) := 1 |Din| ∑ xin∈Din ν(f̃(xin); θ̃MAP)− 1 |Dout| ∑ xout∈Dout ν(f̃(xout); θ̃MAP) , (7)\\nwhere ν(f̃(x); θ̃MAP) is the total variance over the k components of the network output f̃1(x; θ̃), . . . , f̃k(x; θ̃) under the Laplace-approximated posterior p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)), which can be approximated via an S-samples MC-integral\\nν(f̃(x);θ̃MAP) := k∑ i=1 varp(θ̃|D) f̃i(x; θ̃)\\n≈ k∑ i=1\\n( 1\\nS S∑ s=1 f̃i(x; θ̃s) 2\\n) − ( 1\\nS S∑ s=1 f̃i(x; θ̃s)\\n)2 ; with θ̃s ∼ p(θ̃ | D).\\n(8)\\nHere, for clarity, we have shown explicitly the dependency of Σ̃ on θ̃MAP. Note that we can simply set Din to be the training set of a dataset. Furthermore, throughout this paper, we use the simple OOD dataset proposed by Hein et al. (2019) which is constructed via permutation, blurring, and contrast rescaling of the in-distribution datasetDin. As we shall show in Section 5, this artificial, uninformative OOD dataset is sufficient for obtaining good results across benchmark problems. More complex dataset as Dout might improve LULA’s performance further but is not strictly necessary. Since our aim is to solely improve the uncertainty, we must maintain the structure of all weights and biases in θ̃MAP, in accordance to (6). This can simply be enforced via gradient masking: For all l = 1, . . . , L− 1, set the gradients of the blocks of W̃ (l) and b̃(l) not corresponding to Ŵ (l) and b̂(l), respectively, to zero. Furthermore, since the covariance matrix Σ̃(θ̃) of the Laplace-approximated posterior is a function of θ̃, it needs to be updated at every iteration during the optimization ofLLULA.\\nAlgorithm 1 Training LULA units. Input:\\nMAP-trained network f . Dataset D, OOD dataset Dout. Learning rate α. Number of epochs E. 1: Construct f̃ from f by following Section 3.1. 2: for i = 1, . . . , E do 3: p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)) . Obtain a Laplace-approximated posterior of f̃ 4: Compute LLULA(θ̃MAP) via (7) using p(θ̃ | D), D, and Dout 5: g = ∇LLULA(θ̃MAP) 6: ĝ = mask gradient(g) . Zero out the derivatives not corresponding to θ̂ 7: θ̃MAP = θ̃MAP − αĝ 8: end for 9: p(θ̃ | D) ≈ N (θ̃ | θ̃MAP, Σ̃(θ̃MAP)) . Obtain the final Laplace approximation\\n10: return f̃ and p(θ̃ | D)\\nThe cost scales in the network’s depth and can thus be expensive. Inspired by recent findings that last-layer Bayesian methods are competitive to all-layer alternatives (Ober & Rasmussen, 2019), we thus consider a last-layer LA (Kristiadi et al., 2020) as a proxy: We apply a LA only at the last hidden layer and assume that the first L− 2 layers are learnable feature extractor. We use a diagonal last-layer Fisher matrix to approximate the last-layer Hessian. Note that backpropagation through this matrix does not pose a difficulty since modern deep learning libraries such as PyTorch and TensorFlow supports “double backprop” (backpropagation through a gradient) efficiently. Finally, the loss LLULA can be minimized using standard gradient-based optimizers. Refer to Algorithm 1 for a summary.\\nLast but not least, the intuition of LULA training is as follows. By adding LULA units, we obtain an augmented version of the network’s loss landscape. The goal of LULA training is then to exploit the weight-space symmetry (i.e. different parameters but induce the same output) arisen from the construction as shown by Proposition 1 and pick one of these parameters that is symmetric to the original parameter but has “better” curvatures. Here, we define a “good curvature” in terms of the above objective. These curvatures, then, when used in a Laplace approximation, could yield better uncertainty estimates compared to the standard non-LULA-augmented Laplace approximations.\"\n },\n {\n \"heading\": \"4 RELATED WORK\",\n \"text\": \"While traditionally hyperparameter optimization in a LA requires re-training of the network—under the evidence framework (MacKay, 1992b) or empirical Bayes (Bernardo & Smith, 2009), tuning it in a post-hoc manner has increasingly becomes a common practice. Ritter et al. (2018a;b) tune the prior precision of a LA by maximizing the predictive log-likelihood. Kristiadi et al. (2020) extend this procedure by also using OOD data to better calibrate the uncertainty. However, they are limited in terms of flexibility since the prior precision of the LAs constitutes to just a single parameter. LULA can be seen as an extension of these approaches with greater flexibility and is complementary to them since LULA is independent to the prior precision.\\nConfidence calibration via OOD data has achieved state-of-the-art performance in non-Bayesian outlier detection. Hendrycks et al. (2019); Hein et al. (2019); Meinke & Hein (2020) use OOD data to regularize the standard maximum-likelihood training. Malinin & Gales (2018; 2019) use OOD data to train probabilistic models based on the Dirichlet distribution. All these methods are non-Bayesian and non-post-hoc. Our work is thus orthogonal since we aim at improving a class of Bayesian models in a post-hoc manner. LULA can be seen as bringing the best of both worlds: Bayesian uncertainty that is tuned via the state-of-the-art non-Bayesian technique.\"\n },\n {\n \"heading\": \"5 EXPERIMENTS\",\n \"text\": \"In this section, we focus on classification using the standard OOD benchmark problems. Supplementary results on regression are in Section C.2.\"\n },\n {\n \"heading\": \"5.1 IMAGE CLASSIFICATIONS\",\n \"text\": \"Here, we aim to show that LULA units and the proposed training procedure are (i) a significantly better method for tuning the uncertainty of a LA than previous methods and (ii) able to make a vanilla LA better than non-post-hoc (thus more expensive) UQ methods. For the purpose of (i), we compare\\nLULA-augmented KFL (KFL+LULA) against the vanilla KFL with exact prior precision (KFL), KFL where the prior precision is found by maximizing validation log-likelihood (KFL+LL; Ritter et al., 2018b), and KFL with OOD training (KFL+OOD; Kristiadi et al., 2020). Moreover, for the purpose of (ii), we also compare our method with the following baselines, which have been found to yield strong results in UQ (Ovadia et al., 2019): Monte-Carlo dropout (MCD; Gal & Ghahramani, 2016) and deep ensemble (DE; Lakshminarayanan et al., 2017).\\nWe use 5- and 8-layer CNNs for MNIST and CIFAR-10, SVHN, CIFAR-100, respectively. These networks achieve around 99%, 90%, and 50% accuracies for MNIST, CIFAR-10, and CIFAR-100, respectively. For MC-integration during the predictions, we use 10 posterior samples. We quantify the results using the standard metrics: the mean-maximum-confidence (MMC) and area-under-ROC (AUR) metrics (Hendrycks & Gimpel, 2017). All results reported are averages over ten prediction runs. Finally, we use standard test OOD datasets along with the “asymptotic” dataset introduced by Kristiadi et al. (2020) where random uniform noise images are scaled with a large number (5000).\\nFor simplicity, we add LULA units and apply only at the last layer of the network. For each indistribution dataset, the training of the free parameters of LULA units is performed for a single epoch using Adam over the respective validation dataset. The number of these units is obtained via a grid-search over the set {64, 128, 256, 512}, balancing both the in- and out-distribution confidences (see Appendix B for the details).\\nOOD Detection The results for MMC and AUR are shown in Table 1 and Table 2, respectively. First, we would like to direct the reader’s attention to the last four columns of the table. We can see that in general KFL+LULA performs the best among all LA tuning methods. These results validate the effectiveness of the additional flexibility given by LULA units and the proposed training procedure. Indeed, without this additional flexibility, OOD training on just the prior precision becomes less effective, as shown by the results of KFL+OOD. Finally, as we can see in the results on the Asymptotic OOD dataset, LULA makes KFL significantly better at mitigating overconfidence far away from the training data. Now, compared to the contemporary baselines (MCD, DE), we can see that the vanilla KFL yields somewhat worse results. Augmenting the base KFL with LULA makes it competitive to MCD and DE in general. Keep in mind that both KFL and LULA are post-hoc methods.\\nDataset Shift We use the corrupted CIFAR-10 dataset (Hendrycks & Dietterich, 2019) for measuring the robustness of LULA-augmented LA to dataset shifts, following Ovadia et al. (2019). Note that dataset shift is a slightly different concept to OOD data: it concerns about small perturbations of the true data, while OOD data are data that do not come from the true data distribution. Intuitively, humans perceive that the data under a dataset shift lies near the true data while OOD data are farther away. We present the results in Table 3. Focusing first on the last four columns in the table, we see that LULA yields the best results compared other tuning methods for KFL. Furthermore, we see that KFL+LULA outperforms DE, which has been shown by Ovadia et al. (2019) to give the state-of-the-art results in terms of robustness to dataset shifts. Finally, while MCD achieve the best results in this experiment, considering its performance in the previous OOD experiment, we draw a conclusion that KFL+LULA provides a more consistent performance over different tasks.\\nComparison with DPN Finally we compare KFL+LULA with the (non-Bayesian) Dirichlet prior network (DPN, Malinin & Gales, 2018) in the Rotated-MNIST benchmark (Ovadia et al., 2019)\\n(Figure 3). We found that LULA makes the performance of KFL competitive to DPN. We stress that KFL and LULA are post-hoc methods, while DPN requires training from scratch.\"\n },\n {\n \"heading\": \"5.2 COST ANALYSIS\",\n \"text\": \"The cost of constructing a LULA network is negligible even for our deepest network: on both the 5- and 8-layer CNNs, the wall-clock time required (with a standard consumer GPU) to add additional LULA units is on average 0.01 seconds (over ten trials). For training, using the last-layer LA as a proxy of the true LA posterior, it took on average 7.25 seconds and 35 seconds for MNIST and CIFAR-10, SVHN, CIFAR-100, respectively. This tuning cost is cheap relative to the training time of the base network, which ranges between several minutes to more than an hour. We refer the reader to Table 9 (Appendix C) for the detail. All in all, LULA is not only effective, but also cost-efficient.\"\n },\n {\n \"heading\": \"6 CONCLUSION\",\n \"text\": \"We have proposed LULA units: hidden units that can be added to any pre-trained MAP network for the purpose of exclusively tuning the uncertainty of a Laplace approximation without affecting its predictive performance. They can be trained via an objective that depends on both inlier and outlier datasets to minimize (resp. maximize) the network’s output variance, bringing the state-ofthe-art technique from non-Bayesian robust learning literature to the Bayesian world. Even with very simple outlier dataset for training, we show in extensive experiments that LULA units provide more effective post-hoc uncertainty tuning for Laplace approximations and make their performance competitive to more expensive baselines which require re-training of the whole network.\"\n },\n {\n \"heading\": \"APPENDIX A PROOFS\",\n \"text\": \"Proposition 1. Let f : Rn × Rd → Rk be a MAP-trained L-layer network parametrized by θMAP. Suppose f̃ : Rn × Rd̃ → R and θ̃MAP ∈ Rd̃ are obtained via the previous construction. For any input x ∈ Rn, we have f̃(x; θ̃MAP) = f(x; θMAP).\\nProof. Let x ∈ Rn be arbitrary. For each layer l = 1, . . . , L we denote the hidden units and preactivations of f̃ as h̃(l) and ã(l), respectively. We need to show that the output of f̃ , i.e. the last pre-activations ã(L), is equal to the last pre-activations a(L) of f .\\nFor the first layer, we have that\\nã(1) = W̃ (1)x+ b̃(1) =\\n( W (1)\\nŴ (1)\\n) x+ ( b(1)\\nb̂(1)\\n) = ( W (1)x+ b(1)\\nŴ (1)x+ b̂(1)\\n) =: ( a(1)\\nâ(1)\\n) .\\nFor every layer l = 1, . . . , L− 1, we denote the hidden units as the block vector\\nh̃(l) = ( ϕ(a(l)) ϕ(â(l)) ) = ( h(l) ĥ(l) ) .\\nNow, for the intermediate layer l = 2, . . . , L− 1, we observe that\\nã(l) = W̃ (l)h̃(l−1) + b̃(l) =\\n( W (l) 0\\nŴ (l) 1 Ŵ (l) 2\\n)( h(l−1)\\nĥ(l−1)\\n) + ( b(l)\\nb̂(l) ) = ( W (l)h(l−1) + 0 + b(l)\\nŴ (l) 1 h (l−1) + Ŵ (l) 2 ĥ (l−1) + b̂(l)\\n) =: ( a(l)\\nâ(l)\\n) .\\nFinally, for the last-layer, we get\\nã(L) = W̃ (L)x+ b̃(L) = ( W (L) 0 )(h(L−1) ĥ(L−1) ) + b(L) = W (L)h(L−1) + 0 + b(L) = a(L) .\\nThis ends the proof.\\nProposition 2. Suppose f : Rn × Rd → R is a real-valued network and f̃ is as constructed above. Suppose further that diagonal Laplace-approximated posteriors N (θ | θMAP,diag(σ)), N (θ̃ | θ̃,diag(σ̃)) are employed. Using the linearization (5), for any input x ∈ Rn, the variance over the output f̃(x; θ̃) is at least that of f(x; θ).\\nProof. W.l.o.g. we arrange the parameters θ̃ := (θ>, θ̂>)> where θ̂ ∈ Rd̃−d contains the weights corresponding to the the additional LULA units. If g(x) is the gradient of the output f(x; θ) w.r.t. θ, then the gradient of f̃(x; θ̃) w.r.t. θ̃, say g̃(x), can be written as the concatenation (g(x)>, ĝ(x)>)>\\nwhere ĝ(x) is the corresponding gradient of θ̂. Furthermore, diag(σ̃) has diagonal elements( σ11, . . . , σdd, σ̂11, . . . , σ̂d̃−d,d̃−d )> =: (σ>, σ̂>)> .\\nHence we have\\nṽ(x) = g̃(x)>diag(σ̃)g̃(x)\\n= g(x)>diag(σ)g(x)︸ ︷︷ ︸ =v(x) +ĝ(x)>diag(σ̂)ĝ(x)\\n≥ v(x) ,\\nsince diag(σ̂) is positive-definite.\\nAlgorithm 2 Adding LULA units. Input:\\nL-layer net with a MAP estimate θMAP = (W (l) MAP, b (l) MAP) L l=1. Sequence of non-negative integers\\n(ml) L l=1.\\n1: for l = 1, . . . , L− 1 do 2: vec Ŵ (l) ∼ p(vec Ŵ (l)) . Draw from a prior 3: b̂(l) ∼ p(̂b(l)) . Draw from a prior\\n4: W̃ (l)MAP = W (l)MAP 0 Ŵ\\n(l) 1 Ŵ (l) 2 . The zero submatrix 0 is of size nl ×ml−1 5: b̃(l)MAP := b(l)MAP b̂(l)\\n 6: end for\\n7: W̃ (L)MAP = (W (L) MAP, 0) . The zero submatrix is of size k ×mL−1 8: b̃(L)MAP = b (L) MAP 9: θ̃MAP = (W̃ (l) MAP, b̃ (l) MAP) L l=1\\n10: return θ̃MAP\"\n },\n {\n \"heading\": \"APPENDIX B IMPLEMENTATION\",\n \"text\": \"We summarize the augmentation of a network with LULA units in Algorithm 2. Note that the priors of the free parameters Ŵ (l), b̂(l) (lines 2 and 3) can be chosen as independent Gaussians—this reflects the standard procedure for initializing NNs’ parameters.\\nWe train LULA units for a single epoch (since for each dataset, we have a large amount of training points) with learning rate 0.01. For each dataset, the number of the additional last-layer units mL is obtained via a grid search over the set {64, 128, 256, 512} =: ML, minimizing the absolute distance to the optimal MMC, i.e. 1 and 1/k for the in- and out-distribution validation set, respectively:\\nmL = arg min m∈ML\\n|1−MMCin(m)|+ |1/k −MMCout(m)| , (9)\\nwhere MMCin and MMCout are the validation in- and out-distribution MMC of the Laplaceapproximated, trained-LULA, respectively.\"\n },\n {\n \"heading\": \"APPENDIX C ADDITIONAL EXPERIMENT RESULTS\",\n \"text\": \"C.1 TOY DATASET\\nIn practice, one can simply set {Ŵ (l), b̂(l)}L−1l=1 randomly given a prior, e.g. the standard Gaussian (see also Algorithm 2). To validate this practice, we show the vanilla Laplace, untrained LULA,\\nand fine-tuned LULA’s confidence in Figure 4. Even when set randomly from the standard Gaussian prior, LULA weights provide a significant improvement over the vanilla Laplace. Moreover, training them yields even better predictive confidence estimates. In particular, far from the data, the confidence becomes even lower, while still maintaining high confidence in the data regions.\\nC.2 UCI REGRESSION\\nTo validate the performance of LULA in regressions, we employ a subset of the UCI regression benchmark datasets. Following previous works, the network architecture used here is a singlehidden-layer ReLU network with 50 hidden units. The data are standardized to have zero mean and unit variance. We use 50 LULA units and optimize them for 40 epochs using OOD data sampled uniformly from [−10, 10]n. For MCD, KFL, and KFL+LULA, each prediction is done via MCintegration with 100 samples. For the evaluation of each dataset, we use a 60-20-20 train-validationtest split. We repeat each train-test process 10 times and take the average.\\nIn Table 5 we report the average predictive standard deviation for each dataset. Note that this metric is the direct generalization of the 1D error bar in Figure 1 (top) to multi dimension. The outliers are sampled uniformly from [−10, 10]n. Note that since the inlier data are centered around the origin and have unit variance, they lie approximately in a Euclidean ball with radius 2. Therefore, these outliers are very far away from them. Thus, naturally, high uncertainty values over these outliers are desirable. Uncertainties over the test sets are generally low for all methods, although KFL+LULA has slightly higher uncertainties compared to the base KFL. However, KFL+LULA yield much higher uncertainties over outliers across all datasets, significantly more than the baselines. Moreover, in Table 4, we show that KFL+LULA maintains the predictive performance of the base KFL. Altogether, they imply that KFL+LULA can detect outliers better than other methods without costing the predictive performance.\\nC.3 IMAGE CLASSIFICATION\\nIn Table 6 we present the sensitivity analysis of confidences under a Laplace approximation w.r.t. the number of additional LULA units. Generally, we found that small number of additional LULA units,\\ne.g. 32 and 64, is optimal. It is clear that increasing the number of LULA units decreases both the in- and out-distribution confidences. In the case of larger networks, we found that larger values (e.g. 512 in CIFAR-10) make the Hessian badly conditioned, resulting in numerical instability during its inversion. One might be able to alleviate this issue by additionally tuning the prior precision hyperparameter of the Laplace approximation (as in (Ritter et al., 2018b; Kristiadi et al., 2020)), which corresponds to varying the strength of the diagonal correction of the Hessian. However, we emphasize that even with small amounts of additional LULA units, we can already improve vanilla Laplace approximations significantly, as shown in the main text (Section 5.1).\\nWe present the predictive performances of all methods in Table 7. LULA achieves similar accuracies to the base MAP and KFL baselines. Differences in their exact values likely due to various approximations used (e.g. MC-integral). In the case of CIFAR-100, we found that MAP underperforms compared to MCD and DE. This might be because of overfitting, since only weight decay is used for regularization, in contrast to MCD where dropout is used on top of weight decay. Due to MAP’s underperformance, LULA also underperform. However, we stress that whenever the base MAP model performs well, by construction LULA will also perform well.\\nAs a supplement we show the performance of KFL+LULA against DPN in OOD detection on MNIST (Table 8). We found that KFL+LULA performance on OOD detection are competitive or better than DPN.\\nC.4 COMPUTATIONAL COST\\nTo supplement the cost analysis in the main text, we show the wall-clock times required for the construction and training of LULA units in Table 9.\\nC.5 DEEPER NETWORKS\\nWe also asses the performance of LULA in larger network. We use a 20-layer CNN on CIFAR10, SVHN, and CIFAR-100. Both the KFL and LULA are applied only at the last-layer of the network. The results, in terms of MMC, expected calibration error (ECE), and AUR, are presented in Table 10 and Table 11. We observe that LULA is the best method for uncertainty tuning in LA: It makes KFL better calibrated in both in- and out-distribution settings. Moreover, the LULA-imbued KFL is competitive to DE, which has been shown by Ovadia et al. (2019) to be the best Bayesian method for uncertainty quantification. Note that, KFL+LULA is a post-hoc method and thus can be applied to any pre-trained network. In contrast, DE requires training multiple networks (usually 5) from scratch which can be very expensive.\\nWe additionally show the performance of LULA when applied on top of a KFL-approximated DenseNet-121 (Huang et al., 2017) in Tables 12 and 13. LULA generally outperforms previous uncertainty tuning methods for LA and is competitive to DE. However, we observe in SVHN that LULA do not improve KFL significantly. This issue is due to the usage the Smooth Noise dataset, which has already assigned low confidence in this case, for training LULA. Thus, we re-train LULA with the Uniform Noise dataset and present the result in Table 14. We show that using this dataset, we obtain better OOD calibration performance, outperforming DE.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":197,"cells":{"paper_id":{"kind":"string","value":"SP:2a2368b5bc6b59f66af75ea37f4cbc19c8fcf50f"},"summaries":{"kind":"list like","value":["In this paper, the authors studied the possibility of sparsity exploration in Recurrent Neural Networks (RNNs) training. The main contributions include two parts: (1) Selfish-RNN training algorithm in Section 3.1 (2) SNT-ASGD optimizer in Section 3.2. The key idea of the Selfish-RNN training algorithm is a non-uniform redistribution across cell weights for better regularization. The authors mentioned previous sparse training techniques mainly focus on Multilayer Perceptron Networks (MLPs) and Convolutional Neural Networks (CNNs) rather than RNNs. This claim seems to be doubtful because one-time SVD + fine-tuning usually works very well for most RNN training applications in the industry."],"string":"[\n \"In this paper, the authors studied the possibility of sparsity exploration in Recurrent Neural Networks (RNNs) training. The main contributions include two parts: (1) Selfish-RNN training algorithm in Section 3.1 (2) SNT-ASGD optimizer in Section 3.2. The key idea of the Selfish-RNN training algorithm is a non-uniform redistribution across cell weights for better regularization. The authors mentioned previous sparse training techniques mainly focus on Multilayer Perceptron Networks (MLPs) and Convolutional Neural Networks (CNNs) rather than RNNs. This claim seems to be doubtful because one-time SVD + fine-tuning usually works very well for most RNN training applications in the industry.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Sparse neural networks have been widely applied to reduce the necessary resource requirements to train and deploy over-parameterized deep neural networks. For inference acceleration, methods that induce sparsity from a pre-trained dense network (dense-to-sparse) work effectively. Recently, dynamic sparse training (DST) has been proposed to train sparse neural networks without pre-training a large and dense network (sparse-to-sparse), so that the training process can also be accelerated. However, previous sparse-to-sparse methods mainly focus on Multilayer Perceptron Networks (MLPs) and Convolutional Neural Networks (CNNs), failing to match the performance of dense-to-sparse methods in Recurrent Neural Networks (RNNs) setting. In this paper, we propose an approach to train sparse RNNs with a fixed parameter count in one single run, without compromising performance. During training, we allow RNN layers to have a non-uniform redistribution across cell weights for a better regularization. Further, we introduce SNT-ASGD, a variant of the averaged stochastic gradient optimizer, which significantly improves the performance of all sparse training methods for RNNs. Using these strategies, we achieve state-of-the-art sparse training results, even better than dense model results, with various types of RNNs on Penn TreeBank and Wikitext-2 datasets."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"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":["Rui Dai","Lefei Li","Wenjian Yu"],"title":"Fast training and model compression of gated rnns via singular value decomposition","venue":"In 2018 International Joint Conference on Neural Networks (IJCNN),","year":2018},{"authors":["Tim Dettmers","Luke Zettlemoyer"],"title":"Sparse networks from scratch: Faster training without losing performance","venue":"arXiv preprint arXiv:1907.04840,","year":2019},{"authors":["Jeffrey L Elman"],"title":"Finding 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For a fair comparison, we use exactly the same model hyper-parameters and regularization used in ON-LSTM. We set the sparsity of each layer to 55% and the initial removing rate to 0.5. We train the model for 1000 epochs and rerun SNT-ASGD","venue":null,"year":2019},{"authors":["Evci"],"title":"2020) that while state-of-the-art sparse training method (RigL) achieves promising performance in terms of CNNs, it fails to match the performance of pruning in RNNs. Given the fact that magnitude pruning has become a widely-used and strong baseline for model compression, we also report a comparison between Selfish-RNN and iterative magnitude","venue":"I COMPARISON BETWEEN SELFISH-RNN","year":2020},{"authors":["Moradi"],"title":"The results of our paper provide motivation for new types of hardware accelerators and libraries with better support for sparse neural networks. 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Cohen\"\n ],\n \"title\": \"Breaking the softmax bottleneck: A high-rank RNN language model\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Wojciech Zaremba\",\n \"Ilya Sutskever\",\n \"Oriol Vinyals\"\n ],\n \"title\": \"Recurrent neural network regularization\",\n \"venue\": \"arXiv preprint arXiv:1409.2329,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Hattie Zhou\",\n \"Janice Lan\",\n \"Rosanne Liu\",\n \"Jason Yosinski\"\n ],\n \"title\": \"Deconstructing lottery tickets: Zeros, signs, and the supermask\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Michael Zhu\",\n \"Suyog Gupta\"\n ],\n \"title\": \"To prune, or not to prune: exploring the efficacy of pruning for model compression\",\n \"venue\": \"arXiv preprint arXiv:1710.01878,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Julian Georg Zilly\",\n \"Rupesh Kumar Srivastava\",\n \"Jan Koutnı́k\",\n \"Jürgen Schmidhuber\"\n ],\n \"title\": \"Recurrent highway networks\",\n \"venue\": \"In Proceedings of the 34th International Conference on Machine LearningVolume\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Shen\"\n ],\n \"title\": \"ON-LSTM can learn the latent tree structure of natural language by learning the order of neurons. For a fair comparison, we use exactly the same model hyper-parameters and regularization used in ON-LSTM. We set the sparsity of each layer to 55% and the initial removing rate to 0.5. We train the model for 1000 epochs and rerun SNT-ASGD\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Evci\"\n ],\n \"title\": \"2020) that while state-of-the-art sparse training method (RigL) achieves promising performance in terms of CNNs, it fails to match the performance of pruning in RNNs. Given the fact that magnitude pruning has become a widely-used and strong baseline for model compression, we also report a comparison between Selfish-RNN and iterative magnitude\",\n \"venue\": \"I COMPARISON BETWEEN SELFISH-RNN\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Moradi\"\n ],\n \"title\": \"The results of our paper provide motivation for new types of hardware accelerators and libraries with better support for sparse neural networks. Nevertheless, many recent works have been developed to accelerate sparse neural networks including Gray et al\",\n \"venue\": null,\n \"year\": 2020\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Recurrent neural networks (RNNs) (Elman, 1990), with a variant of long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997), have been highly successful in various fields, including language modeling (Mikolov et al., 2010), machine translation (Kalchbrenner & Blunsom, 2013), question answering (Hirschman et al., 1999; Wang & Jiang, 2017), etc. As a standard task to evaluate models’ ability to capture long-range context, language modeling has witnessed great progress in RNNs. Mikolov et al. (2010) demonstrated that RNNs perform much better than backoff models for language modeling. After that, various novel RNN architectures such as Recurrent Highway Networks (RHNs) (Zilly et al., 2017), Pointer Sentinel Mixture Models (Merity et al., 2017), Neural Cache Model (Grave et al., 2017), Mixture of Softmaxes (AWD-LSTM-MoS) (Yang et al., 2018), ordered neurons LSTM (ON-LSTM) (Shen et al., 2019), and effective regularization like variational dropout (Gal & Ghahramani, 2016), weight tying (Inan et al., 2017), DropConnect (Merity et al., 2018) have been proposed to significantly improve the performance of RNNs.\nAt the same time, as the performance of deep neural networks (DNNs) improves, the resources required to train and deploy deep models are becoming prohibitively large. To tackle this problem, various dense-to-sparse methods have been developed, including but not limited to pruning (LeCun et al., 1990; Han et al., 2015), Bayesian methods (Louizos et al., 2017a; Molchanov et al., 2017), distillation (Hinton et al., 2015), L1 Regularization (Wen et al., 2018), and low-rank decomposition (Jaderberg et al., 2014). Given a pre-trained model, these methods work effectively to accelerate the inference. Recently, some dynamic sparse training (DST) approaches (Mocanu et al., 2018; Mostafa & Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2020) have been proposed to bring efficiency for both, the training phase and the inference phase by dynamically changing the sparse connectivity during training. However, previous approaches are mainly for CNNs. For RNNs, the long-term dependencies and repetitive usage of recurrent cells make them more difficult to be sparsified (Kalchbrenner et al., 2018; Evci et al., 2020). More importantly, the state-of-the-art performance achieved by RNNs on language modeling is mainly associated with the optimizer, averaged stochastic gradient descent (ASGD) (Polyak & Juditsky, 1992), which is not compatible with the existing DST approaches. The above-mentioned problems heavily limit the performance\nof the off-the-shelf sparse training methods in the RNN field. For instance, while “The Rigged Lottery” (RigL) achieves state-of-the-art sparse training results with various CNNs, it fails to match the performance of the iterative pruning method in the RNN setting (Evci et al., 2020). In this paper, we introduce an algorithm to train sparse RNNs with a fixed number of computational costs throughout training. We abbreviate our sparse RNN training method as Selfish-RNN because our method encourages cell weights to obtain their parameters selfishly. The main contributions of this work are five-fold:\n• We propose an algorithm to train sparse RNNs from scratch with a fixed number of parameters. This advantage constrains the training costs to a fraction of the costs needed for training a dense model, allowing us to choose suitable sparsity levels for different types of training platforms. • We introduce SNT-ASGD, a sparse variant of the non-monotonically triggered averaged\nstochastic gradient descent optimizer, which overcomes the over-sparsified problem of the original NT-ASGD (Merity et al., 2018) caused by dynamic sparse training. • We demonstrate state-of-the-art sparse training performance with various RNN models,\nincluding stacked LSTMs (Zaremba et al., 2014), RHNs, ordered neurons LSTM (ONLSTM) on Penn TreeBank (PTB) dataset (Marcus et al., 1993) and AWD-LSTM-MoS on WikiText-2 dataset (Melis et al., 2018). • We present an approach to analyze the evolutionary trajectory of the sparse connectivity\noptimized by dynamic sparse training from the perspective of graph. With this approach, we show that there exist many good structural local optima (sparse sub-networks having equally good performance) in RNNs, which can be found in an efficient and robust manner. • Our analysis shows two surprising phenomena in the setting of RNNs contrary to CNNs:\n(1) random-based weight growth performs better than gradient-based weight growth, (2) uniform sparse distribution performs better than Erdős-Rényi (ER) sparse initialization. These results highlight the need to choose different sparse training methods for different architectures."},{"heading":"2 RELATED WORK","text":"Dense-to-Sparse. There are a large amount of works operating on a dense network to yield a sparse network. We divide them into three categories based on the training cost in terms of memory and computation. (1) Iterative Pruning and Retraining. To the best of our knowledge, pruning was first proposed by Janowsky (1989) and Mozer & Smolensky (1989) to yield a sparse network from a pre-trained network. Recently, Han et al. (2015) brought it back to people’s attention based on the idea of iterative pruning and retraining with modern architectures. Some recent works were proposed to further reduce the number of iterative retraining e.g., Narang et al. (2017); Zhu & Gupta (2017). Frankle & Carbin (2019) proposed the Lottery Ticket Hypothesis showing that the sub-networks (“winning tickets”) obtained via iterative pruning combined with their “lucky” initialization can outperform the dense networks. Zhou et al. (2019) discovered that the sign of their initialization is the crucial factor that\nmakes the “winning tickets” work. Our work shows that there exists a much more efficient and robust way to find those “winning ticketts” without any special initialization. The aforementioned methods require at least the same training cost as training a dense model, sometimes even more, as a pre-trained dense model is involved. We compare our method with state-of-the-art pruning method proposed by Zhu & Gupta (2017) in Appendix I. With fewer training costs, our method is able to discover sparse networks that can achieve lower test perplexity than iterative pruning. (2) Learning Sparsity During Training. There are also some works attempting to learn the sparse networks during training. Louizos et al. (2017b) and Wen et al. (2018) are examples that gradually enforce the network weights to zero via L0 and L1 regularization, respectively. Dai et al. (2018) proposed a singular value decomposition (SVD) based method to accelerate the training process for LSTMs. Liu et al. (2020a) proposed Dynamic Sparse Training to discover sparse structure by learning binary masks associated with network weights. However, these methods start with a fully dense network, and hence are not memory efficient. (3) One-Shot Pruning. Some works aim to find sparse neural networks by pruning once prior to the main training phase based on some salience criteria, such as connection sensitivity (Lee et al., 2019), signal propagation, (Lee et al., 2020), and gradient signal preservation (Wang et al., 2020). These techniques can find sparse networks before the standard training, but at least one iteration of dense model needs to be trained to identify the sparse sub-networks, and therefore the pruning process is not applicable to memory-limited scenarios. Additionally, one-shot pruning generally cannot match the performance of dynamic sparse training, especially at extreme sparsity levels (Wang et al., 2020).\nSparse-to-Sparse. Recently, many works have emerged to train intrinsically sparse neural networks from scratch to obtain efficiency both for training and inference. (1) Static Sparse Training. Mocanu et al. (2016) introduced intrinsically sparse networks by exploring the scale-free and small-world topological properties in Restricted Boltzmann Machines. Later, some works expand static sparse training into CNNs based on expander graphs and show comparable performance (Prabhu et al., 2018; Kepner & Robinett, 2019). (2) Dynamic Sparse Training. Mocanu et al. (2018) introduced Sparse Evolutionary Training (SET) which initializes a sparse network and dynamically changes the sparse connectivity by a simple remove-and-regrow strategy. At the same time, DeepR (Bellec et al., 2018) trained very sparse networks by sampling the sparse connectivity based on a Bayesian posterior. The iterative configuration updates have been proved to converge to a stationary distribution. Mostafa & Wang (2019) introduced Dynamic Sparse Reparameterization (DSR) to train sparse neural networks while dynamically adjusting the sparsity levels of different layers. Sparse Networks from Scratch (SNFS) (Dettmers & Zettlemoyer, 2019) improved the sparse training performance by growing free weights according to their momentum. It requires extra computation and memory to update the dense momentum tensor for each iteration. Further, Evci et al. (2020) introduced RigL which activates weights with the highest magnitude gradients. This approach grows weights expected to receive gradients with high magnitudes, while amortizing a large number of memory requirements and computational cost caused by momentum. Due to the inherent limitations of deep learning software and hardware libraries, all of the above works simulate sparsity using a binary mask over weights. More recently, Liu et al. (2020b) proved the potentials of DST by developing for the first time an independent software framework to train very large truly sparse MLPs trained with SET. However, all these works mainly focus on CNNs and MLPs, and they are not designed to match state-of-the-art performance for RNNs.\nWe summarize the properties of all approaches compared in this paper in Table 1. Same with SET, our method can guarantee Backward Sparse, which does not require any extra information from the removed weights. Additionally, we discuss the differences among SET, pruning techniques, and our method in Appendix H."},{"heading":"3 SPARSE RNN TRAINING","text":"Our sparse RNN training method is illustrated in Figure 1 with LSTM as a specific case of RNNs. Note that our method can be easily applied to any other RNN variants. The only difference is the number of cell weights. Before training, we randomly initialize each layer at the same sparsity (the fraction of zero-valued weights), so that the training costs are proportional to the dense model at the beginning. To explore more sparse structures, while to maintain a fixed sparsity level, we need to optimize the sparse connectivity together with the corresponding weights (a combinatorial optimization problem). We apply dynamic sparse connectivity and SNT-ASGD to handle this combinatorial optimization problem. The pseudocode of the full training procedure of our algorithm is shown in Algorithm 1."},{"heading":"3.1 DYNAMIC SPARSE CONNECTIVITY","text":"We consider uniform sparse initialization, magnitude weight removal, random weight growth, cell weight redistribution together as main components of our dynamic sparse connectivity method, which can ensure a fixed number of parameters and a clear sparse backward pass, as discussed next. Notation. Given a dataset of N samples D = {(xi, yi)}Ni=1 and a network f(x; θ) parameterized by θ. We train the network to minimize the loss function ∑N i=1 L(f(xi; θ), yi). The basic mechanism of sparse neural networks is to use a fraction of parameters to reparameterize the whole network, while preserving the performance as much as possible. Hence, a sparse neural network can be denoted as fs(x; θs) with a sparsity level S = 1− ‖θs‖0‖θ‖0 , where ‖ · ‖0 is the `0-norm. Uniform Sparse Initialization. First, the network is uniformly initialized with a sparse distribution in which the sparsity level of each layer is the same S. More precisely, the network is initialized by:\nθs = θ M (1)\nwhere θ is a dense weight tensor initialized in a standard way; M is a binary tensor, in which nonzero elements are sampled uniformly based on the sparsity S; refers to the Hadamard product. Magnitude Weight Removal. For non-RNN layers, we use magnitude weight removal followed by random weight growth to update the sparse connectivity. We remove a fraction p of weights with the smallest magnitude after each training epoch. This step is performed by changing the binary tensor M , as follows: M =M − P (2) where P is a binary tensor with the same shape as M , in which the nonzero elements have the same indices with the top-p smallest-magnitude nonzero weights in θs, with ||P ||0 = p||M ||0. Random Weight Growth. To keep a fixed parameter count, we randomly grow the same number of weights immediately after weight removal, by:\nM =M +R (3)\nwhere R is a binary tensor where the nonzero elements are randomly located at the position of zero elements of M . We choose random growth to get rid of using any information of the non-existing weights, so that both feedforward and backpropagation are completely sparse. It is more desirable to have such pure sparse structures as it enables the possibility of conceiving in the future specialized hardware accelerators for sparse neural networks. Besides, our analysis of growth methods in Section 4.3 shows that random growth can explore more sparse structural degrees of freedom than gradient growth, which might be crucial to the sparse training. Cell Weight Redistribution. Our dynamic sparse connectivity differs from previous methods mainly in cell weight redistribution. For RNN layers, the naive approach is to sparsify all cell weight tensors independently at the same sparsity, as shown in Liu et al. (2019) which is a straightforward extension of applying SET to RNNs. Essentially, it is more desirable to redistribute new parameters to cell weight tensors dependently, as all cell weight tensors collaborate together to regulate information. Intuitively, we redistribute new parameters in a way that weight tensors containing more largemagnitude weights should have more parameters. Large-magnitude weights indicate that their loss\ngradients are large and few oscillations occur. Thus, weight tensors with more large-magnitude connections should be reallocated with more parameters to accelerate training. Concretely, for each RNN layer l, we remove weights dependently given by an ascending sort:\nSortp(|θl1|, |θl2|, .., |θlt|) (4)\nwhere {θl1, θl2, ..., θlt} are all weight tensors within each cell, and Sortp returns p indices of the smallest-magnitude weights. After weight removal, new parameters are uniformly grown to each weight tensor to implement our cell weight redistribution gradually. We also tried other approaches including the mean value of the magnitude of nonzero weights or the mean value of the gradient magnitude of nonzero weights, but our approach achieves the best performance, as shown in Appendix B. We further demonstrate the final sparsity breakdown of cell weights learned by our method in Appendix M and observe that weights of forget gates are consistently sparser than other weights for all models. Note that redistributing parameters across cell weight tensors does not change the FLOP counting, as the sparsity of each layer is not changed. In contrast, the across-layer weight redistribution used by DSR and SNFS affects the sparsity level of each layer. As a result, it will change the number of floating-point operations (FLOPs).\nSimilar with SNFS, We also decay the removing rate p to zero with a cosine annealing. We further use Eq. (1) to enforce the sparse structure before the forward pass and after the backward pass, so that the zero-valued weights will not contribute to the loss. And all the newly activated weights are initialized to zero."},{"heading":"3.2 SPARSE NON-MONOTONICALLY TRIGGERED ASGD","text":"Non-monotonically Triggered ASGD (NT-ASGD) has been shown to achieve surprising performance with various RNNs (Merity et al., 2018; Yang et al., 2018; Shen et al., 2019). However, it becomes less appealing for sparse RNNs training. Unlike dense networks in which every parameter in the model is updated at each iteration, for sparse networks, the zero-valued weights remain zero when they are not activated. Once these zero-valued weights are activated, the original averaging operation of standard NT-ASGD will immediately bring them close to zero. Thereby, after the averaging operation is triggered, the number of valid weights will decrease sharply as shown in Figure 2. To alleviate this problem, we introduce SNT-ASGD as following:\nw̃i =\n{ 0 if mi = 0,∀i,∑K\nt=Ti wi,t (K−Ti+1) if mi = 1,∀i. (5)\nwhere w̃i is the value returned by SNT-ASGD for weight wi; wi,t represents the actual value of weight wi at the tth iteration; mi = 1 if the weight wi exists and mi = 0 means that the weight wi does not exist; Ti is the iteration in which the weight wi grows most recently; and K is the total\nnumber of iterations. We demonstrate the effectiveness of SNT-ASGD in Figure 2. At the beginning, trained with SGD, the number of weights with high magnitude increases fast. However, the trend starts to descend significantly once the optimization switches to NT-ASGD at the 80th epoch, whereas the trend of SNT-ASGD continues to rise after a small drop caused by the averaging operation.\nTo better understand how proposed components, cell weight redistribution and SNT-ASGD, improve the sparse RNN training performance, we further conduct an ablation study in Appendix A. It is clear to see that both of them lead to significant performance improvement."},{"heading":"4 EXPERIMENTAL RESULTS","text":"We evaluate Selfish-RNN with various models including stacked LSTMs, RHNs, ON-LSTM on the Penn TreeBank dataset and AWD-LSTM-MoS on the WikiText-2 dataset. The performance of Selfish-RNN is compared with 5 state-of-the-art sparse inducing techniques, including Intrinsic Sparse Structures (ISS) (Wen et al., 2018), SET, DSR, SNFS, and RigL. ISS is a method to explore sparsity inside RNNs by using group Lasso regularization. We choose Adam (Kingma & Ba, 2014) optimizer for SET, DSR, SNFS, and RigL. We also evaluate our methods with two state-of-the-art RNN models, ON-LSTM on PTB and AWD-LSTM-MoS on Wikitext-2, as reported in Appendix D and Appendix E, respectively."},{"heading":"4.1 STACKED LSTMS","text":"As introduced by Zaremba et al. (2014), stacked LSTMs (large) is a two-layer LSTM model with 1500 hidden units for each LSTM layer. We choose the same sparsity as ISS, 67% and 62%. We empirically found that 0.7 is a safe choice for the removing rate of stacked LSTMs. The clip norm is set to 0.25 and all models are trained for 100 epochs.\nResults are shown in the left side of Table 2. To evaluate our sparse training method fairly, we also provide a new dense baseline trained with the standard NT-ASGD, achieving 6 lower test perplexity than the widely-used baseline. We also test whether a small dense network and a static sparse network\nwith the same number of parameters as Selfish-RNN can match the performance of Selfish-RNN. We train a dense stacked LSTMs with 700 hidden units, named as “Small”. In line with the previous studies (Mocanu et al., 2018; Mostafa & Wang, 2019; Evci et al., 2020), both static sparse networks and the small-dense network fail to match the performance of Selfish-RNN. Training a static sparse network from scratch with uniform distribution performs better than the one with ER distribution. Trained with Adam, all sparse training techniques fail to match the performance of ISS and dense models. Models trained with SNT-ASGD obtain substantially lower perplexity, and Selfish-RNN achieves the lowest one, even better than the new dense baseline with much fewer training costs.\nTo understand better the effect of different optimizers on different DST methods, we report the performance of all DST methods trained with Adam, momentum SGD, and SNT-ASGD. The learning rate of Adam is set as 0.001. The learning rate of momentum SGD is 2 decreased by a factor of 1.33 once the loss fails to decrease and the momentum coefficient is 0.9. The weight decay is set as 1.2e-6 for all optimizers. For SNFS (SNT-ASGD), we replace momentum of weights with their gradients, as SNT-ASGD does not involve any momentum terms. We use the same hyperparameters for all DST methods. The results are shown in Table 3. It is clear that SNT-ASGD brings significant perplexity improvements to all sparse training techniques. This further stands as empirical evidence that SNT-ASGD is crucial to improve the sparse training performance in the RNN setting. Moreover, compared with other DST methods, Selfish-RNN is quite robust to the choice of optimizers due to its simple scheme to update sparse connectivity. Advanced strategies such as across-layer weight redistribution used in DSR and SNFS, gradient-based weight growth used in RigL and SNFS heavily depend on optimizers. They might work decently for some optimization methods but may not work for others.\nAdditionally, note that different DST methods use different sparse distributions, leading to very different computational costs even with the same sparsity. We also report the approximated training and inference FLOPs for all methods. The FLOP gap between Selfish-RNN and RigL is very small, whereas SNFS requires more FLOPs than our method for both training and inference (see Appendix L for details on how FLOPs are calculated). ISS achieves a lower number of FLOPs, since it does not sparsify the embedding layer and therefore, their LSTM layers are much more sparse than LSTM layers obtained by other methods. This would cause a fewer number of FLOPs as LSTM layers typically require more FLOPs than other layers."},{"heading":"4.2 RECURRENT HIGHWAY NETWORKS","text":"Recurrent Highway Networks (Zilly et al., 2017) is a variant of RNNs allowing RNNs to explore deeper architectures inside the recurrent transition. See Appendix C for experimental settings of RHN. The results are shown in the right side of Table 2. Selfish-RNN achieves better performance than the dense model with half FLOPs. Unlike the large FLOP discrepancy of stacked LSTMs, the FLOP gap between different sparse training techniques for RHNs is very small, except SNFS which requires computing dense momentum for each iteration. Additionally, ISS has similar FLOPs with Selfish-RNN for RHN, as it sparsifies the embedding layer as well."},{"heading":"4.3 ANALYZING THE PERFORMANCE OF SELFISH-RNN","text":"Analysis of Evolutionary Trajectory of Sparse Connectivity. The fact that Selfish-RNN consistently achieves good performance with different runs naturally raises some questions: e.g., are final sparse connectivities obtained by different runs similar or very different? Is the distance between the original sparse connectivity and the final sparse connectivity large or small? To answer these questions, we investigate a method based on graph edit distance (GED) (Sanfeliu & Fu, 1983) to measure the topological distance between different sparse connectivities learned by different runs. The distance is scaled between 0 and 1. The smaller the distance is, the more similar the two sparse topologies are (See Appendix J for details on how we measure the sparse topological distance).\nThe results are demonstrated in Figure 3. Figure 3-left shows how the topology of one randominitialized network evolves when trained with Selfish-RNN. We compare the topological distance between the sparse connectivity obtained at the 5th epoch and the sparse connectivities obtained in the following epochs. We can see that the distance gradually increases from 0 to a very high value 0.8, meaning that Selfish-RNN optimizes the initial topology to a very different one after training. Moreover, Figure 3-right illustrates that the topological distance between two same-initialized networks trained with different seeds after the 4th epoch. We can see that starting from the same sparse topology, they evolve to completely different sparse connectivities. Note that even when leading to completely different sparse connectivities, different runs achieve similarly good performance, which indicates that in the case of RNNs there exist many good local optima in terms of sparse connectivity that can have equally good performance. This phenomenon complements the findings of Liu et al. (2020c) which show that there are numerous sparse sub-networks performing similarly well in the context of MLPs.\nAnalysis of Sparse Initialization. We compare two types of sparse initialization, ER distribution and uniform distribution. Uniform distribution namely enforces the sparsity level of each layer to be the same as S. ER distribution allocates higher sparsity to larger layers than smaller ones. Note that its variant Erdős-Rényi-kernel proposed by Evci et al. (2020) scales back to ER for RNNs, as no kernels are involved. The results are shown as the Static group in Table 2. We can see that uniform distribution outperforms ER distribution consistently. Moreover, ER usually causes RNN layers to be less sparse than other layers, resulting in a small increase of FLOPs.\nAnalysis of Growth Methods. Methods that leverage gradient-based weight growth (SNFS and RigL) have shown superiority over the methods using random-based weight growth for CNNs. However, we observe a different behavior with RNNs. We set up a controlled experiment to compare these two methods with SNT-ASGD and momentum SGD. We report the results with various update intervals (the number of iterations between sparse connectivity updates) in Figure 4. Surprisingly, gradient-based growth performs worse than random-based growth in most cases. And there is an increased performance gap as the update interval increases. Our hypothesis is that random growth helps in exploring better the search space, as it naturally considers a large number of various sparse connectivities during training, which is crucial to the performance of dynamic sparse training. Differently, gradient growth drives the network topology towards some similar local optima for the sparse connectivity as it uses a greedy search strategy (highest gradient magnitude) at every topological change. However, benefits provided by high-magnitude gradients might change dynamically afterwards due to complicated interactions between weights. We empirically illustrate our hypothesis via the proposed distance measure between sparse connectivities in Appendix K.\nAnalysis of Hyper-parameters. The sparsity S and the initial removing rate p are two hyperparameters of our method. We show their sensitivity analysis in Appendix F and Appendix G. We find that Selfish Stacked LSTMs, RHNs, ON-LSTM, and AWD-LSTM-MoS need around 25%, 40%, 45%, and 40% parameters to reach the performance of their dense counterparts, respectively. And our method is quite robust to the choice of the initial removing rate."},{"heading":"5 CONCLUSION","text":"In this paper, we proposed an approach to train sparse RNNs from scratch with a fixed parameter count throughout training. Further, we introduced SNT-ASGD, a specially designed sparse optimizer for training sparse RNNs and we showed that it substantially improves the performance of all dynamic sparse training methods in RNNs. We observed that random-based growth achieves lower perplexity than gradient-based growth in the case of RNNs. Further, we developed an approach to compare two different sparse connectivities from the perspective of graph theory. Using this approach, we found that random-based growth explores better the topological search space for optimal sparse connectivities, whereas gradient-based growth is prone to drive the network towards similar sparse connectivity patterns. opening the path for a better understanding of sparse training."},{"heading":"A ABLATION STUDY","text":"To verify if the improvement shown above is caused by the cell weight redistribution or the Sparse NT-ASGD, we conduct an ablation study for all architectures. To avoid distractive factors, all models use the same hyper-parameters with the ones reported in the paper. And the use of finetuning is not excluded. We present the validation and testing perplexity for variants of our model without these two contributions, as shown in Table 4. Not surprisingly, removing either of these two novelties degrades the performance. There is a significant degradation in the performance for all models, up to 13 perplexity point, if the optimizer switches to the standard NT-ASGD. This stands as empirical evidence regarding the benefit of SNT-ASGD. Without cell weight redistribution, the testing perplexity also rises. The only exception is RHN whose number of redistributed weights in each layer is only two. This empirically shows that cell weight redistribution is more effective for the models with more cell weights."},{"heading":"B COMPARISON OF DIFFERENT CELL WEIGHT REDISTRIBUTION METHODS","text":"In Table 5, we conduct a small experiment to compare different methods of cell weight redistribution with stacked LSTMs, including redistributing based on the mean value of the magnitude of nonzero weights from different cell weights and the mean value of the gradient magnitude of nonzero weights."},{"heading":"C EXPERIMENTAL DETAILS FOR RHN","text":"Recurrent Highway Networks (Zilly et al., 2017) is a variant of RNNs allowing RNNs to explore deeper architecture inside the recurrent transition. Instead of stacking recurrent layers directly, RHN stacks multiple highway layers on top of recurrent state transition. Within each highway layer, free weights are redistributed across the input weight and the state weight. The sparsity level is set the same as ISS, 67.7% and 52.8%. Dropout rates are set to be 0.20 for the embedding layer, 0.65 for the input, 0.25 for the hidden units, and 0.65 for the output layer. The model is trained for 500 epochs with a learning rate of 15, a batch size of 20, and a sequence length to of 35. At the end of each training epoch, new weights are redistributed across the weights of the H nonlinear transform and the T gate."},{"heading":"D EXPERIMENTAL RESULTS WITH ON-LSTM","text":"Table 6: Single model perplexity on validation and test sets for the Penn Treebank language modeling task with ON-LSTM. Methods with “ASGD” are trained with SNT-ASGD. The numbers reported are averaged over five runs.\nModels #Param Val Test\nDense1000 25M 58.29± 0.10 56.17± 0.12 Dense1300 25M 58.55± 0.11 56.28± 0.19 SET 11.3M 65.90± 0.08 63.56± 0.14 DSR 11.3M 65.22± 0.07 62.55± 0.06 SNFS 11.3M 68.00± 0.10 65.52± 0.15 RigL 11.3M 64.41± 0.05 62.01± 0.13 RigL1000 (ASGD) 11.3M 59.17± 0.08 57.23± 0.09 RigL1300 (ASGD) 11.3M 59.10± 0.05 57.44± 0.15 Selfish-RNN1000 11.3M 58.17± 0.06 56.31± 0.10 Selfish-RNN1300 11.3M 57.67 ± 0.03 55.82 ± 0.11\nTable 7: Single model perplexity on validation and test sets for the WikiText-2 language modeling task with AWD-LSTMMoS. Baseline is AWD-LSTM-MoS obtained from Yang et al. (2018). Methods with “ASGD” are trained with SNTASGD.\nModels #Param Val Test\nDense 35M 66.01 63.33 SET 15.6M 72.82 69.61 DSR 15.6M 69.95 66.93 SNFS 15.6M 79.97 76.18 RigL 15.6M 71.36 68.52 RigL (ASGD) 15.6M 68.84 65.18\nSelfish-RNN 15.6M 65.96 63.05\nProposed by Shen et al. (2019) recently, ON-LSTM can learn the latent tree structure of natural language by learning the order of neurons. For a fair comparison, we use exactly the same model hyper-parameters and regularization used in ON-LSTM. We set the sparsity of each layer to 55% and the initial removing rate to 0.5. We train the model for 1000 epochs and rerun SNT-ASGD as a fine-tuning step once at the 500th epoch, dubbed as Selfish-RNN1000. As shown in Table 6, Selfish-RNN outperforms the dense model while reducing the model size to 11.3M. Without SNT-ASGD, sparse training techniques can not reduce the test perplexity to 60. SNT-ASGD is able to improve the performance of RigL by 5 perplexity. Moreover, one interesting observation is that one of the regularizations used in the standard ON-LSTM, DropConnect, is perfectly compatible with our method, although it also drops the hidden-to-hidden weights out randomly during training.\nIn our experiments we observe that Selfish-RNN benefits significantly from the second fine-tuning operation. We scale the learning schedule to 1300 epochs with two fine-tuning operations after 500 and 1000 epochs, respectively, dubbed as Selfish-RNN1300. It is interesting that Selfish-RNN1300 can achieve lower testing perplexity after the second fine-tuning step, whereas the dense model Dense1300 can not even reach again the perplexity that it had before the second fine-tuning. The heuristic explanation here is that our method helps the optimization escape the local optima or a local saddle point by optimizing the sparse structure, while for dense models whose energy landscape is fixed, it is very difficult for the optimizer to find its way off the saddle point or the local optima."},{"heading":"E EXPERIMENTAL RESULTS WITH AWD-LSTM-MOS","text":"We also evaluate Selfish-RNN on the WikiText-2 dataset. The model we choose is AWD-LSTM-MoS (Yang et al., 2018), which is the state-of-the-art RNN-based language model. It replaces Softmax with Mixture of Softmaxes (MoS) to alleviate the Softmax bottleneck issue in modeling natural language. For a fair comparison, we exactly follow the model hyper-parameters and regularization used in AWD-LSTM-MoS. We sparsify all layers with 55% sparsity except for the prior layer as its number of parameters is negligible. We train our model for 1000 epochs without finetuning or dynamical evaluation (Krause et al., 2018) to simply show the effectiveness of our method. As demonstrated in Table 7. Selfish AWD-LSTM-MoS can reach dense performance with 15.6M parameters."},{"heading":"F EFFECT OF SPARSITY","text":"There is a trade-off between the sparsity level S and the test perplexity of Selfish-RNN. When there are too few parameters, the sparse neural network will not have enough capacity to model the data. If the sparsity level is too small, the training acceleration will be small. Here, we analyze this trade-off by varying the sparsity level while keeping the other experimental setup the same, as shown in\nFigure 5a. We find that Selfish Stacked LSTMs, RHNs, ON-LSTM, and AWD-LSTM-MoS need around 25%, 40%, 45%, and 40% parameters to reach the performance of their dense counterparts, respectively. Generally, the performance of sparsified models is decreasing as the sparsity level increases."},{"heading":"G EFFECT OF INITIAL REMOVING RATE","text":"The initial removing rate p determines the number of removed weights at each connectivity update. We study the performance sensitivity of our algorithm to the initial removing rate p by varying it ∈ [0.3, 0.5, 0.7]. We set the sparsity level of each model as the one having the best performance in Figure 5a. Results are shown in Figure 5b. We can clearly see that our method is very robust to the choice of the initial removing rate."},{"heading":"H DIFFERENCE AMONG SET, SELFISH-RNN AND ITERATIVE PRUNING METHODS","text":"The topology update strategy of Selfish-RNN differs from SET in several important features. (1) we automatically redistribute weights across cell weights for better regularization, (2) we use magnitudebased removal instead of removing a fraction of the smallest positive weights and the largest negative weights, (3) we use uniform initialization rather than non-uniform sparse distribution like ER or ERK, as it consistently achieves better performance. Additionally, the optimizer proposed in this work, SNT-ASGD, brings substantial perplexity improvement to the sparse RNN training.\nFigure 6-left illustrates a high-level overview from an efficiency perspective of the difference between Selfish-RNN and iterative pruning techniques (Han et al., 2016; Zhu & Gupta, 2017; Frankle & Carbin, 2019). The conventional pruning and re-training techniques usually involve three steps: (1) pre-training a dense model, (2) pruning unimportant weights, and (3) re-training the pruned model to improve performance. The pruning and re-training cycles can be iterated. This iteration is taking place at least once, but it may also take place several times depending on the specific algorithms used. Therefore, the sparse networks obtained via iterative pruning at least involve pre-training a dense model. Different from the aforementioned three-step techniques, FLOPs required by Selfish-RNN is proportional to the density of the model, as it allows us to train a sparse network with a fixed number of parameters throughout training in one single run, without any re-training phases. Moreover, the overhead caused by the adaptive sparse connectivity operation is negligible, as it is operated only once per epoch."},{"heading":"I COMPARISON BETWEEN SELFISH-RNN AND PRUNING","text":"It has been shown by Evci et al. (2020) that while state-of-the-art sparse training method (RigL) achieves promising performance in terms of CNNs, it fails to match the performance of pruning in RNNs. Given the fact that magnitude pruning has become a widely-used and strong baseline for model compression, we also report a comparison between Selfish-RNN and iterative magnitude pruning with stacked LSTMs. The pruning baseline here is the Tensorflow Model Pruning library (Zhu & Gupta, 2017). The results are demonstrated in Figure 6-right.\nWe can see that Selfish-RNN exceeds the performance of pruning in most cases. An interesting phenomenon is that, with increased sparsity, we see a decreased performance gap between SelfishRNN and pruning. Especially, Selfish-RNN performs worse than pruning when the sparsity level is 95%. This can be attributed to the poor trainability problem of sparse models with extreme sparsity levels. Noted in Lee et al. (2020), the extreme sparse structure can break dynamical isometry (Saxe et al., 2014) of sparse networks, which degrades the trainability of sparse neural networks. Different from sparse training methods, pruning operates from a dense network and thus, does not have this problem."},{"heading":"J SPARSE TOPOLOGY DISTANCE MEASUREMENT","text":"Our sparse topology distance measurement considers the unit alignment based on a semi-matching technique introduced by Li et al. (2016) and a graph distance measurement based on graph edit distance (GED) (Sanfeliu & Fu, 1983). More specifically, our measurement includes the following steps:\nStep 1: We train two sparse networks with dynamic sparse training on the training dataset and store the sparse topology after each epoch. Let Wil be the set of sparse topologies for the l-th layer of network i.\nStep 2: Using the saved model, we compute the activity output on the test data, Oil ∈ Rn×m, where n is the number of hidden units and m is the number of samples.\nStep 3: We leverage the activity units of each layer to pair-wisely match topologies Wil . We achieve unit matching between a pair of networks by finding the unit in one network with the maximum correlation to the one in the other network.\nStep 4: After alignment, we apply graph edit distance (GED) to measure the similarity between pairwise Wil . Eventually, the distance is scaled to lie between 0 and 1. The smaller the distance is, the more similar the two sparse topologies are.\nHere, We choose stacked LSTMs on PTB dataset as a specific case to analyze. Specifically, we train two stacked LSTMs for 100 epochs with different random seeds. We choose a relatively small removing rate of 0.1. We start alignment at the 5th epoch to ensure a good alignment result, as at the very beginning of training networks do not learn very well. We then use the matched order of output tensors to align the pairwise topologies Wil ."},{"heading":"K TOPOLOGICAL DISTANCE OF GROWTH METHODS","text":"In this section, we empirically illustrate that gradient growth drives different networks into some similar connectivity patterns based on the proposed distance measurement between sparse connectivities. The initial removing rates are set as 0.1 for all training runs in this section. First, we measure the topological distance between two different training runs trained with gradient growth and random growth, respectively, as shown in Figure 7. We can see that, starting with very different sparse connectivity topologies, two networks trained with random growth end up at the same distance, whereas the topological distance between networks trained with gradient growth is continuously decreasing and this tendency is likely to continue as the training goes on. We further report the distance between two networks with same initialization but different training seeds when trained with gradient growth and random growth, respectively. As shown in Figure 8, the distance between sparse networks discovered by gradient growth is smaller than the distance between sparse networks discovered by random growth. These observations are in line with our hypothesis that gradient growth drives networks into some similar structures, whereas random growth explores more sparse structures spanned over the dense networks."},{"heading":"L FLOPS ANALYSIS OF DIFFERENT APPROACHES","text":"We follow the way of calculating training FLOPs layer by layer based on sparsity level sl, proposed by Evci et al. (2020). We split the process of training a sparse recurrent neural network into two steps: forward pass and backward pass.\nForward pass In order to calculate the loss of the current models given a batch of input data, the output of each layer is needed to be calculated based on a linear transformation and a non-linear activation function. Within each RNN layer, different cell weights are used to regulate information in sequence using the output of the previous time step and the input of this time step.\nBackward pass In order to update weights, during the backward pass, each layer calculates 2 quantities: the gradient of the loss function with respect to the activations of the previous layer and the gradient of the loss function with respect to its own weights. Therefore, the computational expense of backward pass is twice that of forward pass. Given that RNN models usually contain an embedding layer from which it is very efficient to pick a word vector, for models not using weight tying, we\nonly count the computations to calculate the gradient of its parameters as the training FLOPs and we omit its inference FLOPs. For models using weight tying, both the training FLOPs and the inference FLOPs are omitted.\nGiven a specific architecture, we denote fD as dense FLOPs required to finish one training iteration and fS as the corresponding sparse FLOPs (fS ≈ (1− S)fD), where S is the sparsity level. Thus fS fD for very sparse networks. Since different sparse training methods cause different sparse distribution, their FLOPs fS are also different from each other. We omit the FLOPs used to update the sparse connectivity, as it is only performed once per epoch. Overall, the total FLOPs required for one training update on one single sample are given in Table 8. The training FLOPs of dense-to-sparse methods like, ISS and pruning, are 3fD ∗ st, where st is the sparsity of the model at iteration t. Since dense-to-sparse methods require to train a dense model for a while, their training FLOPs and memory requirement are higher than our method. For methods that allow the sparsity of each layer dynamically changing e.g., DSR and SNFS, we approximate their training FLOPs via their final distribution, as their sparse distribution converge to the final distribution in the first few epochs. ER distribution causes a bit more inference FLOPs than uniform distribution because is allocates more weights to the RNN layers than other layers. SNFS requires extra FLOPs to calculate dense gradients during the backward pass. Although RigL also uses the dense gradients to assist weight growth, it only needs to calculate dense gradients every ∆T iterations, thus its averaged FLOPs is given by 3fS∆T+2fS+fD ∆T+1 . Here, we simply omit the extra FLOPs required by gradient-based growth, as it is negligible compared with the whole training FLOPs.\nFor inference, we calculate the inference FLOPs on single sample based on the final sparse distribution learned by different methods."},{"heading":"M FINAL CELL WEIGHT SPARSITY BREAKDOWN","text":"We further study the final sparsity level across cell weights learned automatically by our method. We find a consistent observation that the weight of forget gates, either the forget gate in the standard LSTM or the master forget gate in ON-LSTM, tend to be sparser than the weight of other gates, whereas the weight of cell gates and output gates are denser than the average, as shown in Figure 9. However, there is no big difference between weights in RHN, although the H nonlinear transform weight is slightly sparser than the T gate weight in most RHN layers. This phenomenon is in line with the Ablation analysis where the cell weight redistribution does not provide performance improvement for RHNs. Cell weight redistribution is more important for models with more regulating weights."},{"heading":"N LIMITATION","text":"The aforementioned training benefits have not been fully explored, as off-the-shelf software and hardware have limited support for sparse operations. The unstructured sparsity is difficult to be efficiently mapped to the existing parallel processors. The results of our paper provide motivation for new types of hardware accelerators and libraries with better support for sparse neural networks. Nevertheless, many recent works have been developed to accelerate sparse neural networks including Gray et al. (2017); Moradi et al. (2019); Ma et al. (2019); Yang & Ma (2019); Liu et al. (2020b). For instance, NVIDIA introduces the A100 GPU enabling the Fine-Grained Structured Sparsity (NVIDIA, 2020). The sparse structure is enforced by allowing two nonzero values in every four-entry vector to reduce memory storage and bandwidth by almost 2×. We do not claim that Selfish-RNN is the best way to obtain sparse recurrent neural networks, but simply highlights that it is an important future research direction to develop more efficient hardware and software to benefit from sparse neural networks."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Recurrent neural networks (RNNs) (Elman, 1990), with a variant of long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997), have been highly successful in various fields, including language modeling (Mikolov et al., 2010), machine translation (Kalchbrenner & Blunsom, 2013), question answering (Hirschman et al., 1999; Wang & Jiang, 2017), etc. As a standard task to evaluate models’ ability to capture long-range context, language modeling has witnessed great progress in RNNs. Mikolov et al. (2010) demonstrated that RNNs perform much better than backoff models for language modeling. After that, various novel RNN architectures such as Recurrent Highway Networks (RHNs) (Zilly et al., 2017), Pointer Sentinel Mixture Models (Merity et al., 2017), Neural Cache Model (Grave et al., 2017), Mixture of Softmaxes (AWD-LSTM-MoS) (Yang et al., 2018), ordered neurons LSTM (ON-LSTM) (Shen et al., 2019), and effective regularization like variational dropout (Gal & Ghahramani, 2016), weight tying (Inan et al., 2017), DropConnect (Merity et al., 2018) have been proposed to significantly improve the performance of RNNs.\\nAt the same time, as the performance of deep neural networks (DNNs) improves, the resources required to train and deploy deep models are becoming prohibitively large. To tackle this problem, various dense-to-sparse methods have been developed, including but not limited to pruning (LeCun et al., 1990; Han et al., 2015), Bayesian methods (Louizos et al., 2017a; Molchanov et al., 2017), distillation (Hinton et al., 2015), L1 Regularization (Wen et al., 2018), and low-rank decomposition (Jaderberg et al., 2014). Given a pre-trained model, these methods work effectively to accelerate the inference. Recently, some dynamic sparse training (DST) approaches (Mocanu et al., 2018; Mostafa & Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2020) have been proposed to bring efficiency for both, the training phase and the inference phase by dynamically changing the sparse connectivity during training. However, previous approaches are mainly for CNNs. For RNNs, the long-term dependencies and repetitive usage of recurrent cells make them more difficult to be sparsified (Kalchbrenner et al., 2018; Evci et al., 2020). More importantly, the state-of-the-art performance achieved by RNNs on language modeling is mainly associated with the optimizer, averaged stochastic gradient descent (ASGD) (Polyak & Juditsky, 1992), which is not compatible with the existing DST approaches. The above-mentioned problems heavily limit the performance\\nof the off-the-shelf sparse training methods in the RNN field. For instance, while “The Rigged Lottery” (RigL) achieves state-of-the-art sparse training results with various CNNs, it fails to match the performance of the iterative pruning method in the RNN setting (Evci et al., 2020). In this paper, we introduce an algorithm to train sparse RNNs with a fixed number of computational costs throughout training. We abbreviate our sparse RNN training method as Selfish-RNN because our method encourages cell weights to obtain their parameters selfishly. The main contributions of this work are five-fold:\\n• We propose an algorithm to train sparse RNNs from scratch with a fixed number of parameters. This advantage constrains the training costs to a fraction of the costs needed for training a dense model, allowing us to choose suitable sparsity levels for different types of training platforms. • We introduce SNT-ASGD, a sparse variant of the non-monotonically triggered averaged\\nstochastic gradient descent optimizer, which overcomes the over-sparsified problem of the original NT-ASGD (Merity et al., 2018) caused by dynamic sparse training. • We demonstrate state-of-the-art sparse training performance with various RNN models,\\nincluding stacked LSTMs (Zaremba et al., 2014), RHNs, ordered neurons LSTM (ONLSTM) on Penn TreeBank (PTB) dataset (Marcus et al., 1993) and AWD-LSTM-MoS on WikiText-2 dataset (Melis et al., 2018). • We present an approach to analyze the evolutionary trajectory of the sparse connectivity\\noptimized by dynamic sparse training from the perspective of graph. With this approach, we show that there exist many good structural local optima (sparse sub-networks having equally good performance) in RNNs, which can be found in an efficient and robust manner. • Our analysis shows two surprising phenomena in the setting of RNNs contrary to CNNs:\\n(1) random-based weight growth performs better than gradient-based weight growth, (2) uniform sparse distribution performs better than Erdős-Rényi (ER) sparse initialization. These results highlight the need to choose different sparse training methods for different architectures.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Dense-to-Sparse. There are a large amount of works operating on a dense network to yield a sparse network. We divide them into three categories based on the training cost in terms of memory and computation. (1) Iterative Pruning and Retraining. To the best of our knowledge, pruning was first proposed by Janowsky (1989) and Mozer & Smolensky (1989) to yield a sparse network from a pre-trained network. Recently, Han et al. (2015) brought it back to people’s attention based on the idea of iterative pruning and retraining with modern architectures. Some recent works were proposed to further reduce the number of iterative retraining e.g., Narang et al. (2017); Zhu & Gupta (2017). Frankle & Carbin (2019) proposed the Lottery Ticket Hypothesis showing that the sub-networks (“winning tickets”) obtained via iterative pruning combined with their “lucky” initialization can outperform the dense networks. Zhou et al. (2019) discovered that the sign of their initialization is the crucial factor that\\nmakes the “winning tickets” work. Our work shows that there exists a much more efficient and robust way to find those “winning ticketts” without any special initialization. The aforementioned methods require at least the same training cost as training a dense model, sometimes even more, as a pre-trained dense model is involved. We compare our method with state-of-the-art pruning method proposed by Zhu & Gupta (2017) in Appendix I. With fewer training costs, our method is able to discover sparse networks that can achieve lower test perplexity than iterative pruning. (2) Learning Sparsity During Training. There are also some works attempting to learn the sparse networks during training. Louizos et al. (2017b) and Wen et al. (2018) are examples that gradually enforce the network weights to zero via L0 and L1 regularization, respectively. Dai et al. (2018) proposed a singular value decomposition (SVD) based method to accelerate the training process for LSTMs. Liu et al. (2020a) proposed Dynamic Sparse Training to discover sparse structure by learning binary masks associated with network weights. However, these methods start with a fully dense network, and hence are not memory efficient. (3) One-Shot Pruning. Some works aim to find sparse neural networks by pruning once prior to the main training phase based on some salience criteria, such as connection sensitivity (Lee et al., 2019), signal propagation, (Lee et al., 2020), and gradient signal preservation (Wang et al., 2020). These techniques can find sparse networks before the standard training, but at least one iteration of dense model needs to be trained to identify the sparse sub-networks, and therefore the pruning process is not applicable to memory-limited scenarios. Additionally, one-shot pruning generally cannot match the performance of dynamic sparse training, especially at extreme sparsity levels (Wang et al., 2020).\\nSparse-to-Sparse. Recently, many works have emerged to train intrinsically sparse neural networks from scratch to obtain efficiency both for training and inference. (1) Static Sparse Training. Mocanu et al. (2016) introduced intrinsically sparse networks by exploring the scale-free and small-world topological properties in Restricted Boltzmann Machines. Later, some works expand static sparse training into CNNs based on expander graphs and show comparable performance (Prabhu et al., 2018; Kepner & Robinett, 2019). (2) Dynamic Sparse Training. Mocanu et al. (2018) introduced Sparse Evolutionary Training (SET) which initializes a sparse network and dynamically changes the sparse connectivity by a simple remove-and-regrow strategy. At the same time, DeepR (Bellec et al., 2018) trained very sparse networks by sampling the sparse connectivity based on a Bayesian posterior. The iterative configuration updates have been proved to converge to a stationary distribution. Mostafa & Wang (2019) introduced Dynamic Sparse Reparameterization (DSR) to train sparse neural networks while dynamically adjusting the sparsity levels of different layers. Sparse Networks from Scratch (SNFS) (Dettmers & Zettlemoyer, 2019) improved the sparse training performance by growing free weights according to their momentum. It requires extra computation and memory to update the dense momentum tensor for each iteration. Further, Evci et al. (2020) introduced RigL which activates weights with the highest magnitude gradients. This approach grows weights expected to receive gradients with high magnitudes, while amortizing a large number of memory requirements and computational cost caused by momentum. Due to the inherent limitations of deep learning software and hardware libraries, all of the above works simulate sparsity using a binary mask over weights. More recently, Liu et al. (2020b) proved the potentials of DST by developing for the first time an independent software framework to train very large truly sparse MLPs trained with SET. However, all these works mainly focus on CNNs and MLPs, and they are not designed to match state-of-the-art performance for RNNs.\\nWe summarize the properties of all approaches compared in this paper in Table 1. Same with SET, our method can guarantee Backward Sparse, which does not require any extra information from the removed weights. Additionally, we discuss the differences among SET, pruning techniques, and our method in Appendix H.\"\n },\n {\n \"heading\": \"3 SPARSE RNN TRAINING\",\n \"text\": \"Our sparse RNN training method is illustrated in Figure 1 with LSTM as a specific case of RNNs. Note that our method can be easily applied to any other RNN variants. The only difference is the number of cell weights. Before training, we randomly initialize each layer at the same sparsity (the fraction of zero-valued weights), so that the training costs are proportional to the dense model at the beginning. To explore more sparse structures, while to maintain a fixed sparsity level, we need to optimize the sparse connectivity together with the corresponding weights (a combinatorial optimization problem). We apply dynamic sparse connectivity and SNT-ASGD to handle this combinatorial optimization problem. The pseudocode of the full training procedure of our algorithm is shown in Algorithm 1.\"\n },\n {\n \"heading\": \"3.1 DYNAMIC SPARSE CONNECTIVITY\",\n \"text\": \"We consider uniform sparse initialization, magnitude weight removal, random weight growth, cell weight redistribution together as main components of our dynamic sparse connectivity method, which can ensure a fixed number of parameters and a clear sparse backward pass, as discussed next. Notation. Given a dataset of N samples D = {(xi, yi)}Ni=1 and a network f(x; θ) parameterized by θ. We train the network to minimize the loss function ∑N i=1 L(f(xi; θ), yi). The basic mechanism of sparse neural networks is to use a fraction of parameters to reparameterize the whole network, while preserving the performance as much as possible. Hence, a sparse neural network can be denoted as fs(x; θs) with a sparsity level S = 1− ‖θs‖0‖θ‖0 , where ‖ · ‖0 is the `0-norm. Uniform Sparse Initialization. First, the network is uniformly initialized with a sparse distribution in which the sparsity level of each layer is the same S. More precisely, the network is initialized by:\\nθs = θ M (1)\\nwhere θ is a dense weight tensor initialized in a standard way; M is a binary tensor, in which nonzero elements are sampled uniformly based on the sparsity S; refers to the Hadamard product. Magnitude Weight Removal. For non-RNN layers, we use magnitude weight removal followed by random weight growth to update the sparse connectivity. We remove a fraction p of weights with the smallest magnitude after each training epoch. This step is performed by changing the binary tensor M , as follows: M =M − P (2) where P is a binary tensor with the same shape as M , in which the nonzero elements have the same indices with the top-p smallest-magnitude nonzero weights in θs, with ||P ||0 = p||M ||0. Random Weight Growth. To keep a fixed parameter count, we randomly grow the same number of weights immediately after weight removal, by:\\nM =M +R (3)\\nwhere R is a binary tensor where the nonzero elements are randomly located at the position of zero elements of M . We choose random growth to get rid of using any information of the non-existing weights, so that both feedforward and backpropagation are completely sparse. It is more desirable to have such pure sparse structures as it enables the possibility of conceiving in the future specialized hardware accelerators for sparse neural networks. Besides, our analysis of growth methods in Section 4.3 shows that random growth can explore more sparse structural degrees of freedom than gradient growth, which might be crucial to the sparse training. Cell Weight Redistribution. Our dynamic sparse connectivity differs from previous methods mainly in cell weight redistribution. For RNN layers, the naive approach is to sparsify all cell weight tensors independently at the same sparsity, as shown in Liu et al. (2019) which is a straightforward extension of applying SET to RNNs. Essentially, it is more desirable to redistribute new parameters to cell weight tensors dependently, as all cell weight tensors collaborate together to regulate information. Intuitively, we redistribute new parameters in a way that weight tensors containing more largemagnitude weights should have more parameters. Large-magnitude weights indicate that their loss\\ngradients are large and few oscillations occur. Thus, weight tensors with more large-magnitude connections should be reallocated with more parameters to accelerate training. Concretely, for each RNN layer l, we remove weights dependently given by an ascending sort:\\nSortp(|θl1|, |θl2|, .., |θlt|) (4)\\nwhere {θl1, θl2, ..., θlt} are all weight tensors within each cell, and Sortp returns p indices of the smallest-magnitude weights. After weight removal, new parameters are uniformly grown to each weight tensor to implement our cell weight redistribution gradually. We also tried other approaches including the mean value of the magnitude of nonzero weights or the mean value of the gradient magnitude of nonzero weights, but our approach achieves the best performance, as shown in Appendix B. We further demonstrate the final sparsity breakdown of cell weights learned by our method in Appendix M and observe that weights of forget gates are consistently sparser than other weights for all models. Note that redistributing parameters across cell weight tensors does not change the FLOP counting, as the sparsity of each layer is not changed. In contrast, the across-layer weight redistribution used by DSR and SNFS affects the sparsity level of each layer. As a result, it will change the number of floating-point operations (FLOPs).\\nSimilar with SNFS, We also decay the removing rate p to zero with a cosine annealing. We further use Eq. (1) to enforce the sparse structure before the forward pass and after the backward pass, so that the zero-valued weights will not contribute to the loss. And all the newly activated weights are initialized to zero.\"\n },\n {\n \"heading\": \"3.2 SPARSE NON-MONOTONICALLY TRIGGERED ASGD\",\n \"text\": \"Non-monotonically Triggered ASGD (NT-ASGD) has been shown to achieve surprising performance with various RNNs (Merity et al., 2018; Yang et al., 2018; Shen et al., 2019). However, it becomes less appealing for sparse RNNs training. Unlike dense networks in which every parameter in the model is updated at each iteration, for sparse networks, the zero-valued weights remain zero when they are not activated. Once these zero-valued weights are activated, the original averaging operation of standard NT-ASGD will immediately bring them close to zero. Thereby, after the averaging operation is triggered, the number of valid weights will decrease sharply as shown in Figure 2. To alleviate this problem, we introduce SNT-ASGD as following:\\nw̃i =\\n{ 0 if mi = 0,∀i,∑K\\nt=Ti wi,t (K−Ti+1) if mi = 1,∀i. (5)\\nwhere w̃i is the value returned by SNT-ASGD for weight wi; wi,t represents the actual value of weight wi at the tth iteration; mi = 1 if the weight wi exists and mi = 0 means that the weight wi does not exist; Ti is the iteration in which the weight wi grows most recently; and K is the total\\nnumber of iterations. We demonstrate the effectiveness of SNT-ASGD in Figure 2. At the beginning, trained with SGD, the number of weights with high magnitude increases fast. However, the trend starts to descend significantly once the optimization switches to NT-ASGD at the 80th epoch, whereas the trend of SNT-ASGD continues to rise after a small drop caused by the averaging operation.\\nTo better understand how proposed components, cell weight redistribution and SNT-ASGD, improve the sparse RNN training performance, we further conduct an ablation study in Appendix A. It is clear to see that both of them lead to significant performance improvement.\"\n },\n {\n \"heading\": \"4 EXPERIMENTAL RESULTS\",\n \"text\": \"We evaluate Selfish-RNN with various models including stacked LSTMs, RHNs, ON-LSTM on the Penn TreeBank dataset and AWD-LSTM-MoS on the WikiText-2 dataset. The performance of Selfish-RNN is compared with 5 state-of-the-art sparse inducing techniques, including Intrinsic Sparse Structures (ISS) (Wen et al., 2018), SET, DSR, SNFS, and RigL. ISS is a method to explore sparsity inside RNNs by using group Lasso regularization. We choose Adam (Kingma & Ba, 2014) optimizer for SET, DSR, SNFS, and RigL. We also evaluate our methods with two state-of-the-art RNN models, ON-LSTM on PTB and AWD-LSTM-MoS on Wikitext-2, as reported in Appendix D and Appendix E, respectively.\"\n },\n {\n \"heading\": \"4.1 STACKED LSTMS\",\n \"text\": \"As introduced by Zaremba et al. (2014), stacked LSTMs (large) is a two-layer LSTM model with 1500 hidden units for each LSTM layer. We choose the same sparsity as ISS, 67% and 62%. We empirically found that 0.7 is a safe choice for the removing rate of stacked LSTMs. The clip norm is set to 0.25 and all models are trained for 100 epochs.\\nResults are shown in the left side of Table 2. To evaluate our sparse training method fairly, we also provide a new dense baseline trained with the standard NT-ASGD, achieving 6 lower test perplexity than the widely-used baseline. We also test whether a small dense network and a static sparse network\\nwith the same number of parameters as Selfish-RNN can match the performance of Selfish-RNN. We train a dense stacked LSTMs with 700 hidden units, named as “Small”. In line with the previous studies (Mocanu et al., 2018; Mostafa & Wang, 2019; Evci et al., 2020), both static sparse networks and the small-dense network fail to match the performance of Selfish-RNN. Training a static sparse network from scratch with uniform distribution performs better than the one with ER distribution. Trained with Adam, all sparse training techniques fail to match the performance of ISS and dense models. Models trained with SNT-ASGD obtain substantially lower perplexity, and Selfish-RNN achieves the lowest one, even better than the new dense baseline with much fewer training costs.\\nTo understand better the effect of different optimizers on different DST methods, we report the performance of all DST methods trained with Adam, momentum SGD, and SNT-ASGD. The learning rate of Adam is set as 0.001. The learning rate of momentum SGD is 2 decreased by a factor of 1.33 once the loss fails to decrease and the momentum coefficient is 0.9. The weight decay is set as 1.2e-6 for all optimizers. For SNFS (SNT-ASGD), we replace momentum of weights with their gradients, as SNT-ASGD does not involve any momentum terms. We use the same hyperparameters for all DST methods. The results are shown in Table 3. It is clear that SNT-ASGD brings significant perplexity improvements to all sparse training techniques. This further stands as empirical evidence that SNT-ASGD is crucial to improve the sparse training performance in the RNN setting. Moreover, compared with other DST methods, Selfish-RNN is quite robust to the choice of optimizers due to its simple scheme to update sparse connectivity. Advanced strategies such as across-layer weight redistribution used in DSR and SNFS, gradient-based weight growth used in RigL and SNFS heavily depend on optimizers. They might work decently for some optimization methods but may not work for others.\\nAdditionally, note that different DST methods use different sparse distributions, leading to very different computational costs even with the same sparsity. We also report the approximated training and inference FLOPs for all methods. The FLOP gap between Selfish-RNN and RigL is very small, whereas SNFS requires more FLOPs than our method for both training and inference (see Appendix L for details on how FLOPs are calculated). ISS achieves a lower number of FLOPs, since it does not sparsify the embedding layer and therefore, their LSTM layers are much more sparse than LSTM layers obtained by other methods. This would cause a fewer number of FLOPs as LSTM layers typically require more FLOPs than other layers.\"\n },\n {\n \"heading\": \"4.2 RECURRENT HIGHWAY NETWORKS\",\n \"text\": \"Recurrent Highway Networks (Zilly et al., 2017) is a variant of RNNs allowing RNNs to explore deeper architectures inside the recurrent transition. See Appendix C for experimental settings of RHN. The results are shown in the right side of Table 2. Selfish-RNN achieves better performance than the dense model with half FLOPs. Unlike the large FLOP discrepancy of stacked LSTMs, the FLOP gap between different sparse training techniques for RHNs is very small, except SNFS which requires computing dense momentum for each iteration. Additionally, ISS has similar FLOPs with Selfish-RNN for RHN, as it sparsifies the embedding layer as well.\"\n },\n {\n \"heading\": \"4.3 ANALYZING THE PERFORMANCE OF SELFISH-RNN\",\n \"text\": \"Analysis of Evolutionary Trajectory of Sparse Connectivity. The fact that Selfish-RNN consistently achieves good performance with different runs naturally raises some questions: e.g., are final sparse connectivities obtained by different runs similar or very different? Is the distance between the original sparse connectivity and the final sparse connectivity large or small? To answer these questions, we investigate a method based on graph edit distance (GED) (Sanfeliu & Fu, 1983) to measure the topological distance between different sparse connectivities learned by different runs. The distance is scaled between 0 and 1. The smaller the distance is, the more similar the two sparse topologies are (See Appendix J for details on how we measure the sparse topological distance).\\nThe results are demonstrated in Figure 3. Figure 3-left shows how the topology of one randominitialized network evolves when trained with Selfish-RNN. We compare the topological distance between the sparse connectivity obtained at the 5th epoch and the sparse connectivities obtained in the following epochs. We can see that the distance gradually increases from 0 to a very high value 0.8, meaning that Selfish-RNN optimizes the initial topology to a very different one after training. Moreover, Figure 3-right illustrates that the topological distance between two same-initialized networks trained with different seeds after the 4th epoch. We can see that starting from the same sparse topology, they evolve to completely different sparse connectivities. Note that even when leading to completely different sparse connectivities, different runs achieve similarly good performance, which indicates that in the case of RNNs there exist many good local optima in terms of sparse connectivity that can have equally good performance. This phenomenon complements the findings of Liu et al. (2020c) which show that there are numerous sparse sub-networks performing similarly well in the context of MLPs.\\nAnalysis of Sparse Initialization. We compare two types of sparse initialization, ER distribution and uniform distribution. Uniform distribution namely enforces the sparsity level of each layer to be the same as S. ER distribution allocates higher sparsity to larger layers than smaller ones. Note that its variant Erdős-Rényi-kernel proposed by Evci et al. (2020) scales back to ER for RNNs, as no kernels are involved. The results are shown as the Static group in Table 2. We can see that uniform distribution outperforms ER distribution consistently. Moreover, ER usually causes RNN layers to be less sparse than other layers, resulting in a small increase of FLOPs.\\nAnalysis of Growth Methods. Methods that leverage gradient-based weight growth (SNFS and RigL) have shown superiority over the methods using random-based weight growth for CNNs. However, we observe a different behavior with RNNs. We set up a controlled experiment to compare these two methods with SNT-ASGD and momentum SGD. We report the results with various update intervals (the number of iterations between sparse connectivity updates) in Figure 4. Surprisingly, gradient-based growth performs worse than random-based growth in most cases. And there is an increased performance gap as the update interval increases. Our hypothesis is that random growth helps in exploring better the search space, as it naturally considers a large number of various sparse connectivities during training, which is crucial to the performance of dynamic sparse training. Differently, gradient growth drives the network topology towards some similar local optima for the sparse connectivity as it uses a greedy search strategy (highest gradient magnitude) at every topological change. However, benefits provided by high-magnitude gradients might change dynamically afterwards due to complicated interactions between weights. We empirically illustrate our hypothesis via the proposed distance measure between sparse connectivities in Appendix K.\\nAnalysis of Hyper-parameters. The sparsity S and the initial removing rate p are two hyperparameters of our method. We show their sensitivity analysis in Appendix F and Appendix G. We find that Selfish Stacked LSTMs, RHNs, ON-LSTM, and AWD-LSTM-MoS need around 25%, 40%, 45%, and 40% parameters to reach the performance of their dense counterparts, respectively. And our method is quite robust to the choice of the initial removing rate.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"In this paper, we proposed an approach to train sparse RNNs from scratch with a fixed parameter count throughout training. Further, we introduced SNT-ASGD, a specially designed sparse optimizer for training sparse RNNs and we showed that it substantially improves the performance of all dynamic sparse training methods in RNNs. We observed that random-based growth achieves lower perplexity than gradient-based growth in the case of RNNs. Further, we developed an approach to compare two different sparse connectivities from the perspective of graph theory. Using this approach, we found that random-based growth explores better the topological search space for optimal sparse connectivities, whereas gradient-based growth is prone to drive the network towards similar sparse connectivity patterns. opening the path for a better understanding of sparse training.\"\n },\n {\n \"heading\": \"A ABLATION STUDY\",\n \"text\": \"To verify if the improvement shown above is caused by the cell weight redistribution or the Sparse NT-ASGD, we conduct an ablation study for all architectures. To avoid distractive factors, all models use the same hyper-parameters with the ones reported in the paper. And the use of finetuning is not excluded. We present the validation and testing perplexity for variants of our model without these two contributions, as shown in Table 4. Not surprisingly, removing either of these two novelties degrades the performance. There is a significant degradation in the performance for all models, up to 13 perplexity point, if the optimizer switches to the standard NT-ASGD. This stands as empirical evidence regarding the benefit of SNT-ASGD. Without cell weight redistribution, the testing perplexity also rises. The only exception is RHN whose number of redistributed weights in each layer is only two. This empirically shows that cell weight redistribution is more effective for the models with more cell weights.\"\n },\n {\n \"heading\": \"B COMPARISON OF DIFFERENT CELL WEIGHT REDISTRIBUTION METHODS\",\n \"text\": \"In Table 5, we conduct a small experiment to compare different methods of cell weight redistribution with stacked LSTMs, including redistributing based on the mean value of the magnitude of nonzero weights from different cell weights and the mean value of the gradient magnitude of nonzero weights.\"\n },\n {\n \"heading\": \"C EXPERIMENTAL DETAILS FOR RHN\",\n \"text\": \"Recurrent Highway Networks (Zilly et al., 2017) is a variant of RNNs allowing RNNs to explore deeper architecture inside the recurrent transition. Instead of stacking recurrent layers directly, RHN stacks multiple highway layers on top of recurrent state transition. Within each highway layer, free weights are redistributed across the input weight and the state weight. The sparsity level is set the same as ISS, 67.7% and 52.8%. Dropout rates are set to be 0.20 for the embedding layer, 0.65 for the input, 0.25 for the hidden units, and 0.65 for the output layer. The model is trained for 500 epochs with a learning rate of 15, a batch size of 20, and a sequence length to of 35. At the end of each training epoch, new weights are redistributed across the weights of the H nonlinear transform and the T gate.\"\n },\n {\n \"heading\": \"D EXPERIMENTAL RESULTS WITH ON-LSTM\",\n \"text\": \"Table 6: Single model perplexity on validation and test sets for the Penn Treebank language modeling task with ON-LSTM. Methods with “ASGD” are trained with SNT-ASGD. The numbers reported are averaged over five runs.\\nModels #Param Val Test\\nDense1000 25M 58.29± 0.10 56.17± 0.12 Dense1300 25M 58.55± 0.11 56.28± 0.19 SET 11.3M 65.90± 0.08 63.56± 0.14 DSR 11.3M 65.22± 0.07 62.55± 0.06 SNFS 11.3M 68.00± 0.10 65.52± 0.15 RigL 11.3M 64.41± 0.05 62.01± 0.13 RigL1000 (ASGD) 11.3M 59.17± 0.08 57.23± 0.09 RigL1300 (ASGD) 11.3M 59.10± 0.05 57.44± 0.15 Selfish-RNN1000 11.3M 58.17± 0.06 56.31± 0.10 Selfish-RNN1300 11.3M 57.67 ± 0.03 55.82 ± 0.11\\nTable 7: Single model perplexity on validation and test sets for the WikiText-2 language modeling task with AWD-LSTMMoS. Baseline is AWD-LSTM-MoS obtained from Yang et al. (2018). Methods with “ASGD” are trained with SNTASGD.\\nModels #Param Val Test\\nDense 35M 66.01 63.33 SET 15.6M 72.82 69.61 DSR 15.6M 69.95 66.93 SNFS 15.6M 79.97 76.18 RigL 15.6M 71.36 68.52 RigL (ASGD) 15.6M 68.84 65.18\\nSelfish-RNN 15.6M 65.96 63.05\\nProposed by Shen et al. (2019) recently, ON-LSTM can learn the latent tree structure of natural language by learning the order of neurons. For a fair comparison, we use exactly the same model hyper-parameters and regularization used in ON-LSTM. We set the sparsity of each layer to 55% and the initial removing rate to 0.5. We train the model for 1000 epochs and rerun SNT-ASGD as a fine-tuning step once at the 500th epoch, dubbed as Selfish-RNN1000. As shown in Table 6, Selfish-RNN outperforms the dense model while reducing the model size to 11.3M. Without SNT-ASGD, sparse training techniques can not reduce the test perplexity to 60. SNT-ASGD is able to improve the performance of RigL by 5 perplexity. Moreover, one interesting observation is that one of the regularizations used in the standard ON-LSTM, DropConnect, is perfectly compatible with our method, although it also drops the hidden-to-hidden weights out randomly during training.\\nIn our experiments we observe that Selfish-RNN benefits significantly from the second fine-tuning operation. We scale the learning schedule to 1300 epochs with two fine-tuning operations after 500 and 1000 epochs, respectively, dubbed as Selfish-RNN1300. It is interesting that Selfish-RNN1300 can achieve lower testing perplexity after the second fine-tuning step, whereas the dense model Dense1300 can not even reach again the perplexity that it had before the second fine-tuning. The heuristic explanation here is that our method helps the optimization escape the local optima or a local saddle point by optimizing the sparse structure, while for dense models whose energy landscape is fixed, it is very difficult for the optimizer to find its way off the saddle point or the local optima.\"\n },\n {\n \"heading\": \"E EXPERIMENTAL RESULTS WITH AWD-LSTM-MOS\",\n \"text\": \"We also evaluate Selfish-RNN on the WikiText-2 dataset. The model we choose is AWD-LSTM-MoS (Yang et al., 2018), which is the state-of-the-art RNN-based language model. It replaces Softmax with Mixture of Softmaxes (MoS) to alleviate the Softmax bottleneck issue in modeling natural language. For a fair comparison, we exactly follow the model hyper-parameters and regularization used in AWD-LSTM-MoS. We sparsify all layers with 55% sparsity except for the prior layer as its number of parameters is negligible. We train our model for 1000 epochs without finetuning or dynamical evaluation (Krause et al., 2018) to simply show the effectiveness of our method. As demonstrated in Table 7. Selfish AWD-LSTM-MoS can reach dense performance with 15.6M parameters.\"\n },\n {\n \"heading\": \"F EFFECT OF SPARSITY\",\n \"text\": \"There is a trade-off between the sparsity level S and the test perplexity of Selfish-RNN. When there are too few parameters, the sparse neural network will not have enough capacity to model the data. If the sparsity level is too small, the training acceleration will be small. Here, we analyze this trade-off by varying the sparsity level while keeping the other experimental setup the same, as shown in\\nFigure 5a. We find that Selfish Stacked LSTMs, RHNs, ON-LSTM, and AWD-LSTM-MoS need around 25%, 40%, 45%, and 40% parameters to reach the performance of their dense counterparts, respectively. Generally, the performance of sparsified models is decreasing as the sparsity level increases.\"\n },\n {\n \"heading\": \"G EFFECT OF INITIAL REMOVING RATE\",\n \"text\": \"The initial removing rate p determines the number of removed weights at each connectivity update. We study the performance sensitivity of our algorithm to the initial removing rate p by varying it ∈ [0.3, 0.5, 0.7]. We set the sparsity level of each model as the one having the best performance in Figure 5a. Results are shown in Figure 5b. We can clearly see that our method is very robust to the choice of the initial removing rate.\"\n },\n {\n \"heading\": \"H DIFFERENCE AMONG SET, SELFISH-RNN AND ITERATIVE PRUNING METHODS\",\n \"text\": \"The topology update strategy of Selfish-RNN differs from SET in several important features. (1) we automatically redistribute weights across cell weights for better regularization, (2) we use magnitudebased removal instead of removing a fraction of the smallest positive weights and the largest negative weights, (3) we use uniform initialization rather than non-uniform sparse distribution like ER or ERK, as it consistently achieves better performance. Additionally, the optimizer proposed in this work, SNT-ASGD, brings substantial perplexity improvement to the sparse RNN training.\\nFigure 6-left illustrates a high-level overview from an efficiency perspective of the difference between Selfish-RNN and iterative pruning techniques (Han et al., 2016; Zhu & Gupta, 2017; Frankle & Carbin, 2019). The conventional pruning and re-training techniques usually involve three steps: (1) pre-training a dense model, (2) pruning unimportant weights, and (3) re-training the pruned model to improve performance. The pruning and re-training cycles can be iterated. This iteration is taking place at least once, but it may also take place several times depending on the specific algorithms used. Therefore, the sparse networks obtained via iterative pruning at least involve pre-training a dense model. Different from the aforementioned three-step techniques, FLOPs required by Selfish-RNN is proportional to the density of the model, as it allows us to train a sparse network with a fixed number of parameters throughout training in one single run, without any re-training phases. Moreover, the overhead caused by the adaptive sparse connectivity operation is negligible, as it is operated only once per epoch.\"\n },\n {\n \"heading\": \"I COMPARISON BETWEEN SELFISH-RNN AND PRUNING\",\n \"text\": \"It has been shown by Evci et al. (2020) that while state-of-the-art sparse training method (RigL) achieves promising performance in terms of CNNs, it fails to match the performance of pruning in RNNs. Given the fact that magnitude pruning has become a widely-used and strong baseline for model compression, we also report a comparison between Selfish-RNN and iterative magnitude pruning with stacked LSTMs. The pruning baseline here is the Tensorflow Model Pruning library (Zhu & Gupta, 2017). The results are demonstrated in Figure 6-right.\\nWe can see that Selfish-RNN exceeds the performance of pruning in most cases. An interesting phenomenon is that, with increased sparsity, we see a decreased performance gap between SelfishRNN and pruning. Especially, Selfish-RNN performs worse than pruning when the sparsity level is 95%. This can be attributed to the poor trainability problem of sparse models with extreme sparsity levels. Noted in Lee et al. (2020), the extreme sparse structure can break dynamical isometry (Saxe et al., 2014) of sparse networks, which degrades the trainability of sparse neural networks. Different from sparse training methods, pruning operates from a dense network and thus, does not have this problem.\"\n },\n {\n \"heading\": \"J SPARSE TOPOLOGY DISTANCE MEASUREMENT\",\n \"text\": \"Our sparse topology distance measurement considers the unit alignment based on a semi-matching technique introduced by Li et al. (2016) and a graph distance measurement based on graph edit distance (GED) (Sanfeliu & Fu, 1983). More specifically, our measurement includes the following steps:\\nStep 1: We train two sparse networks with dynamic sparse training on the training dataset and store the sparse topology after each epoch. Let Wil be the set of sparse topologies for the l-th layer of network i.\\nStep 2: Using the saved model, we compute the activity output on the test data, Oil ∈ Rn×m, where n is the number of hidden units and m is the number of samples.\\nStep 3: We leverage the activity units of each layer to pair-wisely match topologies Wil . We achieve unit matching between a pair of networks by finding the unit in one network with the maximum correlation to the one in the other network.\\nStep 4: After alignment, we apply graph edit distance (GED) to measure the similarity between pairwise Wil . Eventually, the distance is scaled to lie between 0 and 1. The smaller the distance is, the more similar the two sparse topologies are.\\nHere, We choose stacked LSTMs on PTB dataset as a specific case to analyze. Specifically, we train two stacked LSTMs for 100 epochs with different random seeds. We choose a relatively small removing rate of 0.1. We start alignment at the 5th epoch to ensure a good alignment result, as at the very beginning of training networks do not learn very well. We then use the matched order of output tensors to align the pairwise topologies Wil .\"\n },\n {\n \"heading\": \"K TOPOLOGICAL DISTANCE OF GROWTH METHODS\",\n \"text\": \"In this section, we empirically illustrate that gradient growth drives different networks into some similar connectivity patterns based on the proposed distance measurement between sparse connectivities. The initial removing rates are set as 0.1 for all training runs in this section. First, we measure the topological distance between two different training runs trained with gradient growth and random growth, respectively, as shown in Figure 7. We can see that, starting with very different sparse connectivity topologies, two networks trained with random growth end up at the same distance, whereas the topological distance between networks trained with gradient growth is continuously decreasing and this tendency is likely to continue as the training goes on. We further report the distance between two networks with same initialization but different training seeds when trained with gradient growth and random growth, respectively. As shown in Figure 8, the distance between sparse networks discovered by gradient growth is smaller than the distance between sparse networks discovered by random growth. These observations are in line with our hypothesis that gradient growth drives networks into some similar structures, whereas random growth explores more sparse structures spanned over the dense networks.\"\n },\n {\n \"heading\": \"L FLOPS ANALYSIS OF DIFFERENT APPROACHES\",\n \"text\": \"We follow the way of calculating training FLOPs layer by layer based on sparsity level sl, proposed by Evci et al. (2020). We split the process of training a sparse recurrent neural network into two steps: forward pass and backward pass.\\nForward pass In order to calculate the loss of the current models given a batch of input data, the output of each layer is needed to be calculated based on a linear transformation and a non-linear activation function. Within each RNN layer, different cell weights are used to regulate information in sequence using the output of the previous time step and the input of this time step.\\nBackward pass In order to update weights, during the backward pass, each layer calculates 2 quantities: the gradient of the loss function with respect to the activations of the previous layer and the gradient of the loss function with respect to its own weights. Therefore, the computational expense of backward pass is twice that of forward pass. Given that RNN models usually contain an embedding layer from which it is very efficient to pick a word vector, for models not using weight tying, we\\nonly count the computations to calculate the gradient of its parameters as the training FLOPs and we omit its inference FLOPs. For models using weight tying, both the training FLOPs and the inference FLOPs are omitted.\\nGiven a specific architecture, we denote fD as dense FLOPs required to finish one training iteration and fS as the corresponding sparse FLOPs (fS ≈ (1− S)fD), where S is the sparsity level. Thus fS fD for very sparse networks. Since different sparse training methods cause different sparse distribution, their FLOPs fS are also different from each other. We omit the FLOPs used to update the sparse connectivity, as it is only performed once per epoch. Overall, the total FLOPs required for one training update on one single sample are given in Table 8. The training FLOPs of dense-to-sparse methods like, ISS and pruning, are 3fD ∗ st, where st is the sparsity of the model at iteration t. Since dense-to-sparse methods require to train a dense model for a while, their training FLOPs and memory requirement are higher than our method. For methods that allow the sparsity of each layer dynamically changing e.g., DSR and SNFS, we approximate their training FLOPs via their final distribution, as their sparse distribution converge to the final distribution in the first few epochs. ER distribution causes a bit more inference FLOPs than uniform distribution because is allocates more weights to the RNN layers than other layers. SNFS requires extra FLOPs to calculate dense gradients during the backward pass. Although RigL also uses the dense gradients to assist weight growth, it only needs to calculate dense gradients every ∆T iterations, thus its averaged FLOPs is given by 3fS∆T+2fS+fD ∆T+1 . Here, we simply omit the extra FLOPs required by gradient-based growth, as it is negligible compared with the whole training FLOPs.\\nFor inference, we calculate the inference FLOPs on single sample based on the final sparse distribution learned by different methods.\"\n },\n {\n \"heading\": \"M FINAL CELL WEIGHT SPARSITY BREAKDOWN\",\n \"text\": \"We further study the final sparsity level across cell weights learned automatically by our method. We find a consistent observation that the weight of forget gates, either the forget gate in the standard LSTM or the master forget gate in ON-LSTM, tend to be sparser than the weight of other gates, whereas the weight of cell gates and output gates are denser than the average, as shown in Figure 9. However, there is no big difference between weights in RHN, although the H nonlinear transform weight is slightly sparser than the T gate weight in most RHN layers. This phenomenon is in line with the Ablation analysis where the cell weight redistribution does not provide performance improvement for RHNs. Cell weight redistribution is more important for models with more regulating weights.\"\n },\n {\n \"heading\": \"N LIMITATION\",\n \"text\": \"The aforementioned training benefits have not been fully explored, as off-the-shelf software and hardware have limited support for sparse operations. The unstructured sparsity is difficult to be efficiently mapped to the existing parallel processors. The results of our paper provide motivation for new types of hardware accelerators and libraries with better support for sparse neural networks. Nevertheless, many recent works have been developed to accelerate sparse neural networks including Gray et al. (2017); Moradi et al. (2019); Ma et al. (2019); Yang & Ma (2019); Liu et al. (2020b). For instance, NVIDIA introduces the A100 GPU enabling the Fine-Grained Structured Sparsity (NVIDIA, 2020). The sparse structure is enforced by allowing two nonzero values in every four-entry vector to reduce memory storage and bandwidth by almost 2×. We do not claim that Selfish-RNN is the best way to obtain sparse recurrent neural networks, but simply highlights that it is an important future research direction to develop more efficient hardware and software to benefit from sparse neural networks.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":198,"cells":{"paper_id":{"kind":"string","value":"SP:60d704b4a1555e24c09963617c879a15d8f3c805"},"summaries":{"kind":"list like","value":["This paper proposes a spatial-temporal graph neural network, which is designed to adaptively capture the complex spatial-temporal dependency. Further, the authors design a spatial-temporal attention module, which aims to capture multi-scale correlations. For multi-step prediction instead of one-step prediction, they further propose the sequence transform block to solve the problem of error accumulations. The authors conducted experiments on three real-world datasets (traffic on highways and mobile traffic), which shows their method achieves the best performance."],"string":"[\n \"This paper proposes a spatial-temporal graph neural network, which is designed to adaptively capture the complex spatial-temporal dependency. Further, the authors design a spatial-temporal attention module, which aims to capture multi-scale correlations. For multi-step prediction instead of one-step prediction, they further propose the sequence transform block to solve the problem of error accumulations. The authors conducted experiments on three real-world datasets (traffic on highways and mobile traffic), which shows their method achieves the best performance.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"Spatial-temporal data forecasting is of great importance for industries such as telecom network operation and transportation management. However, spatialtemporal data are inherent with complex spatial-temporal correlations and behaves heterogeneities among the spatial and temporal aspects, which makes the forecasting remain as a very challenging task though recently great work has been done. In this paper, we propose a novel model, Adaptive Spatial-Temporal Inception Graph Convolution Networks (ASTI-GCN), to solve the multi-step spatial-temporal data forecasting problem. The model proposes multi-scale spatial-temporal joint graph convolution block to directly model the spatial-temporal joint correlations without introducing elaborately constructed mechanisms. Moreover inception mechanism combined with the graph node-level attention is introduced to make the model capture the heterogeneous nature of the graph adaptively. Our experiments on three real-world datasets from two different fields consistently show ASTI-GCN outperforms the state-of-the-art performance. In addition, ASTI-GCN is proved to generalize well."},"authors":{"kind":"list like","value":[],"string":"[]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Lei Bai","Lina Yao","Salil Kanhere","Xianzhi Wang","Quan Sheng"],"title":"Stg2seq: Spatialtemporal graph to sequence model for multi-step passenger demand forecasting","venue":"arXiv preprint arXiv:1905.10069,","year":2019},{"authors":["Lei Bai","Lina Yao","Can Li","Xianzhi Wang","Can Wang"],"title":"Adaptive graph convolutional recurrent network for traffic forecasting","venue":null,"year":2020},{"authors":["Chao Chen","Karl Petty","Alexander Skabardonis","Pravin Varaiya","Zhanfeng Jia"],"title":"Freeway performance measurement system: Mining loop detector data","venue":"Transportation Research Record Journal of the Transportation Research Board,","year":2001},{"authors":["G Barlacchi","M D Nadai","R Larcher","A Casella","C Chitic","G Torrisi","F Antonelli","A Vespignani","A Pentland","B. Lepri"],"title":"A multi-source dataset of urban life in the city of milan and the province of trentino","venue":"Scientific Data,","year":2015},{"authors":["S Gowrishankar","P.S. Satyanarayana"],"title":"A time series modeling and prediction of wireless network traffic","venue":"International Journal of Interactive Mobile Technologies (iJIM),","year":2009},{"authors":["Shengnan Guo","Youfang Lin","Ning Feng","Chao Song","Huaiyu Wan"],"title":"Attention based spatialtemporal graph convolutional networks for traffic flow forecasting","venue":"In Proceedings of the AAAI Conference on Artificial Intelligence,","year":2019},{"authors":["Kaiwen He","Yufen Huang","Xu Chen","Zhi Zhou","Shuai Yu"],"title":"Graph attention spatial-temporal network for deep learning based mobile traffic prediction","venue":"IEEE Global Communications Conference (GLOBECOM),","year":2019},{"authors":["S Hochreiter","J Schmidhuber"],"title":"Long short-term memory","venue":"Neural Computation,","year":1997},{"authors":["Thomas N Kipf","Max Welling"],"title":"Semi-supervised classification with graph convolutional networks","venue":"arXiv preprint arXiv:1609.02907,","year":2016},{"authors":["Xiang Li","Wenhai Wang","Xiaolin Hu","Jian Yang"],"title":"Selective kernel networks. 2019","venue":null,"year":2019},{"authors":["Yaguang Li","Rose Yu","Cyrus Shahabi","Yan Liu"],"title":"Diffusion convolutional recurrent neural network: Data-driven traffic forecasting","venue":"arXiv preprint arXiv:1707.01926,","year":2017},{"authors":["Yuxuan Liang","Songyu Ke","Junbo Zhang","Xiuwen Yi","Yu Zheng"],"title":"Geoman: Multi-level attention networks for geo-sensory time series prediction","venue":"In IJCAI,","year":2018},{"authors":["Chao Song","Youfang Lin","Shengnan Guo","Huaiyu Wan"],"title":"Spatial-temporal synchronous graph convolutional networks: A new framework for spatial-temporal network data forecasting","venue":"In Proceedings of the AAAI Conference on Artificial Intelligence,","year":2020},{"authors":["Christian Szegedy","Wei Liu","Yangqing Jia","Pierre Sermanet","Scott E. Reed","Dragomir Anguelov","Dumitru Erhan","Vincent Vanhoucke","Andrew Rabinovich"],"title":"Going deeper with convolutions","venue":"In IEEE Conference on Computer Vision and Pattern Recognition,","year":2015},{"authors":["Ashish Vaswani","Noam Shazeer","Niki Parmar","Jakob Uszkoreit","Llion Jones","Aidan N Gomez","Lukasz Kaiser","Illia Polosukhin"],"title":"Attention is all you need","venue":null,"year":2017},{"authors":["Billy M Williams","Lester A Hoel"],"title":"Modeling and forecasting vehicular traffic flow as a seasonal arima process: Theoretical basis and empirical results","venue":"Journal of transportation engineering,","year":2003},{"authors":["SHI Xingjian","Zhourong Chen","Hao Wang","Dit-Yan Yeung","Wai-Kin Wong","Wang-chun Woo"],"title":"Convolutional lstm network: A machine learning approach for precipitation nowcasting","venue":"In Advances in neural information processing systems,","year":2015},{"authors":["Bing Yu","Haoteng Yin","Zhanxing Zhu"],"title":"Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting","venue":"arXiv preprint arXiv:1709.04875,","year":2017},{"authors":["Chuanpan Zheng","Xiaoliang Fan","Cheng Wang","Jianzhong Qi"],"title":"Gman: A graph multi-attention network for traffic prediction","venue":"In Proceedings of the AAAI Conference on Artificial Intelligence,","year":2020},{"authors":["Jie Zhou","Ganqu Cui","Zhengyan Zhang","Cheng Yang","Zhiyuan Liu","Lifeng Wang","Changcheng Li","Maosong Sun"],"title":"Graph neural networks: A review of methods and applications","venue":"arXiv preprint arXiv:1812.08434,","year":2018}],"string":"[\n {\n \"authors\": [\n \"Lei Bai\",\n \"Lina Yao\",\n \"Salil Kanhere\",\n \"Xianzhi Wang\",\n \"Quan Sheng\"\n ],\n \"title\": \"Stg2seq: Spatialtemporal graph to sequence model for multi-step passenger demand forecasting\",\n \"venue\": \"arXiv preprint arXiv:1905.10069,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Lei Bai\",\n \"Lina Yao\",\n \"Can Li\",\n \"Xianzhi Wang\",\n \"Can Wang\"\n ],\n \"title\": \"Adaptive graph convolutional recurrent network for traffic forecasting\",\n \"venue\": null,\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Chao Chen\",\n \"Karl Petty\",\n \"Alexander Skabardonis\",\n \"Pravin Varaiya\",\n \"Zhanfeng Jia\"\n ],\n \"title\": \"Freeway performance measurement system: Mining loop detector data\",\n \"venue\": \"Transportation Research Record Journal of the Transportation Research Board,\",\n \"year\": 2001\n },\n {\n \"authors\": [\n \"G Barlacchi\",\n \"M D Nadai\",\n \"R Larcher\",\n \"A Casella\",\n \"C Chitic\",\n \"G Torrisi\",\n \"F Antonelli\",\n \"A Vespignani\",\n \"A Pentland\",\n \"B. Lepri\"\n ],\n \"title\": \"A multi-source dataset of urban life in the city of milan and the province of trentino\",\n \"venue\": \"Scientific Data,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"S Gowrishankar\",\n \"P.S. Satyanarayana\"\n ],\n \"title\": \"A time series modeling and prediction of wireless network traffic\",\n \"venue\": \"International Journal of Interactive Mobile Technologies (iJIM),\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Shengnan Guo\",\n \"Youfang Lin\",\n \"Ning Feng\",\n \"Chao Song\",\n \"Huaiyu Wan\"\n ],\n \"title\": \"Attention based spatialtemporal graph convolutional networks for traffic flow forecasting\",\n \"venue\": \"In Proceedings of the AAAI Conference on Artificial Intelligence,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Kaiwen He\",\n \"Yufen Huang\",\n \"Xu Chen\",\n \"Zhi Zhou\",\n \"Shuai Yu\"\n ],\n \"title\": \"Graph attention spatial-temporal network for deep learning based mobile traffic prediction\",\n \"venue\": \"IEEE Global Communications Conference (GLOBECOM),\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"S Hochreiter\",\n \"J Schmidhuber\"\n ],\n \"title\": \"Long short-term memory\",\n \"venue\": \"Neural Computation,\",\n \"year\": 1997\n },\n {\n \"authors\": [\n \"Thomas N Kipf\",\n \"Max Welling\"\n ],\n \"title\": \"Semi-supervised classification with graph convolutional networks\",\n \"venue\": \"arXiv preprint arXiv:1609.02907,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Xiang Li\",\n \"Wenhai Wang\",\n \"Xiaolin Hu\",\n \"Jian Yang\"\n ],\n \"title\": \"Selective kernel networks. 2019\",\n \"venue\": null,\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yaguang Li\",\n \"Rose Yu\",\n \"Cyrus Shahabi\",\n \"Yan Liu\"\n ],\n \"title\": \"Diffusion convolutional recurrent neural network: Data-driven traffic forecasting\",\n \"venue\": \"arXiv preprint arXiv:1707.01926,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Yuxuan Liang\",\n \"Songyu Ke\",\n \"Junbo Zhang\",\n \"Xiuwen Yi\",\n \"Yu Zheng\"\n ],\n \"title\": \"Geoman: Multi-level attention networks for geo-sensory time series prediction\",\n \"venue\": \"In IJCAI,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Chao Song\",\n \"Youfang Lin\",\n \"Shengnan Guo\",\n \"Huaiyu Wan\"\n ],\n \"title\": \"Spatial-temporal synchronous graph convolutional networks: A new framework for spatial-temporal network data forecasting\",\n \"venue\": \"In Proceedings of the AAAI Conference on Artificial Intelligence,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Christian Szegedy\",\n \"Wei Liu\",\n \"Yangqing Jia\",\n \"Pierre Sermanet\",\n \"Scott E. Reed\",\n \"Dragomir Anguelov\",\n \"Dumitru Erhan\",\n \"Vincent Vanhoucke\",\n \"Andrew Rabinovich\"\n ],\n \"title\": \"Going deeper with convolutions\",\n \"venue\": \"In IEEE Conference on Computer Vision and Pattern Recognition,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Ashish Vaswani\",\n \"Noam Shazeer\",\n \"Niki Parmar\",\n \"Jakob Uszkoreit\",\n \"Llion Jones\",\n \"Aidan N Gomez\",\n \"Lukasz Kaiser\",\n \"Illia Polosukhin\"\n ],\n \"title\": \"Attention is all you need\",\n \"venue\": null,\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Billy M Williams\",\n \"Lester A Hoel\"\n ],\n \"title\": \"Modeling and forecasting vehicular traffic flow as a seasonal arima process: Theoretical basis and empirical results\",\n \"venue\": \"Journal of transportation engineering,\",\n \"year\": 2003\n },\n {\n \"authors\": [\n \"SHI Xingjian\",\n \"Zhourong Chen\",\n \"Hao Wang\",\n \"Dit-Yan Yeung\",\n \"Wai-Kin Wong\",\n \"Wang-chun Woo\"\n ],\n \"title\": \"Convolutional lstm network: A machine learning approach for precipitation nowcasting\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Bing Yu\",\n \"Haoteng Yin\",\n \"Zhanxing Zhu\"\n ],\n \"title\": \"Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting\",\n \"venue\": \"arXiv preprint arXiv:1709.04875,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Chuanpan Zheng\",\n \"Xiaoliang Fan\",\n \"Cheng Wang\",\n \"Jianzhong Qi\"\n ],\n \"title\": \"Gman: A graph multi-attention network for traffic prediction\",\n \"venue\": \"In Proceedings of the AAAI Conference on Artificial Intelligence,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Jie Zhou\",\n \"Ganqu Cui\",\n \"Zhengyan Zhang\",\n \"Cheng Yang\",\n \"Zhiyuan Liu\",\n \"Lifeng Wang\",\n \"Changcheng Li\",\n \"Maosong Sun\"\n ],\n \"title\": \"Graph neural networks: A review of methods and applications\",\n \"venue\": \"arXiv preprint arXiv:1812.08434,\",\n \"year\": 2018\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"Spatial-temporal data forecasting has attracted attention from researchers due to its wide range of applications and the same specific characteristics of spatial-temporal data. Typical applications include mobile traffic forecast (He et al., 2019), traffic road condition forecast (Song et al., 2020; Yu et al., 2017; Guo et al., 2019; Zheng et al., 2020; Li et al., 2017), on-demand vehicle sharing services passenger demand forecast (Bai et al., 2019) and geo-sensory time series prediction (Liang et al., 2018) etc. The accurate forecast is the foundation of many real-world applications, such as Intelligent Telecom Network Operation and Intelligent Transportation Systems (ITS). Specifically, accurate traffic forecast can help transportation agencies better control traffic scheduling and reduce traffic congestion; The traffic volumes prediction of the wireless telecommunication network plays an important role for the network operation and optimization, for example, it can help to infer the accurate sleep periods (low traffic periods) of the base stations to achieve energy saving without sacrificing customer experience.\nHowever, as we all know, accurate spatialtemporal data forecasting faces multiple challenges. First, it is inherent with complex spatial-temporal correlations. In the spatialtemporal graph, different neighbors may have different impacts on the central location at the same time step, as the bold lines shown in Figure1, which called spatial correlations.Different historical observations of the same location influence the future moments of itself variously due to temporal correlations. The observations\nof different neighbors at historical moments can directly affect the central node at future time steps due to the spatial-temporal joint correlations. As shown in Figure1, the information of the spatialtemporal network can propagate along the spatial and temporal dimensions simultaneously, and the\ntransmission process can be discontinuous due to complex external factors, which result in spatialtemporal joint correlations of the spatial-temporal data in a short period.\nSpatial-temporal data is heterogenous in both spatial and temporal dimensions (Song et al., 2020). Nodes in different regions of the graph have various properties and local spatial structures, so the corresponding data distribution can be different. For example, the traffic flow distribution of urban and suburban areas are quite different, while the traffic of urban area is denser and that of suburban area is relatively sparse. Besides, the traffic flow in the same region also exhibit heterogeneity in different time periods. For example, the mobile traffic in business district would decrease at night compared to the daytime, while it’s opposite in the residential district. In addition, multi-step time series forecasting is often accompanied by error accumulation problem. Typical methods like RNNs often cause error accumulation due to iterative forecasting, leading to rapid deterioration of the long-term prediction accuracy. (Yu et al., 2017; Zheng et al., 2020).\nMost of the previous work is mainly to solve the above challenges. To model the spatial-temporal dependency, STGCN (Yu et al., 2017) and DCRNN (Li et al., 2017) extract spatial and temporal correlations separately. ASTGCN (Guo et al., 2019) introduced spatial and temporal attention to model the dynamic spatial and temporal correlations. STG2Seq (Bai et al., 2019) aimed at using GCN to capture spatial and temporal correlations simultaneously. But they all didn’t consider the spatialtemporal joint correlations and heterogeneity. Different from the above methods, STSGCN (Song et al., 2020) used multiple local spatial-temporal graphs to model the spatial-temporal synchronous correlations and spatial-temporal heterogeneity of the local adjacent time steps. But STSGCN can only model the spatial-temporal synchronous correlations of its defined local spatial-temporal graphs and it is equipped with complex structure.\nIn this paper, we propose a novel model called ASTI-GCN, Adaptive spatial-temporal Inception Graph Convolutional Networks, to address the above issues with multi-step spatial-temporal data forecasting. We propose the spatial-temporal joint convolution to directly model the spatial-temporal joint correlations without introducing elaborately constructed mechanisms. And we introduce the inception mechanism to build multi-scale spatial-temporal features to adapt to graph nodes with different properties. Then, to achieve the heterogeneity modeling, we construct the spatial-temporal Inception Graph Convolution Module, which combined the spatial-temporal inception mechanism with the graph attention to build the adaptive ability of graph nodes with different properties. After multiple spatial-temporal inception-GCMs, two decoder modules named sequence decoder and short-term decoder are designed to directly establish the relationships between the historical and future time steps to alleviate error accumulation.\nOverall, our main contributions are summarized as follows:\n• We propose a novel spatial-temporal joint graph convolution network to directly capture spatial-temporal correlations. Moreover, we introduce inception with graph attention to adaptively model the graph heterogeneity.\n• We propose to combine the sequence decoder and short-term decoder together for multistep forecasting to model direct relationships between historical and future time steps to alleviate the error propagation.\n• We evaluate our model on three real-world datasets from two fields, and the experimental results show that our model achieves the best performances among all the eight baselines with good generalization ability."},{"heading":"2 RELATED WORK","text":"Spatial-temporal data information can be extracted using the deep learning method from European space, such as ConvLSTM (Xingjian et al., 2015), PredRNN (Gowrishankar & Satyanarayana, 2009) and so on. However, most of the spatial-temporal data in real scenes are graph data with complex and changeable relationships.Common timing prediction models, such as HA and ARIMA (Williams & Hoel, 2003), cannot be simply migrated to such scenarios. Graph based methods like DCRNN (Li et al., 2017) modeled traffic flow as a diffusion process on a directed graph. Spatial dependencies and temporal dependencies are captured by bidirectional random walk and DCGRU based\nencoder-decoder sequence to sequence learning framework respectively. STGCN (Yu et al., 2017) constructed an undirected graph of traffic network, which is combined with GCN and CNN to model spatial and temporal correlation respectively. ASTGCN (Guo et al., 2019) innovatively introduced attention mechanisms to capture dynamic spatial and temporal dependencies. Similarly, GMAN (Zheng et al., 2020) used temporal and spatial attention to extract dynamic spatial-temporal correlations with spatial-temporal coding. The above models extract spatial-temporal correlation with two separate modules, which cannot learn the influence of neighbor node at the same time and the influence of center node at the historical moment simultaneously. To address this problem, Bai et al. (2019) proposed STG2seq to learn the influence of spatial and temporal dimensions at the same time, which is a purely relies on graph convolution structure. However, all the above methods fail to take the heterogeneity of spatial-temporal data into account, that is, the scope of each node influencing its neighbor nodes at future time steps is different. To solve this problem, Song et al. (2020) proposed STSGCN with multiple modules for different time periods to effectively extract the heterogeneity in local spatial-temporal maps. However, this method pays more attention to local information and lacks of global information extraction. Besides, STSGCN runs slowly due to too many parameters.\nTherefore, we propose an Adaptive spatial-temporal Inception Graph Convolutional Networks The Temporal and spatial correlations are extracted simultaneously by spatial-temporal convolution, and the node heterogeneity is modeled by Inception mechanism. At the same time, considering the different influences of each node and time step, the attention mechanism is introduced to adjust the influence weight adaptively."},{"heading":"3 METHODOLOGY","text":""},{"heading":"3.1 PRELIMINARIES","text":"In this paper, we define G = (V,E,A) as a static undirected spatial graph network. V represents the set of vertices, |V | = N (N indicates the number of vertices). E is the set of edges representing the connectivity between vertices. A ∈ RN×N is the adjacency matrix of the network graph G where Avi,vj represents the connection between nodes vi and vj . The graph signal matrix is expressed as Xt ∈ RN×C , where t denotes the timestep and C indicates the number of features of vertices. The graph signal matrix represents the observations of graph network G at time step t. Problem Studied Given the graph signal matrix of historical T time steps χ = (Xt1 , Xt2 , . . . , XtT ) ∈ RT×N×C , our goal is to predict the graph signal matrix of the next M time steps Ŷ = ( X̂tT+1 , X̂tT+2 , ..., X̂tT+M ) ∈ RM×N×C . In other words, we need to learn a\nmapping function F to map the graph signal matrix of historical time steps to the future time steps:( X̂tT+1 , X̂tT+2 , · · · , X̂tT+M ) = Fθ (Xt1 , Xt2 , · · · , XtT ) (1)\nwhere θ represents learnable parameters of our model."},{"heading":"3.2 ARCHITECTURE","text":"The architecture of the ASTI-GCN proposed in this paper is shown in Figure 2(a). The main ideas of ASTI-GCN can be summarized as follows: (1) We propose spatial-temporal joint graph convolution to directly extract the spatial-temporal correlations; (2) We build the Spatio-Temporal Inception Graph Convolutional Module (STI-GCM) to adaptively model the graph heterogeneity; (3) We use short-term decoder combined with sequence decoder to achieve accurate multi-step forecast."},{"heading":"3.3 SPATIAL-TEMPORAL INCEPTION-GCM","text":"Spatial-temporal joint graph convolution\nIn order to extract spatial-temporal correlations simultaneously, we propose spatial-temporal joint graph convolution. In this paper, we construct spatial-temporal joint graph convolution based on graph convolution in the spectral domain. The spectral graph convolution implemented by using the graph Fourier transform basis which is from eigenvalue decomposition of the Laplacian matrix (L)\nto transform the graph signals from spatial into the spectral domain. But the computation cost of the eigenvalue decomposition of L is expensive when the graph is large. To reduce the number of parameters and the computation complexity, Chebyshev polynomial Tk (x) is used for approximation. The spectral graph convolution can be written as (Yu et al., 2017; Guo et al., 2019; Kipf & Welling, 2016):\nΘ∗Gx = Θ (L)x ≈ K−1∑ k=0 θkTk ( L̃ ) x (2)\nwhere ∗G is graph convolution operator, Θ is graph convolution kernel, x ∈ RN is the graph signal, Tk ( L̃ ) ∈ RN×N is the Chebyshev polynomial of order k with the scaled Laplacian L̃ =\n2 λmax L − IN (L is the graph Laplacian matrix, λmax is the largest eigenvalue of L, IN is identity matrix) (Yu et al., 2017). θk is the coefficient of the k-th order polynomial.\nBased on spectral domain graph convolution, we propose spatial-temporal joint graph convolution. First, these K-hop Tk ( L̃ ) are concatenated as the furthest receptive field in the spatial dimension.\nThen we construct the spatial-temporal joint graph convolution kernel Θs,t, Θs,t ∈ RKt×Ks×C×F , where Kt represents the kernel size in the temporal dimension and Ks represents the kernel size in the spatial dimension, C is the input feature dimensions, F is the number of filters. So the kernel Θs,t has the local spatial-temporal receptive field of Kt × Ks, and Ks should be lower than K( which can be written as Ks < K), because of the largest graph convolution perceived field of Khop. And the spatial-temporal joint graph convolution can be formulated as:\nTK\n( L̃ )\n= Concat(T0 ( L̃ ) , T1 ( L̃ ) , . . . , TK−1 ( L̃ ) ) (3)\nXout = Θs,t ∗X = Θs,tTK ( L̃ ) X (4)\nwhere TK ( L̃ ) ∈ RK×N×N is the concatenation of all Chebyshev polynomials in (K-1) hop. ∗\nis the convolution operation between Θs,t and X , X ∈ RN×T×C is the spatial-temporal signal of the input graph, T is the input time steps. After the spatial-temporal joint graph convolution, the output can be written as Xout ∈ RN×(T−Kt+1)×(K−Ks+1)×F . Besides, the neighbors have various influences on the central node, so we implement a learnable spatial mask matrix Wmask ∈ RN×N (Song et al., 2020) to adjust the graph adjacency relationship for assigning weights to different neighbors.\nSpatial-temporal inception-attention\nDifferent from the images, each node of the spatial-temporal graph usually represents a road or eNodeB etc. Then, affected by external factors like geographic location and surrounding environment, the spatial-temporal data properties of different nodes are various, namely the heterogeneity. To solve this problem, an intuitive method is to learn different models for each node, but this method could cause extensive parameters and maintain low generalization ability. So we take another way in this paper. We understand that heterogeneity is manifested in the differences of local spatialtemporal receptive fields of each node of the graph which result from nodes’ various properties and local spatial structures. Inspired by (Song et al., 2020; Zheng et al., 2020; Zhou et al., 2018; Vaswani et al., 2017), we apply a learnable graph node embedding Se ∈ RN×E to represent the properties of each node. Meanwhile, we introduce inception (Szegedy et al., 2015) to extract multi-scale spatialtemporal correlations through spatial-temporal joint graph convolution. Then, we combine the graph attention with inception to achieve node-level attention for modeling the heterogeneity.\nFirstly, we implement inception, as shown in Figure 2(b). For example, the 3×2 block represents that it involves the kernel θs,t ∈ R3×2×C×F , which means it can extract the spatial-temporal correlations of the node itself and its neighbors in the three adjacent time steps by one layer, which is needed by two layers STSGCM in STSGCN (Song et al., 2020). We use the padding method when implement inception, so after B branches, the output of the inception module can be Cout ∈ RN×T×K×(F×B), where we set the number of output filters of each branch to be the same. Then we combined with the graph node attention, which has being widely used (Vaswani et al., 2017). We use the Q = SeWq , Wq ∈ RE×F to get the queries of graph nodes. For each branch in inception, we apply the idea\nof SKNET (Li et al., 2019) to do global pooling Cgl = T∑ i=1 K∑ j=1 Cout,Cg ∈ RN×(F×B) as the corresponding keys of the branches (one can also use Wk ∈ RF×F to do transform as (Vaswani et al., 2017), here we omit it for simplicity). Then we compute the attention by S = QKT (Vaswani et al., 2017). Take a graph node vi as an example,Svi,b = qvi•cg,vi,b√ F\n, qvi ∈ R1×F , cg,vi,b ∈ R1×F denotes the attention of vi and each branchCg,vi,b which represents the corresponding spatialtemporal receptive field perception. Then we concatenate the inception branch results adjusted by attention score to obtain the output. The calculation can be formulated as follows:\nαvi,b = exp(Svi,b) B∑ bc=1 exp(Svi,bc)\n(5)\nAttvi = ||Bb=1 {αvi,b · Cout,vi,b} (6)\nwhere αvi,b ∈ R, Cout,vi,b ∈ RT×K×F , Attvi ∈ RT×K×F×B represent the output of node vi from inception-attention block. Therefore, the final output of spatial-temporal inception-attention block is Catt ∈ RN×T×K×F×B . STI-GCM output Layer\nThen, we use spatial convolution to generate the output of STI-GCM. We first reshape Catt into Catt ∈ RN×T×K×(F ·B). Next, the learnable weight matrix Ws ∈ RK×(F ·B)×(F ·B) is used to convert Catt to Csatt ∈ RN×T×(F ·B). We also implement SE-net (Hu et al., 2020) to model the channel attention. Finally, the output is converted to Cstatt ∈ RN×T×F using the full connection layer with Wo ∈ R(F ·B)×F . And the process can be formulated as Cstatt = CattWsWo."},{"heading":"3.4 FUSION OUTPUT MODULE","text":"The iterative forecasting can achieve high accuracy in short-term forecasting, but in long term prediction, its accuracy would decrease rapidly due to the accumulation of errors. So, we propose the sequence transform block named sequence decoder, to build direct relationships between historical and future multi time steps. Since different historical time steps have different effects on different future time steps, we introduce temporal attention by Ŷs out = CstattWFWt(WF ∈ RF×1,Wt ∈ Rt×M ) to adjust the corresponding weights of historical temporal features. At last, in order to benefit from both, we adopt the fusion of iterative prediction and sequence transform prediction results:\nŶ = g(Ŷm out, Ŷs out) (7)\nwhere Ŷm out,Ŷs out represent the prediction of the short-term decoder and sequence decoder prediction result respectively, g(•) represents the weighted fusion function, Ŷ is the final multi-step forecasting result.\nIn this paper, we use the mean square error (MSE) between the predicted value and the true value as the loss function and minimize it through backpropagation.\nL(θ) = 1\nτ i=t+τ∑ i=t+1 (Yi−Ŷi)2 (8)\nwhere θ represents all learnable parameters of our model, Yi is the ground truth. Ŷi denotes the model’s prediction of all nodes at time step i."},{"heading":"4 EXPERIMENT","text":"We evaluate ASTI-GCN on two highway datasets (PEMSD4, PEMSD8) and one telecommunications dataset. The traffic datasets come from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). The network traffic dataset comes from the mobile traffic volume records of Milan provided by Telecom Italia (G et al., 2015).\nPEMSD4: PEMSD4 comes from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). It refers to the traffic data in San Francisco Bay Area from January to February 2018.\nPEMSD8: PEMSD8 comes from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). It is the traffic data in San Bernardino from July to August 2016.\nMobile traffic: It contains records of mobile traffic volume over 10-minute intervals in Milan where is divided into 100*100 regions. We use the data in November 2013.\nWe divide the road traffic datasets at a ratio of 6:2:2 for train set, validation set and test set. For the mobile traffic dataset, we use the first 20 days of data to train models and the next 10 days for testing. All datasets are normalized by standard normalization method before training and renormalized before evaluation. We implemented our model in Python with TensorFlow 1.14. In the experiment, we use one-hour historical data to predict the data of the next hour. The hyperparameters of ASTI-GCN are determined by the performance on the validation set. The best size of spatial-temporal kernels is set to 3×1, 1×3, 5×2, 3×2, 2×3, which are used in 4 spatial-temporal inception layers respectively. We evaluate the performance of our network with three evaluation metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). For the road traffic datasets, if two nodes are connected, the corresponding value in the adjacency matrix is set to 1, otherwise 0. For the Milan dataset, Spearman correlation coefficient was used to define the adjacency matrix. We compare ASTI-GCN with other widely used forecasting models, including HA, ARIMA (Williams & Hoel, 2003), LSTM (Hochreiter & Schmidhuber, 1997), DCRNN, STGCN, ASTGN, STSGCN, AGCRN. See Appendix A for more detail."},{"heading":"4.1 BASELINES","text":"HA: Historical Average method. We use the average of the traffic over last hour to predict the next time slice.\nARIMA (Williams & Hoel, 2003): ARIMA is a well-known model for time series prediction.\nLSTM (Hochreiter & Schmidhuber, 1997): LSTM can extract long-term and short-term temporal dependency and is widely used in time series forecasting.\nDCRNN (Li et al., 2017): A deep learning framework for traffic forecasting that incorporates both spatial and temporal dependency in the traffic flow.\nSTGCN (Yu et al., 2017): STGCN uses spectral GCN and CNN to capture spatial and temporal dependencies respectively.\nASTGN (Guo et al., 2019): ASTGCN introduces a temporal and spatial attention mechanism to capture dynamic temporal and spatial correlations.\nSTSGCN (Song et al., 2020): STSGCN purely uses GCN to extract spatial-temporal information simultaneously.\nAGCRN (Bai et al., 2020): AGCRN uses an adaptive graph convolutional recurrent network to capture node information."},{"heading":"4.2 COMPARISON AND RESULT ANALYSIS","text":"Table 1 shows the overall prediction results including the average MAE, RMSE and MAPE of our proposed method and baselines. Due to the huge computation cost, it is hard to measure the performance of STSGCN on the Milan dataset. Compared with the models that can model spatial correlation, other models that only model temporal correlation (Historical Average method, ARIMA, LSTM) have poor performance in the three datasets. This is because such models ignore the spatial influence between nodes and only use the historical information of a single node. Among spatial-temporal models, ASTI-GCN achieves the best performance in all indicators except MAPE of PEMSD4 and MILAN dataset. Because STGCN, ASTGCN and DCRNN did not consider the heterogeneity in the spatial-temporal data, while STSGCN only considered the local spatial-temporal heterogeneity of three adjacent time steps, which was insufficient for the global information extraction, our model performed better. Besides, we find AGCRN performs well on road datasets, which is close to our results, but has poor performance on the mobile traffic dataset. We conjecture that the reason is large number of nodes and the large difference in node distribution in MILAN dataset. Meanwhile, it further indicates that ASTI-GCN has a stronger generalization ability than AGCRN.\nFigure 3 shows the changes of different metrics on the three datasets with the increase of predicted time steps. As we can see from the Figure 3, the prediction error increases over time, indicating that the prediction becomes more difficult. Those methods that only consider temporal correlation (ARIMA, HA, LSTM) perform well in short-term forecasting, but as the time interval increases their performance deteriorates sharply. The performance of GCN-based methods is relatively stable, which shows the effectiveness of capturing spatial information. Although our model has no outstanding performance in short-term forecasting tasks, it shows the best performance in medium and long-term forecasting tasks. This benefits from our spatial-temporal inception-GCN module which reduces the accumulation of errors significantly."},{"heading":"4.3 COMPONENT ANALYSIS","text":"In order to prove the effectiveness of each key component, we carried out component analysis experiments on PEMSD8 dataset. The basic information of each model is as follows:\nBasic model: the model consists of two blocks. In each block, spatial and temporal convolution are performed respectively without the multi-scale spatial-temporal joint graph convolutions and graph attention. It does not use the sequence decoder with temporal attention as well. The output prediction result of the next time step simply uses a convolutional layer and the fully connected layers to generate.\n+ STI-GCM: Based on the basic model, multi-scale spatial temporal joint graph convolutions are used to replace the separate spatial and temporal convolutions to extract spatial-temporal correlations. In addition, the graph attention is introduced to capture the heterogeneity.\n+ Mask: We equip the basic model with the Mask matrix to learn the influence weights of different neighbors on the central node adaptively.\n+ Sequence decoder with attention: We add to the basic model with the proposed sequence decoder and the temporal attention for multi-step forecasting.\n+ alternant sequence decoder and short-term decoder: The output layer is modified into the fusion of the sequence forecasting result and short-term decoder forecasting result to benefit from both of them.\nExcept for the different control variables, the settings of each experiment were the same, and each experiment was repeated 10 times. The results which are shown in Table2 indicate that the model using STI-GCM has better performance, because the network can adaptively capture the heterogeneity of spatial-temporal network data. The sequence decoder with temporal attention shows good performance. The Mask mechanism also contributes some improvements. In the process of fusion, we train the short-term decoder and sequence decoder model circularly, and fuse the results of the two models as the final prediction output. It can not only ensure the short-term forecasting accuracy, but also avoid the influence of error accumulation on the long-term forecasting accuracy."},{"heading":"5 CONCLUSION","text":"This paper introduces a new deep learning framework for multi-step spatial-temporal forecasting. We propose spatial-temporal joint convolution to directly capture spatial-temporal correlation. At the same time, we employ the inception mechanisms to extract multi-scale spatial-temporal correlation, and introduce graph attention to model graph heterogeneity. Moreover, we combine short-term decoding and sequence decoding to map historical data to future time steps directly, avoiding the accumulation of errors. We evaluate our method on 3 real-world datasets from 2 different fields, and the results show that our method is superior to other baselines with good generalization ability."},{"heading":"A APPENDIX","text":"A.1 DATASETS\nWe evaluate ASTI-GCN on two highway datasets (PEMSD4, PEMSD8) and one telecommunications dataset. PEMSD4 refers to the traffic data in San Francisco Bay Area from January to February 2018. PEMSD8 is the traffic data in San Bernardino from July to August 2016. The telecommunications dataset contains records of mobile traffic volume over 10-minute intervals in Milan. We summarize the statistics of the datasets in Table 3.\nA.2 ADJACENCY MATRIX DEFINITION\nFor the road traffic datasets, if two nodes are connected, the corresponding value in the adjacency matrix is set to 1, otherwise 0. The spatial adjacency matrix can be expressed as:\nAi,j = { 1, if vi connects to vj 0, otherwise (9)\nwhere Ai,j is the edge weight of node i and node j, vi is node i.\nFor the Milan dataset, Spearman correlation coefficient was used to define the adjacency matrix:\nAi,j =\n{ 1, i 6= j and Spearman ( tvi , tvj ) > δ\n0, otherwise (10)\nwhere tvi and tvj are time series in training data of node i and node j, respectively. Spearman(·) is non-parametric indicator that measure the correlation between the two time series, δ is the threshold that controls the distribution and sparsity of the matrix, which is set to 0.92 in this article.\nA.3 METRICS\nWe evaluate the performance of our network with three evaluation metrics : Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE).\nMAE = 1\nm m∑ i=1 |ŷi − yi| (11)\nMAPE = 100%\nm m∑ i=1 ∣∣∣∣ ŷi − yiyi ∣∣∣∣ (12)\nRMSE = √√√√ 1 m m∑ i=1 (ŷi − yi)2 (13)\nwhere ŷi and yi represent the predicted value and the ground truth respectively. m is the total number of predicted values."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"Spatial-temporal data forecasting has attracted attention from researchers due to its wide range of applications and the same specific characteristics of spatial-temporal data. Typical applications include mobile traffic forecast (He et al., 2019), traffic road condition forecast (Song et al., 2020; Yu et al., 2017; Guo et al., 2019; Zheng et al., 2020; Li et al., 2017), on-demand vehicle sharing services passenger demand forecast (Bai et al., 2019) and geo-sensory time series prediction (Liang et al., 2018) etc. The accurate forecast is the foundation of many real-world applications, such as Intelligent Telecom Network Operation and Intelligent Transportation Systems (ITS). Specifically, accurate traffic forecast can help transportation agencies better control traffic scheduling and reduce traffic congestion; The traffic volumes prediction of the wireless telecommunication network plays an important role for the network operation and optimization, for example, it can help to infer the accurate sleep periods (low traffic periods) of the base stations to achieve energy saving without sacrificing customer experience.\\nHowever, as we all know, accurate spatialtemporal data forecasting faces multiple challenges. First, it is inherent with complex spatial-temporal correlations. In the spatialtemporal graph, different neighbors may have different impacts on the central location at the same time step, as the bold lines shown in Figure1, which called spatial correlations.Different historical observations of the same location influence the future moments of itself variously due to temporal correlations. The observations\\nof different neighbors at historical moments can directly affect the central node at future time steps due to the spatial-temporal joint correlations. As shown in Figure1, the information of the spatialtemporal network can propagate along the spatial and temporal dimensions simultaneously, and the\\ntransmission process can be discontinuous due to complex external factors, which result in spatialtemporal joint correlations of the spatial-temporal data in a short period.\\nSpatial-temporal data is heterogenous in both spatial and temporal dimensions (Song et al., 2020). Nodes in different regions of the graph have various properties and local spatial structures, so the corresponding data distribution can be different. For example, the traffic flow distribution of urban and suburban areas are quite different, while the traffic of urban area is denser and that of suburban area is relatively sparse. Besides, the traffic flow in the same region also exhibit heterogeneity in different time periods. For example, the mobile traffic in business district would decrease at night compared to the daytime, while it’s opposite in the residential district. In addition, multi-step time series forecasting is often accompanied by error accumulation problem. Typical methods like RNNs often cause error accumulation due to iterative forecasting, leading to rapid deterioration of the long-term prediction accuracy. (Yu et al., 2017; Zheng et al., 2020).\\nMost of the previous work is mainly to solve the above challenges. To model the spatial-temporal dependency, STGCN (Yu et al., 2017) and DCRNN (Li et al., 2017) extract spatial and temporal correlations separately. ASTGCN (Guo et al., 2019) introduced spatial and temporal attention to model the dynamic spatial and temporal correlations. STG2Seq (Bai et al., 2019) aimed at using GCN to capture spatial and temporal correlations simultaneously. But they all didn’t consider the spatialtemporal joint correlations and heterogeneity. Different from the above methods, STSGCN (Song et al., 2020) used multiple local spatial-temporal graphs to model the spatial-temporal synchronous correlations and spatial-temporal heterogeneity of the local adjacent time steps. But STSGCN can only model the spatial-temporal synchronous correlations of its defined local spatial-temporal graphs and it is equipped with complex structure.\\nIn this paper, we propose a novel model called ASTI-GCN, Adaptive spatial-temporal Inception Graph Convolutional Networks, to address the above issues with multi-step spatial-temporal data forecasting. We propose the spatial-temporal joint convolution to directly model the spatial-temporal joint correlations without introducing elaborately constructed mechanisms. And we introduce the inception mechanism to build multi-scale spatial-temporal features to adapt to graph nodes with different properties. Then, to achieve the heterogeneity modeling, we construct the spatial-temporal Inception Graph Convolution Module, which combined the spatial-temporal inception mechanism with the graph attention to build the adaptive ability of graph nodes with different properties. After multiple spatial-temporal inception-GCMs, two decoder modules named sequence decoder and short-term decoder are designed to directly establish the relationships between the historical and future time steps to alleviate error accumulation.\\nOverall, our main contributions are summarized as follows:\\n• We propose a novel spatial-temporal joint graph convolution network to directly capture spatial-temporal correlations. Moreover, we introduce inception with graph attention to adaptively model the graph heterogeneity.\\n• We propose to combine the sequence decoder and short-term decoder together for multistep forecasting to model direct relationships between historical and future time steps to alleviate the error propagation.\\n• We evaluate our model on three real-world datasets from two fields, and the experimental results show that our model achieves the best performances among all the eight baselines with good generalization ability.\"\n },\n {\n \"heading\": \"2 RELATED WORK\",\n \"text\": \"Spatial-temporal data information can be extracted using the deep learning method from European space, such as ConvLSTM (Xingjian et al., 2015), PredRNN (Gowrishankar & Satyanarayana, 2009) and so on. However, most of the spatial-temporal data in real scenes are graph data with complex and changeable relationships.Common timing prediction models, such as HA and ARIMA (Williams & Hoel, 2003), cannot be simply migrated to such scenarios. Graph based methods like DCRNN (Li et al., 2017) modeled traffic flow as a diffusion process on a directed graph. Spatial dependencies and temporal dependencies are captured by bidirectional random walk and DCGRU based\\nencoder-decoder sequence to sequence learning framework respectively. STGCN (Yu et al., 2017) constructed an undirected graph of traffic network, which is combined with GCN and CNN to model spatial and temporal correlation respectively. ASTGCN (Guo et al., 2019) innovatively introduced attention mechanisms to capture dynamic spatial and temporal dependencies. Similarly, GMAN (Zheng et al., 2020) used temporal and spatial attention to extract dynamic spatial-temporal correlations with spatial-temporal coding. The above models extract spatial-temporal correlation with two separate modules, which cannot learn the influence of neighbor node at the same time and the influence of center node at the historical moment simultaneously. To address this problem, Bai et al. (2019) proposed STG2seq to learn the influence of spatial and temporal dimensions at the same time, which is a purely relies on graph convolution structure. However, all the above methods fail to take the heterogeneity of spatial-temporal data into account, that is, the scope of each node influencing its neighbor nodes at future time steps is different. To solve this problem, Song et al. (2020) proposed STSGCN with multiple modules for different time periods to effectively extract the heterogeneity in local spatial-temporal maps. However, this method pays more attention to local information and lacks of global information extraction. Besides, STSGCN runs slowly due to too many parameters.\\nTherefore, we propose an Adaptive spatial-temporal Inception Graph Convolutional Networks The Temporal and spatial correlations are extracted simultaneously by spatial-temporal convolution, and the node heterogeneity is modeled by Inception mechanism. At the same time, considering the different influences of each node and time step, the attention mechanism is introduced to adjust the influence weight adaptively.\"\n },\n {\n \"heading\": \"3 METHODOLOGY\",\n \"text\": \"\"\n },\n {\n \"heading\": \"3.1 PRELIMINARIES\",\n \"text\": \"In this paper, we define G = (V,E,A) as a static undirected spatial graph network. V represents the set of vertices, |V | = N (N indicates the number of vertices). E is the set of edges representing the connectivity between vertices. A ∈ RN×N is the adjacency matrix of the network graph G where Avi,vj represents the connection between nodes vi and vj . The graph signal matrix is expressed as Xt ∈ RN×C , where t denotes the timestep and C indicates the number of features of vertices. The graph signal matrix represents the observations of graph network G at time step t. Problem Studied Given the graph signal matrix of historical T time steps χ = (Xt1 , Xt2 , . . . , XtT ) ∈ RT×N×C , our goal is to predict the graph signal matrix of the next M time steps Ŷ = ( X̂tT+1 , X̂tT+2 , ..., X̂tT+M ) ∈ RM×N×C . In other words, we need to learn a\\nmapping function F to map the graph signal matrix of historical time steps to the future time steps:( X̂tT+1 , X̂tT+2 , · · · , X̂tT+M ) = Fθ (Xt1 , Xt2 , · · · , XtT ) (1)\\nwhere θ represents learnable parameters of our model.\"\n },\n {\n \"heading\": \"3.2 ARCHITECTURE\",\n \"text\": \"The architecture of the ASTI-GCN proposed in this paper is shown in Figure 2(a). The main ideas of ASTI-GCN can be summarized as follows: (1) We propose spatial-temporal joint graph convolution to directly extract the spatial-temporal correlations; (2) We build the Spatio-Temporal Inception Graph Convolutional Module (STI-GCM) to adaptively model the graph heterogeneity; (3) We use short-term decoder combined with sequence decoder to achieve accurate multi-step forecast.\"\n },\n {\n \"heading\": \"3.3 SPATIAL-TEMPORAL INCEPTION-GCM\",\n \"text\": \"Spatial-temporal joint graph convolution\\nIn order to extract spatial-temporal correlations simultaneously, we propose spatial-temporal joint graph convolution. In this paper, we construct spatial-temporal joint graph convolution based on graph convolution in the spectral domain. The spectral graph convolution implemented by using the graph Fourier transform basis which is from eigenvalue decomposition of the Laplacian matrix (L)\\nto transform the graph signals from spatial into the spectral domain. But the computation cost of the eigenvalue decomposition of L is expensive when the graph is large. To reduce the number of parameters and the computation complexity, Chebyshev polynomial Tk (x) is used for approximation. The spectral graph convolution can be written as (Yu et al., 2017; Guo et al., 2019; Kipf & Welling, 2016):\\nΘ∗Gx = Θ (L)x ≈ K−1∑ k=0 θkTk ( L̃ ) x (2)\\nwhere ∗G is graph convolution operator, Θ is graph convolution kernel, x ∈ RN is the graph signal, Tk ( L̃ ) ∈ RN×N is the Chebyshev polynomial of order k with the scaled Laplacian L̃ =\\n2 λmax L − IN (L is the graph Laplacian matrix, λmax is the largest eigenvalue of L, IN is identity matrix) (Yu et al., 2017). θk is the coefficient of the k-th order polynomial.\\nBased on spectral domain graph convolution, we propose spatial-temporal joint graph convolution. First, these K-hop Tk ( L̃ ) are concatenated as the furthest receptive field in the spatial dimension.\\nThen we construct the spatial-temporal joint graph convolution kernel Θs,t, Θs,t ∈ RKt×Ks×C×F , where Kt represents the kernel size in the temporal dimension and Ks represents the kernel size in the spatial dimension, C is the input feature dimensions, F is the number of filters. So the kernel Θs,t has the local spatial-temporal receptive field of Kt × Ks, and Ks should be lower than K( which can be written as Ks < K), because of the largest graph convolution perceived field of Khop. And the spatial-temporal joint graph convolution can be formulated as:\\nTK\\n( L̃ )\\n= Concat(T0 ( L̃ ) , T1 ( L̃ ) , . . . , TK−1 ( L̃ ) ) (3)\\nXout = Θs,t ∗X = Θs,tTK ( L̃ ) X (4)\\nwhere TK ( L̃ ) ∈ RK×N×N is the concatenation of all Chebyshev polynomials in (K-1) hop. ∗\\nis the convolution operation between Θs,t and X , X ∈ RN×T×C is the spatial-temporal signal of the input graph, T is the input time steps. After the spatial-temporal joint graph convolution, the output can be written as Xout ∈ RN×(T−Kt+1)×(K−Ks+1)×F . Besides, the neighbors have various influences on the central node, so we implement a learnable spatial mask matrix Wmask ∈ RN×N (Song et al., 2020) to adjust the graph adjacency relationship for assigning weights to different neighbors.\\nSpatial-temporal inception-attention\\nDifferent from the images, each node of the spatial-temporal graph usually represents a road or eNodeB etc. Then, affected by external factors like geographic location and surrounding environment, the spatial-temporal data properties of different nodes are various, namely the heterogeneity. To solve this problem, an intuitive method is to learn different models for each node, but this method could cause extensive parameters and maintain low generalization ability. So we take another way in this paper. We understand that heterogeneity is manifested in the differences of local spatialtemporal receptive fields of each node of the graph which result from nodes’ various properties and local spatial structures. Inspired by (Song et al., 2020; Zheng et al., 2020; Zhou et al., 2018; Vaswani et al., 2017), we apply a learnable graph node embedding Se ∈ RN×E to represent the properties of each node. Meanwhile, we introduce inception (Szegedy et al., 2015) to extract multi-scale spatialtemporal correlations through spatial-temporal joint graph convolution. Then, we combine the graph attention with inception to achieve node-level attention for modeling the heterogeneity.\\nFirstly, we implement inception, as shown in Figure 2(b). For example, the 3×2 block represents that it involves the kernel θs,t ∈ R3×2×C×F , which means it can extract the spatial-temporal correlations of the node itself and its neighbors in the three adjacent time steps by one layer, which is needed by two layers STSGCM in STSGCN (Song et al., 2020). We use the padding method when implement inception, so after B branches, the output of the inception module can be Cout ∈ RN×T×K×(F×B), where we set the number of output filters of each branch to be the same. Then we combined with the graph node attention, which has being widely used (Vaswani et al., 2017). We use the Q = SeWq , Wq ∈ RE×F to get the queries of graph nodes. For each branch in inception, we apply the idea\\nof SKNET (Li et al., 2019) to do global pooling Cgl = T∑ i=1 K∑ j=1 Cout,Cg ∈ RN×(F×B) as the corresponding keys of the branches (one can also use Wk ∈ RF×F to do transform as (Vaswani et al., 2017), here we omit it for simplicity). Then we compute the attention by S = QKT (Vaswani et al., 2017). Take a graph node vi as an example,Svi,b = qvi•cg,vi,b√ F\\n, qvi ∈ R1×F , cg,vi,b ∈ R1×F denotes the attention of vi and each branchCg,vi,b which represents the corresponding spatialtemporal receptive field perception. Then we concatenate the inception branch results adjusted by attention score to obtain the output. The calculation can be formulated as follows:\\nαvi,b = exp(Svi,b) B∑ bc=1 exp(Svi,bc)\\n(5)\\nAttvi = ||Bb=1 {αvi,b · Cout,vi,b} (6)\\nwhere αvi,b ∈ R, Cout,vi,b ∈ RT×K×F , Attvi ∈ RT×K×F×B represent the output of node vi from inception-attention block. Therefore, the final output of spatial-temporal inception-attention block is Catt ∈ RN×T×K×F×B . STI-GCM output Layer\\nThen, we use spatial convolution to generate the output of STI-GCM. We first reshape Catt into Catt ∈ RN×T×K×(F ·B). Next, the learnable weight matrix Ws ∈ RK×(F ·B)×(F ·B) is used to convert Catt to Csatt ∈ RN×T×(F ·B). We also implement SE-net (Hu et al., 2020) to model the channel attention. Finally, the output is converted to Cstatt ∈ RN×T×F using the full connection layer with Wo ∈ R(F ·B)×F . And the process can be formulated as Cstatt = CattWsWo.\"\n },\n {\n \"heading\": \"3.4 FUSION OUTPUT MODULE\",\n \"text\": \"The iterative forecasting can achieve high accuracy in short-term forecasting, but in long term prediction, its accuracy would decrease rapidly due to the accumulation of errors. So, we propose the sequence transform block named sequence decoder, to build direct relationships between historical and future multi time steps. Since different historical time steps have different effects on different future time steps, we introduce temporal attention by Ŷs out = CstattWFWt(WF ∈ RF×1,Wt ∈ Rt×M ) to adjust the corresponding weights of historical temporal features. At last, in order to benefit from both, we adopt the fusion of iterative prediction and sequence transform prediction results:\\nŶ = g(Ŷm out, Ŷs out) (7)\\nwhere Ŷm out,Ŷs out represent the prediction of the short-term decoder and sequence decoder prediction result respectively, g(•) represents the weighted fusion function, Ŷ is the final multi-step forecasting result.\\nIn this paper, we use the mean square error (MSE) between the predicted value and the true value as the loss function and minimize it through backpropagation.\\nL(θ) = 1\\nτ i=t+τ∑ i=t+1 (Yi−Ŷi)2 (8)\\nwhere θ represents all learnable parameters of our model, Yi is the ground truth. Ŷi denotes the model’s prediction of all nodes at time step i.\"\n },\n {\n \"heading\": \"4 EXPERIMENT\",\n \"text\": \"We evaluate ASTI-GCN on two highway datasets (PEMSD4, PEMSD8) and one telecommunications dataset. The traffic datasets come from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). The network traffic dataset comes from the mobile traffic volume records of Milan provided by Telecom Italia (G et al., 2015).\\nPEMSD4: PEMSD4 comes from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). It refers to the traffic data in San Francisco Bay Area from January to February 2018.\\nPEMSD8: PEMSD8 comes from the Caltrans Performance Measurement System (PeMS) (Chen et al., 2001). It is the traffic data in San Bernardino from July to August 2016.\\nMobile traffic: It contains records of mobile traffic volume over 10-minute intervals in Milan where is divided into 100*100 regions. We use the data in November 2013.\\nWe divide the road traffic datasets at a ratio of 6:2:2 for train set, validation set and test set. For the mobile traffic dataset, we use the first 20 days of data to train models and the next 10 days for testing. All datasets are normalized by standard normalization method before training and renormalized before evaluation. We implemented our model in Python with TensorFlow 1.14. In the experiment, we use one-hour historical data to predict the data of the next hour. The hyperparameters of ASTI-GCN are determined by the performance on the validation set. The best size of spatial-temporal kernels is set to 3×1, 1×3, 5×2, 3×2, 2×3, which are used in 4 spatial-temporal inception layers respectively. We evaluate the performance of our network with three evaluation metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). For the road traffic datasets, if two nodes are connected, the corresponding value in the adjacency matrix is set to 1, otherwise 0. For the Milan dataset, Spearman correlation coefficient was used to define the adjacency matrix. We compare ASTI-GCN with other widely used forecasting models, including HA, ARIMA (Williams & Hoel, 2003), LSTM (Hochreiter & Schmidhuber, 1997), DCRNN, STGCN, ASTGN, STSGCN, AGCRN. See Appendix A for more detail.\"\n },\n {\n \"heading\": \"4.1 BASELINES\",\n \"text\": \"HA: Historical Average method. We use the average of the traffic over last hour to predict the next time slice.\\nARIMA (Williams & Hoel, 2003): ARIMA is a well-known model for time series prediction.\\nLSTM (Hochreiter & Schmidhuber, 1997): LSTM can extract long-term and short-term temporal dependency and is widely used in time series forecasting.\\nDCRNN (Li et al., 2017): A deep learning framework for traffic forecasting that incorporates both spatial and temporal dependency in the traffic flow.\\nSTGCN (Yu et al., 2017): STGCN uses spectral GCN and CNN to capture spatial and temporal dependencies respectively.\\nASTGN (Guo et al., 2019): ASTGCN introduces a temporal and spatial attention mechanism to capture dynamic temporal and spatial correlations.\\nSTSGCN (Song et al., 2020): STSGCN purely uses GCN to extract spatial-temporal information simultaneously.\\nAGCRN (Bai et al., 2020): AGCRN uses an adaptive graph convolutional recurrent network to capture node information.\"\n },\n {\n \"heading\": \"4.2 COMPARISON AND RESULT ANALYSIS\",\n \"text\": \"Table 1 shows the overall prediction results including the average MAE, RMSE and MAPE of our proposed method and baselines. Due to the huge computation cost, it is hard to measure the performance of STSGCN on the Milan dataset. Compared with the models that can model spatial correlation, other models that only model temporal correlation (Historical Average method, ARIMA, LSTM) have poor performance in the three datasets. This is because such models ignore the spatial influence between nodes and only use the historical information of a single node. Among spatial-temporal models, ASTI-GCN achieves the best performance in all indicators except MAPE of PEMSD4 and MILAN dataset. Because STGCN, ASTGCN and DCRNN did not consider the heterogeneity in the spatial-temporal data, while STSGCN only considered the local spatial-temporal heterogeneity of three adjacent time steps, which was insufficient for the global information extraction, our model performed better. Besides, we find AGCRN performs well on road datasets, which is close to our results, but has poor performance on the mobile traffic dataset. We conjecture that the reason is large number of nodes and the large difference in node distribution in MILAN dataset. Meanwhile, it further indicates that ASTI-GCN has a stronger generalization ability than AGCRN.\\nFigure 3 shows the changes of different metrics on the three datasets with the increase of predicted time steps. As we can see from the Figure 3, the prediction error increases over time, indicating that the prediction becomes more difficult. Those methods that only consider temporal correlation (ARIMA, HA, LSTM) perform well in short-term forecasting, but as the time interval increases their performance deteriorates sharply. The performance of GCN-based methods is relatively stable, which shows the effectiveness of capturing spatial information. Although our model has no outstanding performance in short-term forecasting tasks, it shows the best performance in medium and long-term forecasting tasks. This benefits from our spatial-temporal inception-GCN module which reduces the accumulation of errors significantly.\"\n },\n {\n \"heading\": \"4.3 COMPONENT ANALYSIS\",\n \"text\": \"In order to prove the effectiveness of each key component, we carried out component analysis experiments on PEMSD8 dataset. The basic information of each model is as follows:\\nBasic model: the model consists of two blocks. In each block, spatial and temporal convolution are performed respectively without the multi-scale spatial-temporal joint graph convolutions and graph attention. It does not use the sequence decoder with temporal attention as well. The output prediction result of the next time step simply uses a convolutional layer and the fully connected layers to generate.\\n+ STI-GCM: Based on the basic model, multi-scale spatial temporal joint graph convolutions are used to replace the separate spatial and temporal convolutions to extract spatial-temporal correlations. In addition, the graph attention is introduced to capture the heterogeneity.\\n+ Mask: We equip the basic model with the Mask matrix to learn the influence weights of different neighbors on the central node adaptively.\\n+ Sequence decoder with attention: We add to the basic model with the proposed sequence decoder and the temporal attention for multi-step forecasting.\\n+ alternant sequence decoder and short-term decoder: The output layer is modified into the fusion of the sequence forecasting result and short-term decoder forecasting result to benefit from both of them.\\nExcept for the different control variables, the settings of each experiment were the same, and each experiment was repeated 10 times. The results which are shown in Table2 indicate that the model using STI-GCM has better performance, because the network can adaptively capture the heterogeneity of spatial-temporal network data. The sequence decoder with temporal attention shows good performance. The Mask mechanism also contributes some improvements. In the process of fusion, we train the short-term decoder and sequence decoder model circularly, and fuse the results of the two models as the final prediction output. It can not only ensure the short-term forecasting accuracy, but also avoid the influence of error accumulation on the long-term forecasting accuracy.\"\n },\n {\n \"heading\": \"5 CONCLUSION\",\n \"text\": \"This paper introduces a new deep learning framework for multi-step spatial-temporal forecasting. We propose spatial-temporal joint convolution to directly capture spatial-temporal correlation. At the same time, we employ the inception mechanisms to extract multi-scale spatial-temporal correlation, and introduce graph attention to model graph heterogeneity. Moreover, we combine short-term decoding and sequence decoding to map historical data to future time steps directly, avoiding the accumulation of errors. We evaluate our method on 3 real-world datasets from 2 different fields, and the results show that our method is superior to other baselines with good generalization ability.\"\n },\n {\n \"heading\": \"A APPENDIX\",\n \"text\": \"A.1 DATASETS\\nWe evaluate ASTI-GCN on two highway datasets (PEMSD4, PEMSD8) and one telecommunications dataset. PEMSD4 refers to the traffic data in San Francisco Bay Area from January to February 2018. PEMSD8 is the traffic data in San Bernardino from July to August 2016. The telecommunications dataset contains records of mobile traffic volume over 10-minute intervals in Milan. We summarize the statistics of the datasets in Table 3.\\nA.2 ADJACENCY MATRIX DEFINITION\\nFor the road traffic datasets, if two nodes are connected, the corresponding value in the adjacency matrix is set to 1, otherwise 0. The spatial adjacency matrix can be expressed as:\\nAi,j = { 1, if vi connects to vj 0, otherwise (9)\\nwhere Ai,j is the edge weight of node i and node j, vi is node i.\\nFor the Milan dataset, Spearman correlation coefficient was used to define the adjacency matrix:\\nAi,j =\\n{ 1, i 6= j and Spearman ( tvi , tvj ) > δ\\n0, otherwise (10)\\nwhere tvi and tvj are time series in training data of node i and node j, respectively. Spearman(·) is non-parametric indicator that measure the correlation between the two time series, δ is the threshold that controls the distribution and sparsity of the matrix, which is set to 0.92 in this article.\\nA.3 METRICS\\nWe evaluate the performance of our network with three evaluation metrics : Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE).\\nMAE = 1\\nm m∑ i=1 |ŷi − yi| (11)\\nMAPE = 100%\\nm m∑ i=1 ∣∣∣∣ ŷi − yiyi ∣∣∣∣ (12)\\nRMSE = √√√√ 1 m m∑ i=1 (ŷi − yi)2 (13)\\nwhere ŷi and yi represent the predicted value and the ground truth respectively. m is the total number of predicted values.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2020,"string":"2,020"},"title":{"kind":"null"}}},{"rowIdx":199,"cells":{"paper_id":{"kind":"string","value":"SP:a99af0f9e848f4f9068ad407612745a85a262644"},"summaries":{"kind":"list like","value":["This paper extends NTK to RNN to explain behavior of RNNs in overparametrized case. It’s a good extension study and interesting to see RNN with infinite-width limit converges to a kernel. The paper proves the same RNTK formula when the weights are shared and not shared. The proposed sensitivity for computationally friendly RNTK hyperparameter tuning is also insightful."],"string":"[\n \"This paper extends NTK to RNN to explain behavior of RNNs in overparametrized case. It’s a good extension study and interesting to see RNN with infinite-width limit converges to a kernel. The paper proves the same RNTK formula when the weights are shared and not shared. The proposed sensitivity for computationally friendly RNTK hyperparameter tuning is also insightful.\"\n]","isConversational":false},"abstractText":{"kind":"string","value":"The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which provides new insights into the behavior of overparametrized RNNs. A key property of the RNTK should greatly benefit practitioners is its ability to compare inputs of different length. To this end, we characterize how the RNTK weights different time steps to form its output under different initialization parameters and nonlinearity choices. A synthetic and 56 real-world data experiments demonstrate that the RNTK offers significant performance gains over other kernels, including standard NTKs, across a wide array of data sets."},"authors":{"kind":"list like","value":[{"affiliations":[],"name":"Sina Alemohammad"},{"affiliations":[],"name":"Zichao Wang"},{"affiliations":[],"name":"Randall Balestriero"},{"affiliations":[],"name":"Richard G. Baraniuk"}],"string":"[\n {\n \"affiliations\": [],\n \"name\": \"Sina Alemohammad\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Zichao Wang\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Randall Balestriero\"\n },\n {\n \"affiliations\": [],\n \"name\": \"Richard G. Baraniuk\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"authors":["Zeyuan Allen-Zhu","Yuanzhi Li","Yingyu Liang"],"title":"Learning and generalization in overparameterized neural networks, going beyond two layers","venue":"Advances in Neural Information Processing Systems,","year":2019},{"authors":["Zeyuan Allen-Zhu","Yuanzhi Li","Zhao Song"],"title":"On the convergence rate of training recurrent neural networks","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Martin Arjovsky","Amar Shah","Yoshua Bengio"],"title":"Unitary evolution recurrent neural networks","venue":"In International Conference on Machine Learning,","year":2016},{"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,","year":2019},{"authors":["Sanjeev Arora","Simon S. 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Febrero-Bande"],"title":"An extensive experimental survey of regression methods","venue":"Neural Networks,","year":2019},{"authors":["Adrià Garriga-Alonso","Carl Edward Rasmussen","Laurence Aitchison"],"title":"Deep convolutional networks as shallow gaussian processes","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Mikael Henaff","Arthur Szlam","Yann LeCun"],"title":"Recurrent orthogonal networks and long-memory tasks","venue":"In International Conference on Machine Learning,","year":2016},{"authors":["Sepp Hochreiter","Jürgen Schmidhuber"],"title":"Long short-term memory","venue":"Neural Computation,","year":1997},{"authors":["Wei Hu","Zhiyuan Li","Dingli Yu"],"title":"Simple and effective regularization methods for training on noisily labeled data with generalization guarantee","venue":"In International Conference on Learning Representations,","year":2020},{"authors":["Kaixuan Huang","Yuqing Wang","Molei Tao","Tuo Zhao"],"title":"Why do deep residual networks generalize better than deep feedforward networks?–a neural tangent kernel perspective","venue":null,"year":2002},{"authors":["Arthur Jacot","Franck Gabriel","Clément Hongler"],"title":"Neural tangent kernel: Convergence and generalization in neural networks","venue":"In Advances in neural information processing systems,","year":2018},{"authors":["Li Jing","Yichen Shen","Tena Dubcek","John Peurifoy","Scott Skirlo","Yann LeCun","Max Tegmark","Marin Soljačić"],"title":"Tunable efficient unitary neural networks (eunn) and their application to rnns","venue":"In International Conference on Machine Learning,","year":2017},{"authors":["Quoc V Le","Navdeep Jaitly","Geoffrey E Hinton"],"title":"A simple way to initialize recurrent networks of rectified linear units","venue":"arXiv preprint arXiv:1504.00941,","year":2015},{"authors":["Jaehoon Lee","Jascha Sohl-dickstein","Jeffrey Pennington","Roman Novak","Sam Schoenholz","Yasaman Bahri"],"title":"Deep neural networks as gaussian processes","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["Jaehoon Lee","Lechao Xiao","Samuel Schoenholz","Yasaman Bahri","Roman Novak","Jascha SohlDickstein","Jeffrey Pennington"],"title":"Wide neural networks of any depth evolve as linear models under gradient descent","venue":"In Advances in Neural Information Processing Systems,","year":2019},{"authors":["Jaehoon Lee","Samuel S Schoenholz","Jeffrey Pennington","Ben Adlam","Lechao Xiao","Roman Novak","Jascha Sohl-Dickstein"],"title":"Finite versus infinite neural networks: an empirical study","venue":null,"year":2007},{"authors":["Radford M Neal"],"title":"Bayesian Learning for Neural Networks","venue":"PhD thesis, University of Toronto,","year":1995},{"authors":["Behnam Neyshabur","Zhiyuan Li","Srinadh Bhojanapalli","Yann LeCun","Nathan Srebro"],"title":"The role of over-parametrization in generalization of neural networks","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Roman Novak","Yasaman Bahri","Daniel A. Abolafia","Jeffrey Pennington","Jascha Sohl-Dickstein"],"title":"Sensitivity and generalization in neural networks: an empirical study","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["Roman Novak","Lechao Xiao","Yasaman Bahri","Jaehoon Lee","Greg Yang","Daniel A. Abolafia","Jeffrey Pennington","Jascha Sohl-dickstein"],"title":"Bayesian deep convolutional networks with many channels are gaussian processes","venue":"In International Conference on Learning Representations,","year":2019},{"authors":["Zichao Wang","Randall Balestriero","Richard Baraniuk"],"title":"A max-affine spline perspective of recurrent neural networks","venue":"In International Conference on Learning Representations,","year":2018},{"authors":["Greg Yang"],"title":"Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation","venue":"arXiv preprint arXiv:1902.04760,","year":2019},{"authors":["Greg Yang"],"title":"Tensor programs I: Wide feedforward or recurrent neural networks of any architecture are gaussian processes","venue":"arXiv preprint arXiv:1910.12478,","year":2019},{"authors":["Greg Yang"],"title":"Tensor programs II: Neural tangent kernel for any architecture","venue":"arXiv preprint arXiv:2006.14548,","year":2020},{"authors":["Greg Yang"],"title":"Tensor programs III: Neural matrix laws","venue":"arXiv preprint arXiv:2009.10685,","year":2020},{"authors":["Difan Zou","Yuan Cao","Dongruo Zhou","Quanquan Gu"],"title":"Stochastic gradient descent optimizes over-parameterized deep ReLU networks","venue":"arXiv preprint arXiv:1811.08888,","year":2018},{"authors":["Lin"],"title":"2011) and for hyperparameter selection we performed 10-fold validation for splitting the training data into 90% training set and 10% validation test. We then choose the best performing set of hyperparameters on all the validation sets, retrain the models with the best set of hyperparameters on the entire training data and finally report the performance on the unseen test data. The performance of all kernels on each data set is shown in table","venue":null,"year":2011},{"authors":["Friedman Rank (Fernández-Delgado"],"title":"2019) first ranks the accuracy of each classifier on each dataset and then takes the average of the ranks for each classifier over all","venue":null,"year":2019}],"string":"[\n {\n \"authors\": [\n \"Zeyuan Allen-Zhu\",\n \"Yuanzhi Li\",\n \"Yingyu Liang\"\n ],\n \"title\": \"Learning and generalization in overparameterized neural networks, going beyond two layers\",\n \"venue\": \"Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Zeyuan Allen-Zhu\",\n \"Yuanzhi Li\",\n \"Zhao Song\"\n ],\n \"title\": \"On the convergence rate of training recurrent neural networks\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Martin Arjovsky\",\n \"Amar Shah\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Unitary evolution recurrent neural networks\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Sanjeev Arora\",\n \"Simon S. Du\",\n \"Wei Hu\",\n \"Zhiyuan Li\",\n \"Russ R Salakhutdinov\",\n \"Ruosong Wang\"\n ],\n \"title\": \"On exact computation with an infinitely wide neural net\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Sanjeev Arora\",\n \"Simon S. Du\",\n \"Wei Hu\",\n \"Zhiyuan Li\",\n \"Ruosong Wang\"\n ],\n \"title\": \"Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks\",\n \"venue\": \"arXiv preprint arXiv:1901.08584,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Sanjeev Arora\",\n \"Simon S. Du\",\n \"Zhiyuan Li\",\n \"Ruslan Salakhutdinov\",\n \"Ruosong Wang\",\n \"Dingli Yu\"\n ],\n \"title\": \"Harnessing the power of infinitely wide deep nets on small-data tasks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Mikhail Belkin\",\n \"Daniel Hsu\",\n \"Siyuan Ma\",\n \"Soumik Mandal\"\n ],\n \"title\": \"Reconciling modern machine-learning practice and the classical bias–variance trade-off\",\n \"venue\": \"Proceedings of the National Academy of Sciences,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Yoshua. Bengio\",\n \"Patrice. Simard\",\n \"Paolo. Frasconi\"\n ],\n \"title\": \"Learning long-term dependencies with gradient descent is difficult\",\n \"venue\": \"IEEE Trans. Neural Networks,\",\n \"year\": 1994\n },\n {\n \"authors\": [\n \"Erwin Bolthausen\"\n ],\n \"title\": \"An iterative construction of solutions of the tap equations for the sherrington– kirkpatrick model\",\n \"venue\": \"Communications in Mathematical Physics,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Chih-Chung Chang\",\n \"Chih-Jen Lin\"\n ],\n \"title\": \"LIBSVM: A library for support vector machines\",\n \"venue\": \"ACM Transactions on Intelligent Systems and Technology,\",\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Kyunghyun Cho\",\n \"Bart Van Merriënboer\",\n \"Caglar Gulcehre\",\n \"Dzmitry Bahdanau\",\n \"Fethi Bougares\",\n \"Holger Schwenk\",\n \"Yoshua Bengio\"\n ],\n \"title\": \"Learning phrase representations using rnn encoder-decoder for statistical machine translation\",\n \"venue\": \"arXiv preprint arXiv:1406.1078,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Youngmin Cho\",\n \"Lawrence K Saul\"\n ],\n \"title\": \"Kernel methods for deep learning\",\n \"venue\": \"In Advances in Neural Information Processing Systems, pp\",\n \"year\": 2009\n },\n {\n \"authors\": [\n \"Hoang Anh Dau\",\n \"Eamonn Keogh\",\n \"Kaveh Kamgar\",\n \"Chin-Chia Michael Yeh\",\n \"Yan Zhu\",\n \"Shaghayegh Gharghabi\",\n \"Chotirat Ann Ratanamahatana\",\n \"Yanping Chen\",\n \"Bing Hu\",\n \"Nurjahan Begum\",\n \"Anthony Bagnall\",\n \"Abdullah Mueen\",\n \"Gustavo Batista\",\n \"ML Hexagon\"\n ],\n \"title\": \"The UCR time series classification archive, 2019\",\n \"venue\": null,\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Simon S. Du\",\n \"Kangcheng Hou\",\n \"Russ R Salakhutdinov\",\n \"Barnabas Poczos\",\n \"Ruosong Wang\",\n \"Keyulu Xu\"\n ],\n \"title\": \"Graph neural tangent kernel: Fusing graph neural networks with graph kernels\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Simon S. Du\",\n \"Jason Lee\",\n \"Haochuan Li\",\n \"Liwei Wang\",\n \"Xiyu Zhai\"\n ],\n \"title\": \"Gradient descent finds global minima of deep neural networks\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"David Duvenaud\",\n \"Oren Rippel\",\n \"Ryan Adams\",\n \"Zoubin Ghahramani\"\n ],\n \"title\": \"Avoiding pathologies in very deep networks\",\n \"venue\": \"In Artificial Intelligence and Statistics,\",\n \"year\": 2014\n },\n {\n \"authors\": [\n \"Jeffrey L. Elman\"\n ],\n \"title\": \"Finding structure in time\",\n \"venue\": \"Cognitive Science,\",\n \"year\": 1990\n },\n {\n \"authors\": [\n \"M. Fernández-Delgado\",\n \"M.S. Sirsat\",\n \"E. Cernadas\",\n \"S. Alawadi\",\n \"S. Barro\",\n \"M. Febrero-Bande\"\n ],\n \"title\": \"An extensive experimental survey of regression methods\",\n \"venue\": \"Neural Networks,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Adrià Garriga-Alonso\",\n \"Carl Edward Rasmussen\",\n \"Laurence Aitchison\"\n ],\n \"title\": \"Deep convolutional networks as shallow gaussian processes\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Mikael Henaff\",\n \"Arthur Szlam\",\n \"Yann LeCun\"\n ],\n \"title\": \"Recurrent orthogonal networks and long-memory tasks\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2016\n },\n {\n \"authors\": [\n \"Sepp Hochreiter\",\n \"Jürgen Schmidhuber\"\n ],\n \"title\": \"Long short-term memory\",\n \"venue\": \"Neural Computation,\",\n \"year\": 1997\n },\n {\n \"authors\": [\n \"Wei Hu\",\n \"Zhiyuan Li\",\n \"Dingli Yu\"\n ],\n \"title\": \"Simple and effective regularization methods for training on noisily labeled data with generalization guarantee\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Kaixuan Huang\",\n \"Yuqing Wang\",\n \"Molei Tao\",\n \"Tuo Zhao\"\n ],\n \"title\": \"Why do deep residual networks generalize better than deep feedforward networks?–a neural tangent kernel perspective\",\n \"venue\": null,\n \"year\": 2002\n },\n {\n \"authors\": [\n \"Arthur Jacot\",\n \"Franck Gabriel\",\n \"Clément Hongler\"\n ],\n \"title\": \"Neural tangent kernel: Convergence and generalization in neural networks\",\n \"venue\": \"In Advances in neural information processing systems,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Li Jing\",\n \"Yichen Shen\",\n \"Tena Dubcek\",\n \"John Peurifoy\",\n \"Scott Skirlo\",\n \"Yann LeCun\",\n \"Max Tegmark\",\n \"Marin Soljačić\"\n ],\n \"title\": \"Tunable efficient unitary neural networks (eunn) and their application to rnns\",\n \"venue\": \"In International Conference on Machine Learning,\",\n \"year\": 2017\n },\n {\n \"authors\": [\n \"Quoc V Le\",\n \"Navdeep Jaitly\",\n \"Geoffrey E Hinton\"\n ],\n \"title\": \"A simple way to initialize recurrent networks of rectified linear units\",\n \"venue\": \"arXiv preprint arXiv:1504.00941,\",\n \"year\": 2015\n },\n {\n \"authors\": [\n \"Jaehoon Lee\",\n \"Jascha Sohl-dickstein\",\n \"Jeffrey Pennington\",\n \"Roman Novak\",\n \"Sam Schoenholz\",\n \"Yasaman Bahri\"\n ],\n \"title\": \"Deep neural networks as gaussian processes\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Jaehoon Lee\",\n \"Lechao Xiao\",\n \"Samuel Schoenholz\",\n \"Yasaman Bahri\",\n \"Roman Novak\",\n \"Jascha SohlDickstein\",\n \"Jeffrey Pennington\"\n ],\n \"title\": \"Wide neural networks of any depth evolve as linear models under gradient descent\",\n \"venue\": \"In Advances in Neural Information Processing Systems,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Jaehoon Lee\",\n \"Samuel S Schoenholz\",\n \"Jeffrey Pennington\",\n \"Ben Adlam\",\n \"Lechao Xiao\",\n \"Roman Novak\",\n \"Jascha Sohl-Dickstein\"\n ],\n \"title\": \"Finite versus infinite neural networks: an empirical study\",\n \"venue\": null,\n \"year\": 2007\n },\n {\n \"authors\": [\n \"Radford M Neal\"\n ],\n \"title\": \"Bayesian Learning for Neural Networks\",\n \"venue\": \"PhD thesis, University of Toronto,\",\n \"year\": 1995\n },\n {\n \"authors\": [\n \"Behnam Neyshabur\",\n \"Zhiyuan Li\",\n \"Srinadh Bhojanapalli\",\n \"Yann LeCun\",\n \"Nathan Srebro\"\n ],\n \"title\": \"The role of over-parametrization in generalization of neural networks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Roman Novak\",\n \"Yasaman Bahri\",\n \"Daniel A. Abolafia\",\n \"Jeffrey Pennington\",\n \"Jascha Sohl-Dickstein\"\n ],\n \"title\": \"Sensitivity and generalization in neural networks: an empirical study\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Roman Novak\",\n \"Lechao Xiao\",\n \"Yasaman Bahri\",\n \"Jaehoon Lee\",\n \"Greg Yang\",\n \"Daniel A. Abolafia\",\n \"Jeffrey Pennington\",\n \"Jascha Sohl-dickstein\"\n ],\n \"title\": \"Bayesian deep convolutional networks with many channels are gaussian processes\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Zichao Wang\",\n \"Randall Balestriero\",\n \"Richard Baraniuk\"\n ],\n \"title\": \"A max-affine spline perspective of recurrent neural networks\",\n \"venue\": \"In International Conference on Learning Representations,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Greg Yang\"\n ],\n \"title\": \"Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation\",\n \"venue\": \"arXiv preprint arXiv:1902.04760,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Greg Yang\"\n ],\n \"title\": \"Tensor programs I: Wide feedforward or recurrent neural networks of any architecture are gaussian processes\",\n \"venue\": \"arXiv preprint arXiv:1910.12478,\",\n \"year\": 2019\n },\n {\n \"authors\": [\n \"Greg Yang\"\n ],\n \"title\": \"Tensor programs II: Neural tangent kernel for any architecture\",\n \"venue\": \"arXiv preprint arXiv:2006.14548,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Greg Yang\"\n ],\n \"title\": \"Tensor programs III: Neural matrix laws\",\n \"venue\": \"arXiv preprint arXiv:2009.10685,\",\n \"year\": 2020\n },\n {\n \"authors\": [\n \"Difan Zou\",\n \"Yuan Cao\",\n \"Dongruo Zhou\",\n \"Quanquan Gu\"\n ],\n \"title\": \"Stochastic gradient descent optimizes over-parameterized deep ReLU networks\",\n \"venue\": \"arXiv preprint arXiv:1811.08888,\",\n \"year\": 2018\n },\n {\n \"authors\": [\n \"Lin\"\n ],\n \"title\": \"2011) and for hyperparameter selection we performed 10-fold validation for splitting the training data into 90% training set and 10% validation test. We then choose the best performing set of hyperparameters on all the validation sets, retrain the models with the best set of hyperparameters on the entire training data and finally report the performance on the unseen test data. The performance of all kernels on each data set is shown in table\",\n \"venue\": null,\n \"year\": 2011\n },\n {\n \"authors\": [\n \"Friedman Rank (Fernández-Delgado\"\n ],\n \"title\": \"2019) first ranks the accuracy of each classifier on each dataset and then takes the average of the ranks for each classifier over all\",\n \"venue\": null,\n \"year\": 2019\n }\n]","isConversational":false},"sections":{"kind":"list like","value":[{"heading":"1 INTRODUCTION","text":"The overparameterization of modern deep neural networks (DNNs) has resulted in not only remarkably good generalization performance on unseen data (Novak et al., 2018; Neyshabur et al., 2019; Belkin et al., 2019) but also guarantees that gradient descent learning can find the global minimum of their highly nonconvex loss functions (Du et al., 2019b; Allen-Zhu et al., 2019b;a; Zou et al., 2018; Arora et al., 2019b). From these successes, a natural question arises: What happens when we take overparameterization to the limit by allowing the width of a DNN’s hidden layers to go to infinity? Surprisingly, the analysis of such an (impractical) DNN becomes analytically tractable. Indeed, recent work has shown that the training dynamics of (infinite-width) DNNs under gradient flow is captured by a constant kernel called the Neural Tangent Kernel (NTK) that evolves according to a linear ordinary differential equation (ODE) (Jacot et al., 2018; Lee et al., 2019; Arora et al., 2019a).\nEvery DNN architecture and parameter initialization produces a distinct NTK. The original NTK was derived from the Multilayer Perceptron (MLP)(Jacot et al., 2018) and was soon followed by kernels derived from Convolutional Neural Networks (CNTK) (Arora et al., 2019a; Yang, 2019a), Residual DNNs (Huang et al., 2020), and Graph Convolutional Neural Networks (GNTK) (Du et al., 2019a). In (Yang, 2020a), a general strategy to obtain the NTK of any architecture is provided.\nIn this paper, we extend the NTK concept to the important class of overparametrized Recurrent Neural Networks (RNNs), a fundamental DNN architecture for processing sequential data. We show that RNN in its infinite-width limit converges to a kernel that we dub the Recurrent Neural Tangent Kernel (RNTK). The RNTK provides high performance for various machine learning tasks, and an analysis of the properties of the kernel provides useful insights into the behavior of RNNs in the following overparametrized regime. In particular, we derive and study the RNTK to answer the following theoretical questions:\nQ: Can the RNTK extract long-term dependencies between two data sequences? RNNs are known to underperform at learning long-term dependencies due to the gradient vanishing or exploding (Bengio et al., 1994). Attempted ameliorations have included orthogonal weights (Arjovsky et al., 2016; Jing et al., 2017; Henaff et al., 2016) and gating such as in Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Unit (GRU) (Cho et al., 2014) RNNs. We demonstrate that the RNTK can detect long-term dependencies with proper initialization of the hyperparameters, and moreover, we show how the dependencies are extracted through time via different hyperparameter choices.\nQ: Do the recurrent weights of the RNTK reduce its representation power compared to other NTKs? An attractive property of an RNN that is shared by the RNTK is that it can deal with sequences of different lengths via weight sharing through time. This enables the reduction of the number of learnable parameters and thus more stable training at the cost of reduced representation power. We prove the surprising fact that employing tied vs. untied weights in an RNN does not impact the analytical form of the RNTK.\nQ: Does the RNTK generalize well? A recent study has revealed that the use of an SVM classifier with the NTK, CNTK, and GNTK kernels outperforms other classical kernel-based classifiers and trained finite DNNs on small data sets (typically fewer than 5000 training samples) (Lee et al., 2020; Arora et al., 2019a; 2020; Du et al., 2019a). We extend these results to RNTKs to demonstrate that the RNTK outperforms a variety of classic kernels, NTKs and finite RNNs for time series data sets in both classification and regression tasks. Carefully designed experiments with data of varying lengths demonstrate that the RNTK’s performance accelerates beyond other techniques as the difference in lengths increases. Those results extend the empirical observations from (Arora et al., 2019a; 2020; Du et al., 2019a; Lee et al., 2020) into finite DNNs, NTK, CNTK, and GNTK comparisons by observing that their performance-wise ranking depends on the employed DNN architecture.\nWe summarize our contributions as follows:\n[C1] We derive the analytical form for the RNTK of an overparametrized RNN at initialization using rectified linear unit (ReLU) and error function (erf) nonlinearities for arbitrary data lengths and number of layers (Section 3.1).\n[C2] We prove that the RNTK remains constant during (overparametrized) RNN training and that the dynamics of training are simplified to a set of ordinary differential equations (ODEs) (Section 3.2).\n[C3] When the input data sequences are of equal length, we show that the RNTKs of weight-tied and weight-untied RNNs converge to the same RNTK (Section 3.3).\n[C4] Leveraging our analytical formulation of the RNTK, we empirically demonstrate how correlations between data at different times are weighted by the function learned by an RNN for different sets of hyperparameters. We also offer practical suggestions for choosing the RNN hyperparameters for deep information propagation through time (Section 3.4).\n[C5] We demonstrate that the RNTK is eminently practical by showing its superiority over classical kernels, NTKs, and finite RNNs in exhaustive experiments on time-series classification and regression with both synthetic and 56 real-world data sets (Section 4)."},{"heading":"2 BACKGROUND AND RELATED WORK","text":"Notation. We denote [n] = {1, . . . , n}, and Id the identity matrix of size d. [A]i,j represents the (i, j)-th entry of a matrix, and similarly [a]i represents the i-th entry of a vector. We use φ(·) : R → R to represent the activation function that acts coordinate wise on a vector and φ′ to denote its derivative. We will often use the rectified linear unit (ReLU) φ(x) = max(0, x) and error function (erf) φ(x) = 2√\nπ ∫ x 0 e−z 2\ndz activation functions. N (µ,Σ) represents the multidimensional Gaussian distribution with the mean vector µ and the covariance matrix Σ.\nRecurrent Neural Networks (RNNs). Given an input sequence data x = {xt}Tt=1 of length T with data at time t, xt ∈ Rm, a simple RNN (Elman, 1990) performs the following recursive computation at each layer ` and each time step t\ng(`,t)(x) = W (`)h(`,t−1)(x) +U (`)h(`−1,t)(x) + b(`), h(`,t)(x) = φ ( g(`,t)(x) ) ,\nwhere W (`) ∈ Rn×n, b(`) ∈ Rn for ` ∈ [L], U (1) ∈ Rn×m and U (`) ∈ Rn×n for ` ≥ 2 are the RNN parameters. g(`,t)(x) is the pre-activation vector at layer ` and time step t, and h(`,t)(x) is the after-activation (hidden state). For the input layer ` = 0, we define h(0,t)(x) := xt. h(`,0)(x) as the initial hidden state at layer ` that must be initialized to start the RNN recursive computation.\nThe output of an L-hidden layer RNN with linear read out layer is achieved via\nfθ(x) = V h (L,T )(x),\nwhere V ∈ Rd×n. Figure 1 visualizes an RNN unrolled through time.\nh(2,2)h (2,1)(x) h(2,3)(x)\nh(1,2)(x)h(1,1)(x) h(1,3)(x)\nW (2) W (2)\nW (1) W (1) U (2) U (2) U (2)\nh(2,0)(x)\nh(1,0)(x)\nW (2)\nW (1)\nx1 x2 x3\nU (1) U (1) U (1)\nNeural Tangent Kernel (NTK). Let fθ(x) ∈ Rd be the output of a DNN with parameters θ. For two input data sequences x and x′, the NTK is defined as (Jacot et al., 2018)\nΘ̂s(x,x ′) = 〈∇θsfθs(x),∇θsfθs(x′)〉,\nwhere fθs and θs are the network output and parameters during training at time s. 1 Let X and Y be the set of training inputs and targets, `(ŷ, y) : Rd × Rd → R+ be the loss function, and L = 1|X | ∑ (x,y)∈X×Y `(fθs(x),y) be the the empirical loss. The evolution of the parameters θs and output of the network fθs on a test input using gradient descent with infinitesimal step size (a.k.a gradient flow) with learning rate η is given by\n∂θs ∂s = −η∇θsfθs(X )T∇fθs (X )L (1)\n∂fθs(x)\n∂s = −η∇θsfθs(x)∇θsfθs(X )T∇fθs (X )L = −ηΘ̂s(x,X )∇fθs (X )L. (2)\nGenerally, Θ̂s(x,x′), hereafter referred to as the empirical NTK, changes over time during training, making the analysis of the training dynamics difficult. When fθs corresponds to an infinite-width MLP, (Jacot et al., 2018) showed that Θ̂s(x,x′) converges to a limiting kernel at initialization and stays constant during training, i.e.,\nlim n→∞\nΘ̂s(x,x ′) = lim\nn→∞ Θ̂0(x,x\n′) := Θ(x,x′) ∀s ,\nwhich is equivalent to replacing the outputs of the DNN by their first-order Taylor expansion in the parameter space (Lee et al., 2019). With a mean-square error (MSE) loss function, the training dynamics in (1) and (2) simplify to a set of linear ODEs, which coincides with the training dynamics of kernel ridge regression with respect to the NTK when the ridge term goes to zero. A nonzero ridge regularization can be conjured up by adding a regularization term λ 2\n2 ‖θs − θ0‖ 2 2 to the empirical loss\n(Hu et al., 2020)."},{"heading":"3 THE RECURRENT NEURAL TANGENT KERNEL","text":"We are now ready to derive the RNTK. We first prove the convergence of an RNN at initialization to the RNTK in the infinite-width limit and discuss various insights it provides. We then derive the convergence of an RNN after training to the RNTK. Finally, we analyze the effects of various hyperparameter choices on the RNTK. Proofs of all of our results are provided in the Appendices."},{"heading":"3.1 RNTK FOR AN INFINITE-WIDTH RNN AT INITIALIZATION","text":"First we specify the following parameter initialization scheme that follows previous work on NTKs (Jacot et al., 2018), which is crucial to our convergence results:\nW (`) = σ`w√ n W(`), U (1) = σ1u√ m U(1), U (`) = σ`u√ n U(`)(`≥2), V = σv√ n V, b(`) =σbb (`) , (3)\nwhere\n[W`]i,j , [U (`)]i,j , [V]i,j , [b (`)]i ∼ N (0, 1) . (4)\nWe will refer to (3) and (4) as the NTK initialization. The choices of the hyperparameters σw, σu, σv and σb can significantly impact RNN performance, and we discuss them in detail in Section\n1We use s to denote time here, since t is used to index the time steps of the RNN inputs.\n3.4. For the initial (at time t = 0) hidden state at each layer `, we set h(`,0)(x) to an i.i.d. copy of N (0, σh) (Wang et al., 2018) . For convenience, we collect all of the learnable parameters of the RNN into θ = vect [ {{W(`),U(`),b(`)}L`=1,V} ] .\nThe derivation of the RNTK at initialization is based on the correspondence between Gaussian initialized, infinite-width DNNs and Gaussian Processes (GPs), known as the DNN-GP. In this setting every coordinate of the DNN output tends to a GP as the number of units/neurons in the hidden layer (its width) goes to infinity. The corresponding DNN-GP kernel is computed as\nK(x,x′) = E θ∼N\n[ [fθ(x)]i · [fθ(x′)]i ] , ∀i ∈ [d]. (5)\nFirst introduced for a single-layer, fully-connected neural network by (Neal, 1995), recent works on NTKs have extended the results for various DNN architectures (Lee et al., 2018; Duvenaud et al., 2014; Novak et al., 2019; Garriga-Alonso et al., 2019; Yang, 2019b), where in addition to the output, all pre-activation layers of the DNN tends to a GPs in the infinite-width limit. In the case of RNNs, each coordinate of the RNN pre-activation g(`,t)(x) converges to a centered GP depending on the inputs with kernel\nΣ(`,t,t ′)(x,x′) = E\nθ∼N\n[ [g(`,t)(x)]i · [g(`,t ′)(x′)]i ] ∀i ∈ [n]. (6)\nAs per (Yang, 2019a), the gradients of random infinite-width DNNs computed during backpropagation are also Gaussian distributed. In the case of RNNs, every coordinate of the vector δ(`,t)(x) :=√ n ( ∇g(`,t)(x)fθ(x) ) converges to a GP with kernel\nΠ(`,t,t ′)(x,x′) = E\nθ∼N\n[ [δ(`,t)(x)]i · [δ(`,t ′)(x′)]i ] ∀i ∈ [n]. (7)\nBoth convergences occur independently of the coordinate index i and for inputs of possibly different lengths, i.e., T 6= T ′. With (6) and (7), we now prove that an infinite-width RNN at initialization converges to the limiting RNTK.\nTheorem 1 Let x and x′ be two data sequences of potentially different lengths T and T ′, respectively. Without loss of generality, assume that T ≤ T ′, and let τ := T ′ − T . Let n be the number of units in the hidden layers, the empirical RNTK for an L-layer RNN with NTK initialization converges to the following limiting kernel as n→∞\nlim n→∞\nΘ̂0(x,x ′) = Θ(x,x′) = Θ(L,T,T ′)(x,x′)⊗ Id , (8)\nwhere\nΘ(L,T,T ′)(x,x′) = ( L∑ `=1 T∑ t=1 ( Π(`,t,t+τ)(x,x′) · Σ(`,t,t+τ)(x,x′) )) +K(x,x′) , (9)\nwith K(x,x′), Σ(`,t,t+τ)(x,x′), and Π(`,t,t+τ)(x,x′) defined in (5)–(7).\nRemarks. Theorem 1 holds generally for any two data sequences, including different lengths ones. This highlights the RNTK’s ability to produce a similarity measure Θ(x,x′) even if the inputs are of different lengths, without resorting to heuristics such as zero padding the inputs to the to the max length of both sequences. Dealing with data of different length is in sharp contrast to common kernels such as the classical radial basis functions, polynomial kernels, and current NTKs. We showcase this capability below in Section 4.\nTo visualize Theorem 1, we plot in the left plot in Figure 2 the convergence of a single layer, sufficiently wide RNN to its RNTK with the two simple inputs x = {1,−1, 1} of length 3 and x′ = {cos(α), sin(α)} of length 2, where α = [0, 2π]. For an RNN with a sufficiently large hidden state (n = 1000), we see clearly that it converges to the RNTK (n =∞). RNTK Example for a Single-Layer RNN. We present a concrete example of Theorem 1 by showing how to recursively compute the RNTK for a single-layer RNN; thus we drop the layer index for notational simplicity. We compute and display the RNTK for the general case of a multilayer RNN in Appendix B.3. To compute the RNTK Θ(T,T ′)(x,x′), we need to compute the GP\nkernels Σ(t,t+τ)(x,x′) and Π(t,t+τ)(x,x′). We first define the operator Vφ [ K ]\nthat depends on the nonlinearity φ(·) and a positive semi-definite matrixK ∈ R2×2\nVφ [ K ]\n= E[φ(z1) · φ(z2)], (z1, z2) ∼ N (0,K) . (10) Following (Yang, 2019a), we obtain the analytical recursive formula for the GP kernel Σ(t,t+τ)(x,x′) for a single layer RNN as\nΣ(1,1)(x,x′) = σ2wσ 2 h1(x=x′) + σ2u m 〈x1,x′1〉+ σ2b (11) Σ(t,1)(x,x′) = σ2u m 〈xt,x′1〉+ σ2b t > 1 (12)\nΣ(1,t ′)(x,x′) = σ2u m 〈x1,x′t′〉+ σ2b t′ > 1 (13)\nΣ(t,t ′)(x,x′) = σ2wVφ [ K(t,t ′)(x,x′) ]\n+ σ2u m 〈xt,x′t′〉+ σ2b t, t′ > 1 (14)\nK(x,x′) = σ2vVφ [ K(T+1,T ′+1)(x,x′) ] , (15)\nwhere\nK(t,t ′)(x,x′) =\n[ Σ(t−1,t−1)(x,x) Σ(t−1,t ′−1)(x,x′)\nΣ(t−1,t ′−1)(x,x′) Σ(t ′−1,t′−1)(x′,x′)\n] . (16)\nSimilarly, we obtain the analytical recursive formula for the GP kernel Π(t,t+τ)(x,x′) as\nΠ(T,T ′)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] (17)\nΠ(t,t+τ)(x,x′) = σ2wVφ′ [ K(t+1,t+τ+1)(x,x′) ] Π(t+1,t+1+τ)(x,x′) t ∈ [T − 1] (18)\nΠ(t,t ′)(x,x′) = 0 t′ − t 6= τ. (19) For φ = ReLU and φ = erf , we provide analytical expressions for Vφ [ K ] and Vφ′ [ K ]\nin Appendix B.5. These yield an explicit formula for the RNTK that enables fast and point-wise kernel evaluations. For other activation functions, one can apply the Monte Carlo approximation to obtain Vφ [ K ]\nand Vφ′ [ K ] (Novak et al., 2019)."},{"heading":"3.2 RNTK FOR AN INFINITE-WIDTH RNN DURING TRAINING","text":"We prove that an infinitely-wide RNN, not only at initialization but also during gradient descent training, converges to the limiting RNTK at initialization.\nTheorem 2 Let n be the number of units of each RNN’s layer. Assume that Θ(X ,X ) is positive definite on X such that λmin(Θ(X ,X )) > 0. Let η∗ := 2 ( λmin(Θ(X ,X )) + λmax(Θ(X ,X ) ))−1 . For an L-layer RNN with NTK initialization as in (3), (4) trained under gradient flow (recall (1) and (2)) with η < η∗, we have with high probability\nsup s ‖θs − θ0‖2√ n , sup s ‖Θ̂s(X ,X )− Θ̂0(X ,X )‖2 = O ( 1√ n ) .\nRemarks. Theorem 2 states that the training dynamics of an RNN in the infinite-width limit as in (1), (2) are governed by the RNTK derived from the RNN at its initialization. Intuitively, this is due to the NTK initialization (3), (4) which positions the parameters near a local minima, thus minimizing the amount of update that needs to be applied to the weights to obtain the final parameters."},{"heading":"3.3 RNTK FOR AN INFINITE-WIDTH RNN WITHOUT WEIGHT SHARING","text":"We prove that, in the infinite-width limit, an RNN without weight sharing (untied weights), i.e., using independent new weights W(`,t), U(`,t) and b(`,t) at each time step t, converges to the same RNTK as an RNN with weight sharing (tied weights). First, recall that it is a common practice to use weight-tied RNNs, i.e., in layer `, the weights W(`), U(`) and b(`) are the same across all time steps t. This practice conserves memory and reduces the number of learnable parameters. We demonstrate that, when using untied-weights, the RNTK formula remains unchanged.\nTheorem 3 For inputs of the same length, an RNN with untied weights converges to the same RNTK as an RNN with tied weights in the infinite-width (n→∞) regime.\nRemarks. Theorem 3 implies that weight-tied and weight-untied RNNs have similar behaviors in the infinite-width limit. It also suggests that existing results on the simpler, weight-untied RNN setting may be applicable for the more general, weight-tied RNN. The plot on the right side of Figure 2 empirically demonstrates the convergence of both the weight-tied and weight-untied RNNs to the RNTK with increasing hidden layer size n; moreover, the convergence rates are similar."},{"heading":"3.4 INSIGHTS INTO THE ROLES OF THE RNTK’S HYPERPARAMETERS","text":"Our analytical form for the RNTK is fully determined by a small number of hyperparameters, which contains the various weight variances collected into S = {σw, σu, σb, σh} and the activation function.2 In standard supervised-learning settings, one often performs cross-validation to select the hyperparameters. However, since kernel methods become computationally intractable for large datasets, we seek a more computationally friendly alternative to cross-validation. Here we conduct a novel exploratory analysis that provides new insights into the impact of the RNTK hyperparameters on the RNTK output and suggests a simple method to select them a priori in a deliberate manner.\nTo visualize the role of the RNTK hyperparameters, we introduce the sensitivity s(t) of the RNTK of two input sequences x and x′ with respect to the input xt at time t\ns(t) = ‖∇xtΘ(x,x′)‖2 . (20)\n2From (11) to (18) we emphasize that σv merely scales the RNTK and does not change its overall behavior.\nHere, s(t) indicates how sensitive the RNTK is to the data at time t, i.e., xt, in presence of another data sequence x′. Intuitively, large/small s(t) indicates that the RNTK is relatively sensitive/insensitive to the input xt at time t.\nThe sensitivity is crucial to understanding to which extent the RNTK prediction is impacted by the input at each time step. In the case where some time indices have a small sensitivity, then any input variation in those corresponding times will not alter the RNTK output and thus will produce a metric that is invariant to those changes. This situation can be beneficial or detrimental based on the task at hand. Ideally, and in the absence of prior knowledge on the data, one should aim to have a roughly constant sensitivity across time in order to treat all time steps equally in the RNTK input comparison.\nFigure 3 plots the normalized sensitivity s(t)/maxt(s(t)) for two data sequences of the same length T = 100, with s(t) computed numerically for xt,x′t ∼ N (0, 1). We repeated the experiments 10000 times; the mean of the sensitivity is shown in Figure 3. Each of the plots shows the changes of parameters SReLU = { √ 2, 1, 0, 0} for φ = ReLU and Serf = {1, 0.01, 0.05, 0} for φ = erf .\nFrom Figure 3 we first observe that both ReLU and erf show similar per time step sensitivity measure s(t) behavior around the hyperparameters SReLU and Serf . If one varies any of the weight variance parameters, the sensitivity exhibits a wide range of behavior, and in particular with erf . We observe that σw has a major influence on s(t). For ReLU, a small decrease/increase in σw can lead to over-sensitivity of the RNTK to data at the last/first times steps, whereas for erf , any changes in σw leads to over-sensitivity to the last time steps.\nAnother notable observation is the importance of σh, which is usually set to zero for RNNs. (Wang et al., 2018) showed that a non-zero σh acts as a regularization that improves the performance of RNNs with the ReLU nonlinearity. From the sensitivity perspective, a non-zero σh results in reducing the importance of the first time steps of the input. We also see the same behavior in erf , but with stronger changes as σh increases. Hence whenever one aims at reinforcing the input pairwise comparisons, such parameters should be favored.\nThis sensitivity analysis provides a practical tool for RNTK hyperparameter tuning. In the absence of knowledge about the data, hyperparameters should be chosen to produce the least time varying sensitivity. If given a priori knowledge, hyperparameters can be selected that direct the RNTK to the desired time-steps."},{"heading":"4 EXPERIMENTS","text":"We now empirically validate the performance of the RNTK compared to classic kernels, NTKs, and trained RNNs on both classification and regression tasks using a large number of time series data sets. Of particular interest is the capability of the RNTK to offer high performance even on inputs of different lengths.\nTime Series Classification. The first set of experiments considers time series inputs of the same lengths from 56 datasets in the UCR time-series classification data repository (Dau et al., 2019). We restrict ourselves to selected data sets with fewer than 1000 training samples and fewer than 1000 time steps (T ) as kernel methods become rapidly intractable for larger datasets. We compare the RNTK with a variety of other kernels, including the Radial Basis Kernel (RBF), polynomial kernel, and NTK (Jacot et al., 2018), as well as finite RNNs with Gaussian, identity (Le et al., 2015) initialization, and GRU (Cho et al., 2014). We use φ = ReLU for both the RNTKs and NTKs. For each kernel, we train a C-SVM (Chang & Lin, 2011) classifier, and for each finite RNN we use gradient descent training. For model hyperparameter tuning, we use 10-fold cross-validation. Details on the data sets and experimental setup are available in Appendix A.1.\nWe summarize the classification results over all 56 datasets in Table 1; detailed results on each data set is available in Appendix A.2. We see that the RNTK outperforms not only the classical kernels but also the NTK and trained RNNs in all metrics. The results demonstrate the ability of RNTK to provide increased performances compare to various other methods (kernels and RNNs). The superior performance of RNTK compared to other kernels, including NTK, can be explained by the internal recurrent mechanism present in RNTK, allowing time-series adapted sample comparison. In addition, RNTK also outperforms RNN and GRU. As the datasets we consider are relative small in size, finite RNNs and GRUs that typically require large amount of data to succeed do not perform well in our setting. An interesting future direction would be to compare RNTK to RNN/GRU on larger datasets.\nTime Series Regression. We now validate the performance of the RNTK on time series inputs of different lengths on both synthetic data and real data. For both scenarios, the target is to predict the next time-step observation of the randomly extracted windows of different length using kernel ridge regression.\nWe compare the RNTK to other kernels, the RBF and polynomial kernels and the NTK. We also compare our results with a data independent predictor that requires no training, that is simply to predict the next time step with previous time step (PTS).\nFor the synthetic data experiment, we simulate 1000 samples of one period of a sinusoid and add white Gaussian noise with default σn = 0.05. From this fixed data, we extract training set size Ntrain = 20 segments of uniform random lengths in the range of [Tfixed, Tfixed + Tvar] with Tfixed = 10. We use standard kernel ridge regression for this task. The test set is comprised of Ntest = 5000 obtained from other randomly extracted segments, again of varying lengths. For the real data, we use 975 days of the Google stock value in the years 2014–2018. As in the simulated signal setup above, we extract Ntrain segments of different lengths from the first 700 days and test on the Ntest segments from days 701 to 975. Details of the experiment are available in Appendix A.2.\nWe report the predicted signal-to-noise ratio (SNR) for both datasets in Figures 4a and 4c for various values of Tvar. We vary the noise level and training set size for fixed Tvar = 10 in Figures 4b and 4d. As we see from Figures 4a and 4c, the RNTK offers substantial performance gains compared to the other kernels, due to its ability to naturally deal with variable length inputs. Moreover, the performance gap increases with the amount of length variation of the inputs Tvar. Figure 4d demonstrates that, unlike the other methods, the RNTK maintains its performance even when the training set is small. Finally, Figure 4c demonstrates that the impact of noise in the data on the regression performance is roughly the same for all models but becomes more important for RNTK with a large σn; this might be attributed to the recurrent structure of the model allowing for a time propagation and amplification of the noise for very low SNR. These experiments demonstrate the distinctive advantages of the RNTK over classical kernels, and NTKs for input data sequences of varying lengths.\nIn the case of PTS, we expect the predictor to outperform kernel methods when learning from the training samples is hard, due to noise in the data or small training size which can lead to over fitting. In Figure 4a RNTK and Polynomial kernels outperforms PTS for all values of Tvar, but for larger Tvar, NTK and RBF under perform PTS due to the increasing detrimental effect of zero padding.\nFor the Google stock value, we see a superior performance of PTS with respect to all other kernel methods due to the nature of those data heavily relying on close past data. However, RNTK is able to reduce the effect of over-fitting, and provide the closest results to PTS among all kernel methods we employed, with increasing performance as the number of training samples increase."},{"heading":"5 CONCLUSIONS","text":"In this paper, we have derived the RNTK based on the architecture of a simple RNN. We have proved that, at initialization, after training, and without weight sharing, any simple RNN converges to the same RNTK. This convergence provides new insights into the behavior of infinite-width RNNs, including how they process different-length inputs, their training dynamics, and the sensitivity of their output at every time step to different nonlinearities and initializations. We have highlighted the RNTK’s practical utility by demonstrating its superior performance on time series classification and regression compared to a range of classical kernels, the NTK, and trained RNNs. There are many avenues for future research, including developing RNTKs for gated RNNs such as the LSTM (Hochreiter & Schmidhuber, 1997) and investigating which of our theoretical insights extend to finite RNNs."},{"heading":"ACKNOWLEDGMENTS","text":"This work was supported by NSF grants CCF-1911094, IIS-1838177, and IIS-1730574; ONR grants N00014-18-12571, N00014-20-1-2787, and N00014-20-1-2534; AFOSR grant FA9550-18-1-0478; and a Vannevar Bush Faculty Fellowship, ONR grant N00014-18-1-2047."},{"heading":"A EXPERIMENT DETAILS","text":""},{"heading":"A.1 TIME SERIES CLASSIFICATION","text":"Kernel methods settings. We used RNTK, RBF, polynomial and NTK (Jacot et al., 2018). For data pre-processing, we normalized the norm of each x to 1. For training we used C-SVM in LIBSVM library (Chang & Lin, 2011) and for hyperparameter selection we performed 10-fold validation for splitting the training data into 90% training set and 10% validation test. We then choose the best performing set of hyperparameters on all the validation sets, retrain the models with the best set of hyperparameters on the entire training data and finally report the performance on the unseen test data. The performance of all kernels on each data set is shown in table 2.\nFor C-SVM we chose the cost function value\nC ∈ {0.01, 0.1, 1, 10, 100}\nand for each kernel we used the following hyperparameter sets\n• RNTK: We only used single layer RNTK, we φ = ReLU and the following hyperparameter sets for the variances:\nσw ∈ {1.34, 1.35, 1.36, 1.37, 1.38, 1.39, 1.40, 1.41, 1.42, √\n2, 1.43, 1.44, 1.45, 1.46, 1.47} σu = 1\nσb ∈ {0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.9, 1, 2} σh ∈ {0, 0.01, 0.1, 0.5, 1}\n• NTK: The formula for NTK of L-layer MLP (Jacot et al., 2018) for x,x′ ∈ Rm is:\nΣ(1) = σ2w m 〈x,x′〉+ σ2b\nΣ(`)(x,x′) = σ2wVφ[K (`)(x,x′)] + σ2b ` ∈ [L] Σ̇(`)(x,x′) = σ2wVφ′ [K (`+1)(x,x′)] ` ∈ [L]\nK(`)(x,x′) = [ Σ(`−1)(x,x) Σ(`−1)(x,x′) Σ(`−1)(x,x′) Σ(`−1)(x′,x′) ] K(x,x′) = σ2vVφ[K(L+1)(x,x′)]\nkNTK = L∑ `=1\n( Σ(`)(x,x′)\nL∏ `′=` Σ̇(`)(x,x′)\n) +K(x,x′)\nand we used the following hyperparamters\nL ∈ [10] σw ∈ {0.5, 1, √ 2, 2, 2.5, 3}\nσb ∈ {0, 0.01, 0.1, 0.2, 0.5, 0.8, 1, 2, 5}\n• RBF:\nkRBF(x,x ′) = e(−α‖x−x ′‖22)\nα ∈ {0.01, 0.05, 0.1, 0.2, 0.5, 0.6, 0.7, 0.8, 1, 2, 3, 4, 5, 10, 20, 30, 40, 100}\n• Polynomial:\nkPolynomial(x,x ′) = (r + 〈x,x′〉)d\nd ∈ [5] r ∈ {0, 0.1, 0.2, 0.5, 1, 2}\nFinite-width RNN settings. We used 3 different RNNs. The first is a ReLU RNN with Gaussian initialization with the same NTK initialization scheme, where parameter variances are σw = σv = √ 2,\nσu = 1 and σb = 0. The second is a ReLU RNN with identity initialization following (Le et al., 2015). The third is a GRU (Cho et al., 2014) with uniform initialization. All models are trained with RMSProp algorithm for 200 epochs. Early stopping is implemented when the validation set accuracy does not improve for 5 consecutive epochs.\nWe perform standard 5-fold cross validation. For each RNN architecture we used hyperparamters of number of layer, number of hidden units and learning rate as\nL ∈ {1, 2} n ∈ {50, 100, 200, 500}\nη ∈ {0.01, 0.001, 0.0001, 0.00001}\nMetrics descriptions First, only in this paragraph, let i ∈ {1, 2, ..., N} index a total of N datasets and j ∈ {1, 2, ...,M} index a total of M classifiers. Let yij be the accuracy of the j-th classifer on the i-th dataset. We reported results on 4 metrics: average accuracy (Acc. mean), P90, P95, PMA and Friedman Rank. P90 and P95 is the fraction of datasets that the classifier achieves at least 90% and 95% of the maximum achievable accuracy for each dataset, i.e.,\nP90j = 1\nN ∑ i 1(yij ≥ 0.9(max j yij)) . (21)\nPMA is the accuracy of the classifier on a dataset divided by the maximum achievable accuracy on that dataset, averaged over all datasets:\nPMAj = 1\nN ∑ i yij max j yij . (22)\nFriedman Rank (Fernández-Delgado et al., 2019) first ranks the accuracy of each classifier on each dataset and then takes the average of the ranks for each classifier over all datasets, i.e.,\nFRj = 1\nN ∑ i rij , (23)\nwhere rij is the ranking of the j-th classifier on the i-th dataset.\nNote that a better classifier achieves a lower Friedman Rank, Higher P/90 and PMA.\nRemark. In order to provide insight into the performance of RNTK in long time steps setting, we picked two datasets with more that 1000 times steps: SemgHandSubjectCh2 (T = 1024) and StarLightCurves (T = 1024)."},{"heading":"A.2 TIME SERIES REGRESSION","text":"For time series regression, we used the 5-fold validation of training set and same hyperparamter sets for all kernels. For training we kernel ridge regression with ridge term chosen form\nλ ∈ {0, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.8, 1, 2, 3, 4, 5, 6, 7, 8, 10, 100}"},{"heading":"B PROOFS FOR THEOREMS 1 AND 3: RNTK CONVERGENCE AT INITIALIZATION","text":""},{"heading":"B.1 PRELIMINARY: NETSOR PROGRAMS","text":"Calculation of NTK in any architecture relies on finding the GP kernels that correspond to each pre-activation and gradient layers at initialization. For feedforward neural networks with n1, . . . , nL number of neurons (channels in CNNs) at each layer the form of this GP kernels can be calculated via taking the limit of n1, . . . , nL sequentially one by one. The proof is given by induction, where by conditioning on the previous layers, each entry of the current layer is sum of infinite i.i.d Gaussian random variables, and based on Central Limit Theorem (CLT), it becomes a Gaussian process with\nkernel calculated based on the previous layers. Since the first layer is an affine transformation of input with Gaussian weights, it is a Gaussian process and the proof is completed. See (Lee et al., 2018; Duvenaud et al., 2014; Novak et al., 2019; Garriga-Alonso et al., 2019) for a formal treatment. However, due to weight-sharing, sequential limit is not possible and condoning on previous layers does not result in i.i.d. weights. Hence the aforementioned arguments break. To deal with it, in (Yang, 2019a) a proof using Gaussian conditioning trick (Bolthausen, 2014) is presented which allows use of recurrent weights in a network. More precisely, it has been demonstrated than neural networks (without batch normalization) can be expressed and a series of matrix multiplication and (piece wise) nonlinearity application, generally referred as Netsor programs. It has been shown that any architecture that can be expressed as Netsor programs that converge to GPs as width goes to infinity in the same rate, which a general rule to obtain the GP kernels. For completeness of this paper, we briefly restate the results from (Yang, 2019a) which we will use later for calculation derivation of RNTK.\nThere are 3 types of variables in Netsor programs; A-vars, G-vars and H-vars. A-vars are matrices and vectors with i.i.d Gaussian entries, G-vars are vectors introduced by multiplication of a vector by an A-var and H-vars are vectors after coordinate wise nonlinearities is applied to G-vars. Generally, G-vars can be thought of as pre-activation layers which are asymptotically treated as a Gaussian distributed vectors, H-vars as after-activation layers and A-vars are the weights. Since in neural networks inputs are immediately multiplied by a weight matrix, it can be thought of as an G-var, namely gin. Generally Netsor programs supports G-vars with different dimension, however the asymptotic behavior of a neural networks described by Netsor programs does not change under this degree of freedom, as long as they go to infinity at the same rate. For simplicity, let the G-vars and H-vars have the same dimension n since the network of interest is RNN and all pre-activation layers have the same dimension. We introduce the Netsor programs under this simplification. To produce the output of a neural network, Netsor programs receive a set of G-vars and A-vars as input, and new variables are produced sequentially using the three following operators:\n• Matmul : multiplication of an A-var: A with an H-var: h, which produce a new G-var, g. g = Ah (24)\n• Lincomp: Linear combination of G-vars, gi, 1 ≤ i ≤ k , with coefficients ai ∈ R 1 ≤ i ≤ k which produce of new G-var:\ng = k∑ i=1 aigi (25)\n• Nonlin: creating a new H-var, h, by using a nonlinear function φ : Rk → R that act coordinate wise on a set of G-vars, gi, 1 ≤ i ≤ k :\nh = ϕ(g1, . . . , gk) (26)\nAny output of the neural network y ∈ R should be expressed as inner product of a new A-var which has not been used anywhere else in previous computations and an H-var:\ny = v>h\nAny other output can be produced by another v′ and h′ (possibility the same h or v).\nIt is assumed that each entry of any A-var : A ∈ Rn×n in the netsor programs computations is drawn from N (0, σ 2 a\nn ) and the input G-vars are Gaussian distributed. The collection of a specific entry of all G-vars of in the netsor program converges in probability to a Gaussian vector {[g1]i, . . . , [gk]i} ∼ N (µ,Σ) for all i ∈ [n] as n goes to infinity. Let µ(g) := E [ [g]i ] be the mean of aG-var and Σ(g, g′) := E [ [g]i · [g′]i ] be the covariance between any two G-vars. The general rule for µ(g) is given by the following equations:\nµ(g) = µin(g) if g is input k∑ i=1 aiµ(gi) if g = k∑ i=1 aigi\n0 otherwise\n(27)\nFor g and g′, let G = {g1, . . . , gr} be the set of G-vars that has been introduced before g and g′ with distribution N (µG ,ΣG), where ΣG ∈ R|G|×|G| containing the pairwise covariances between the G-vars. Σ(g, g′) is calculated via the following rules:\nΣ(g, g′) = Σin(g, g′) if g and g′ are inputs k∑ i=1 aiΣ(gi, g′) if g = k∑ i=1 aigi k∑ i=1 aiΣ(g, gi) if g′ = k∑ i=1 aigi σ2A E z∼N (µ,ΣG) [ϕ(z)ϕ̄(z)] if g = Ah and g′ = Ah′\n0 otherwise\n(28)\nWhere h = ϕ(g1, . . . , gr) and h′ = ϕ̄(g1, . . . , gr) are functions of G-vars in G from possibly different nonlinearities. This set of rules presents a recursive method for calculating the GP kernels in a network where the recursive formula starts from data dependent quantities Σin and µin which are given.\nAll the above results holds when the nonlinearities are bounded uniformly by e(cx 2−α) for some α > 0 and when their derivatives exist.\nStandard vs. NTK initialization. The common practice (which netsor programs uses) is to initialize DNNs weights [A]i,j with N (0, σa√n ) (known as standard initialization) where generally n is the number of units in the previous layer. In this paper we have used a different parameterization scheme as used in (Jacot et al., 2018) and we factor the standard deviation as shown in 3 and initialize weights with standard standard Gaussian. This approach does not change the the forward computation of DNN, but normalizes the backward computation (when computing the gradients) by factor 1n , otherwise RNTK will be scales by n. However this problem can be solved by scaling the step size by 1 n and there is no difference between NTK and standard initialization (Lee et al., 2019)."},{"heading":"B.2 PROOF FOR THEOREM 1: SINGLE LAYER CASE","text":"We first derive the RNTK in a simpler setting, i.e., a single layer and single output RNN. We then generalize the results to multi-layer and multi-output RNNs. We drop the layer index ` to simplify notation. From 3 and 4, the forward pass for computing the output under NTK initialization for each input x = {xt}Tt=1 is given by:\ng(t)(x) = σw√ m Wh(t−1)(x) + σu√ n Uxt + σbb (29)\nh(t)(x) = φ ( g(t)(x) ) (30)\nfθ(x) = σv√ n v>h(T )(x) (31)\nNote that (29), (30) and (31) use all the introduced operators introduced in 24, 25 and 26 given input variables W, {Uxt}Tt=1,b,v and h(0)(x).\nFirst, we compute the kernels of forward pass Σ(t,t ′)(x,x′) and backward pass Π(t,t ′)(x,x′) introduced in (6) and (7) for two input x and x′. Note that based on (27) the mean of all variables is zero since the inputs are all zero mean. In the forward pass for the intermediate layers we have:\nΣ(t,t ′)(x,x′) = Σ(g(t)(x), g(t ′)(x′))\n= Σ ( σw√ n Wh(t−1)(x) + σu√ m Uxt + σbb, σw√ n Wh(t ′−1)(x′) + σu√ m Ux′t′ + σbb ) = Σ ( σw√ n Wh(t−1)(x), σw√ n Wh(t ′−1)(x′) ) + Σin ( σu√ m Uxt, σu√ m Ux′t′ ) + Σin (σbb, σbb) .\nWe have used the second and third rule in (28) to expand the formula, We have also used the first and fifth rule to set the cross term to zero, i.e.,\nΣ ( σw√ n Wh(t−1)(x), σu√ n Ux′t′ ) = 0\nΣ ( σw√ n Wh(t−1)(x), σbb ) = 0\nΣ ( σu√ m Uxt, σw√ n Wh(t ′−1)(x′) ) = 0\nΣ ( σbb,\nσw√ n Wh(t ′−1)(x′)\n) = 0\nΣin ( σu√ m Uxt, σbb ) = 0\nΣin ( σbb,\nσu√ m Ux′t′\n) = 0.\nFor the non-zero terms we have\nΣin (σbb, σbb) = σ 2 b Σin ( σu√ m Uxt, σu√ m Ux′t′ ) = σ2u m 〈xt,x′t′〉,\nwhich can be achieved by straight forward computation. If t 6= 1 and t′ 6= 1, by using the forth rule in (28) we have\nΣ ( σw√ n Wh(t−1)(x), σw√ n Wh(t ′−1)(x′) ) = σ2w E z∼N (0,K(t,t′)(x,x′)) [φ(z1)φ(z2)] = Vφ [ K(t,t ′)(x,x′) ] .\nWithK(t,t ′)(x,x′) defined in (16). Otherwise, it will be zero by the fifth rule (if t or t = 1) . Here the set of previously introduced G-vars is G ={ {g(α)(x)},Uxα}t−1α=1, {g(α ′)(x′),Ux′α′} t′−1 α′=1,h (0)(x),h(0)(x′) }\n, but the dependency is only on the last layer G-vars, ϕ({g : g ∈ G}) = φ(g(t−1)(x)), ϕ̄(({g : g ∈ G})) = φ(g(t′−1)(x′)), leading the calculation to the operator defined in (10). As a result\nΣ(t,t ′)(x,x′) = σ2wVφ [ K(t,t ′)(x,x′) ]\n+ σ2u m 〈xt,x′t′〉+ σ2b .\nTo complete the recursive formula, using the same procedure for the first layers we have\nΣ(1,1)(x,x′) = σ2wσ 2 h1(x=x′) + σ2u m 〈x1,x′1〉+ σ2b ,\nΣ(1,t ′)(x,x′) = σ2u m 〈x1,x′t′〉+ σ2b ,\nΣ(t,1)(x,x′) = σ2u m 〈xt,x′1〉+ σ2b .\nThe output GP kernel is calculated via K(x,x′) = σ2vVφ [ K(T+1,T ′+1)(x,x′) ] The calculation of the gradient vectors δ(t)(x) = √ n ( ∇g(t)(x)fθ(x) ) in the backward pass is given by\nδ(T )(x) = σvv φ′(g(T )(x))\nδ(t)(x) = σw√ n\nW> ( φ′(g(t)(x)) δ(t+1)(x) ) t ∈ [T − 1]\nTo calculate the backward pass kernels, we rely on the following Corollary from (Yang, 2020b)\nCorollary 1 In infinitely wide neural networks weights used in calculation of back propagation gradients (W>) is an i.i.d copy of weights used in forward propagation (W) as long as the last layer weight (v) is sampled independently from other parameters and has mean 0.\nThe immediate result of Corollary 1 is that g(t)(x) and δ(t)(x) are two independent Gaussian vector as their covariance is zero based on the fifth rule in (28). Using this result, we have:\nΠ(t,t ′)(x,x′) = Σ ( δ(t)(x), δ(t ′)(x) )\n= E [ [δ(t)(x)]i · [δ(t ′)(x′)]i ] = σ2wE [ [φ′(g(t)(x))]i · [δ(t+1)(x)]i · [φ′(g(t ′)(x′))]i · [δ(t ′+1)(x′)]i\n] = σ2w E\nz∼N (0,K(t+1,t+1′)(x,x′)) [φ′(z1) · φ′(z2)] · E\n[ [δ(t+1)(x)]i · [δ(t ′+1)(x′)]i ] = σ2wVφ′ [ K(t+1,t ′+1)(x,x′) ] Π(t+1,t ′+1)(x,x′).\nIf T ′ − t′ = T − t, then the the formula will lead to\nΠ(T,T ′)(x,x′) = E [ [δ(T )(x)]i, [δ (T ′)(x′)]i ] = σ2vE [ [v]i · [φ′(g(T )(x))]i · [v]i · [φ′(g(T ′)(x′))]i\n] = E [ [φ′(g(T )(x))]i · [φ′(g(T ′)(x′))]i ] · E [[v]i [v]i]\n= σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] .\nOtherwise it will end to either of two cases for some t′′ < T or T ′ and by the fifth rule in (28) we have:\nΣ ( δ(t ′′)(x), δ(T ′)(x) ) = Σ ( σw√ n W> ( φ′(g(t ′′)(x)) δ(t ′′+1)(x′) ) ,v φ′(g(T ′)(x)) ) = 0\nΣ ( δ(T )(x), δ(t ′′)(x) ) = Σ ( v φ′(g(T )(x)), σw√\nn W>\n( φ′(g(t ′′)(x′)) δ(t ′′+1)(x′) )) = 0.\nWithout loss of generality, from now on assume T ′ < T and T ′ − T = τ , the final formula for computing the backward gradients becomes:\nΠ(T,T+τ)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] Π(t,t+τ)(x,x′) = σ2wVφ′ [ K(t+1,t+τ+1)(x,x′) ] Π(t+1,t+1+τ)(x,x′) t ∈ [T − 1]\nΠ(t,t ′)(x,x′) = 0 t′ − t 6= τ\nNow we have derived the single layer RNTK. Recall that θ = Vect [ {W,U,b,v} ] contains all of the network’s learnable parameters. As a result, we have:\n∇θfθ(x) = Vect [ {∂fθ(x) ∂W , ∂fθ(x) ∂U , ∂fθ(x) ∂b , ∂fθ(x) ∂v } ] .\nAs a result 〈∇θfθ(x),∇θfθ(x′)〉 = 〈 ∂fθ(x)\n∂W , ∂fθ(x\n′)\n∂W\n〉 + 〈 ∂fθ(x)\n∂U , ∂fθ(x\n′)\n∂U\n〉 + 〈 ∂fθ(x)\n∂b , ∂fθ(x\n′)\n∂b 〉 + 〈 ∂fθ(x)\n∂v , ∂fθ(x\n′)\n∂v\n〉\nWhere the gradients of output with respect to weights can be formulated as the following compact form:\n∂fθ(x)\n∂W = T∑ t=1 ( 1√ n δ(t)(x) ) · ( σw√ n h(t−1)(x) )> ∂fθ(x)\n∂U = T∑ t=1 ( 1√ n δ(t)(x) ) · ( σu√ m xt )> ∂fθ(x)\n∂b = T∑ t=1 ( σb√ n δ(t)(x) ) ∂fθ(x)\n∂v = σv√ n h(T )(x).\nAs a result we have:〈 ∂fθ(x)\n∂W , ∂fθ(x\n′)\n∂W\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉) · ( σ2w n 〈 h(t−1)(x),h(t ′−1)(x′) 〉)\n〈 ∂fθ(x)\n∂U , ∂fθ(x\n′)\n∂U\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉) · ( σ2u m 〈xt,x′t′〉 ) 〈 ∂fθ(x)\n∂b , ∂fθ(x\n′)\n∂b\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉)\n· σ2b〈 ∂fθ(x)\n∂v , ∂fθ(x\n′)\n∂v\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉) .\nRemember that for any two G-var E [[g]i[g′]i] is independent of index i. Therefore,\n1\nn\n〈 h(t−1)(x),h(t ′−1)(x′) 〉 → Vφ [ K(t,t ′)(x,x′) ]\nt > 1\n1\nn\n〈 h(0)(x),h(0)(x′) 〉 → σ2h.\nHence, by summing the above terms in the infinite-width limit we get\n〈∇θfθ(x),∇θfθ(x′)〉 → T ′∑ t′=1 T∑ t=1 Π(t,t ′)(x,x′) · Σ(t,t ′)(x′,x′) +K(x,x′). (32) Since Π(t,t ′)(x,x′) = 0 for t′ − t 6= τ it is simplified to\n〈∇θfθ(x),∇θfθ(x′)〉 = ( T∑ t=1 Π(t,t+τ)(x,x′) · Σ(t,t+τ)(x′,x′) ) +K(x,x′).\nMulti-dimensional output. For fθ(x) ∈ Rd, the i-th output for i ∈ [d] is obtained via\n[fθ(x)]i = σv√ n v>i h (T )(x),\nwhere vi is independent of vj for i 6= j. As a result, for The RNTK Θ(x,x′) ∈ Rd×d for multidimensional output we have\n[Θ(x,x′)]i,j = 〈 ∇θ [fθ(x)]i ,∇θ [fθ(x ′)]j 〉 For i = j, the kernel is the same as computed in (32) and we denote it as\n〈∇θ [fθ(x)]i ,∇θ [fθ(x ′)]i〉 = Θ (T,T ′)(x,x′).\nFor i 6= j, since vi is independent of vj , Π(T,T ′)(x,x′) and all the backward pass gradients become zero, so 〈 ∇θ [fθ(x)]i ,∇θ [fθ(x ′)]j 〉 = 0 i 6= j\nwhich gives us the following formula\nΘ(x,x′) = Θ(T,T ′)(x,x′)⊗ Id.\nThis concludes the proof for Theorem 1 for single-layer case."},{"heading":"B.3 PROOF FOR THEOREM 1: MULTI-LAYER CASE","text":"Now we drive the RNTK for multi-layer RNTK. We will only study single output case and the generalization to multi-dimensional case is identical as the single layer case. The set of equations for calculation of the output of a L-layer RNN for x = {xt}Tt=1 are\ng(`,t)(x) = σ`w√ n W(`)h(`,t−1)(x) + σ`u√ m U(`)xt + σ ` bb (`) ` = 1\ng(`,t)(x) = σ`w√ n W(`)h(`,t−1)(x) + σ`u√ n U(`)h(`−1,t)(x) + σ`bb (`) ` > 1\nh(`,t)(x) = φ ( g(`,t)(x) ) fθ(x) =\nσv√ n v>h(L,T )(x)\nThe forward pass kernels for the first layer is the same as calculated in B.2. For ` ≥ 2 we have:\nΣ(`,t,t ′)(x,x′) = Σ(g(`,t)(x), g(`,t ′)(x′))\n= Σ ( σ`w√ n W(`)h(`,t−1)(x), σ`w√ n W(`)h(`,t ′−1)(x′) ) + Σ ( σ`u√ n U(`)h(`−1,t)(x), σ`u√ n U(`)h(`−1,t ′)(x′) ) + Σin ( σ`bb (`), σ`bb (`) )\n= (σ`w) 2Vφ\n[ K(`,t,t ′)(x,x′) ]\n+ (σ`u) 2Vφ\n[ K(`−1,t+1,t ′+1)(x,x′) ]\n+ (σ`b) 2,\nwhere\nK(`,t,t ′)(x,x′) =\n[ Σ(`,t−1,t−1)(x,x) Σ(`,t−1,t ′−1)(x,x′)\nΣ(`,t−1,t ′−1)(x,x′) Σ(`,t ′−1,t′−1)(x′,x′)\n] ,\nand Σin is defined in (B.2). For the first first time step we have:\nΣ(`,1,1)(x,x′) = (σ`w) 2σ2h1(x=x′) + (σ ` u) 2Vφ [ K(`,2,2)(x,x′) ] + (σ`b) 2 ,\nΣ(`,t,1)(x,x′) = (σ`u) 2Vφ\n[ K(`,t+1,2)(x,x′) ] + (σ`b) 2 ,\nΣ(`,1,t ′)(x,x′) = (σ`u) 2Vφ [ K(`,2,t ′+1)(x,x′) ] + (σ`b) 2 .\nAnd the output layer\nK(x,x′) = σ2vVφ [ K(L,T+1,T ′+1)(x,x′) ] .\nNote that because of using new weights at each layer we get\nΣ(g(`,t)(x), g(` ′,t′))(x)) = 0 ` 6= `′\nNow we calculate the backward pass kernels in multi-layer RNTK. The gradients at the last layer is calculated via\nδ(L,T )(x) = σvv φ′(g(L,T )(x)). In the last hidden layer for different time steps we have\nδ(L,t)(x) = σLw√ n\n( W(L) )> ( φ′(g(L,t)(x)) δ(L,t+1)(x) ) t ∈ [T − 1]\nIn the last time step for different hidden layers we have\nδ(`,T )(x) = σ`+1u√ n\n( U(`+1) )> ( φ′(g(`,T )(x)) δ(`+1,T )(x) ) ` ∈ [L− 1]\nAt the end for the other layers we have\nδ(`,t)(x) = σ`w√ n\n( W(`) )> ( φ′(g(`,t)(x)) δ(`,t+1)(x) ) + σ`+1u√ n ( U(`+1) )> ( φ′(g(`,t)(x)) δ(`+1,t)(x) ) ` ∈ [L− 1], t ∈ [T − 1]\nThe recursive formula for the Π(L,t,t ′)(x,x′) is the same as the single layer, and it is non-zero for t′ − t = T ′ − T = τ . As a result we have Π(L,T,T+τ)(x,x′) = σ2vVφ′ [ K(L,T+1,T+τ+1) ] (x,x′)\nΠ(L,t,t+τ)(x,x′) = (σLw) 2Vφ′\n[ K(L,t+1,t+τ+1) ] (x,x′) · Π(L,t+1,t+1+τ)(x,x′) t ∈ [T − 1]\nΠ(L,t,t ′)(x,x′) = 0 t′ − t 6= τ\n(33) Similarly by using the same course of arguments used in the single layer setting, for the last time step we have Π(`,T,T+τ)(x,x′) = (σ`+1u ) 2Vφ′ [ K(`,T+1,T+τ+1) ] (x,x′) · Π(`+1,T+1,T+τ+1)(x,x′) ` ∈ [L− 1]\nFor the other layers we have\nΠ(`,t,t ′)(x,x′) = (σ`w) 2Vφ′ [ K(`,t+1,t+τ+1) ] (x,x′) · Π(`,t+1,t+1+τ)(x,x′)\n+ (σ`+1u ) 2Vφ′\n[ K(`,t+1,t+τ+1) ] (x,x′) · Π(`+1,t+1,t ′+1)(x,x′).\nFor t′ − t 6= τ the recursion continues until it reaches Π(L,T,t′′)(x,x′), t′′ < T ′ or Π(L,t ′′,T ′)(x,x′), t′′ < T and as a result based on (33) we get\nΠ(`,t,t ′)(x,x′) = 0 t′ − t 6= τ (34)\nFor t′ − t = τ it leads to Π(L,T,T ′)(x,x′) and has a non-zero value. Now we derive RNTK for multi-layer:\n〈∇θfθ(x),∇θfθ(x′)〉 = L∑ `=1 〈 ∂fθ(x) ∂W(`) , ∂fθ(x ′) ∂W(`) 〉 + L∑ `=1 〈 ∂fθ(x) ∂U(`) , ∂fθ(x ′) ∂U(`) 〉\n+ L∑ `=1 〈 ∂fθ(x) ∂b(`) , ∂fθ(x ′) ∂b(`) 〉 + 〈 ∂fθ(x) ∂v , ∂fθ(x ′) ∂v 〉 ,\nwhere〈 ∂fθ(x)\n∂W(`) , ∂fθ(x\n′)\n∂W(`)\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉) · ( (σ`w) 2 n 〈 h(`,t−1)(x),h(`,t ′−1)(x′) 〉)\n〈 ∂fθ(x)\n∂U(`) , ∂fθ(x\n′)\n∂U(`)\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉) · ( (σ`u) 2 m 〈xt,x′t′〉 ) ` = 1\n〈 ∂fθ(x)\n∂U(`) , ∂fθ(x\n′)\n∂U(`)\n〉 = T ′∑ t′=1 T∑ t=1 [( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉)\n· ( (σ`u) 2\nn\n〈 h(`−1,t)(x),h(`−1,t ′)(x′) 〉)]\n` > 1\n〈 ∂fθ(x)\n∂b(`) , ∂fθ(x\n′)\n∂b(`)\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉)\n· (σ`b)2〈 ∂fθ(x)\n∂v , ∂fθ(x\n′)\n∂v\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉)\nSumming up all the terms and replacing the inner product of vectors with their expectations we get\n〈∇θfθ(x),∇θfθ(x′)〉 = Θ(L,T,T ′) = L∑ `=1 T∑ t=1 T ′∑ t′=1 Π(`,t,t ′)(x,x′) · Σ(`,t,t ′)(x,x′) +K(x,x′). By (34), we can simplify to\nΘ(L,T,T ′) = ( L∑ `=1 T∑ t=1 Π(`,t,t ′)(x,x′) · Σ(`,t,t+τ)(x,x′) ) +K(x,x′).\nFor multi-dimensional output it becomes\nΘ(x,x′) = Θ(L,T,T ′)(x,x′)⊗ Id.\nThis concludes the proof for Theorem 1 for the multi-layer case."},{"heading":"B.4 PROOF FOR THEOREM 3: WEIGHT-UNTIED RNTK","text":"The architecture of a weight-untied single layer RNN is\ng(t)(x) = σw√ m W(t)h(t−1)(x) + σu√ n U(t)xt + σbb (t)\nh(t)(x) = φ ( g(t)(x) ) fθ(x) =\nσv√ n v>h(T )(x)\nWhere we use new weights at each time step and we index it by time. Like previous sections, we first derive the forward pass kernels for two same length data x = {xt}Tt=1,x = {x′t′}Tt′=1\nΣ(t,t)(x,x′) = σ2wVφ [ K(t,t)(x,x′) ] + σ2u m 〈xt,x′t〉+ σ2b .\nΣ(t,t ′)(x,x′) = 0 t 6= t′\nSince we are using same weight at the same time step, Σ(t,t)(x,x′) can be written as a function of the previous kernel, which is exactly as the weight-tied RNN. However for different length, it becomes zero as a consequence of using different weights, unlike weight-tied which has non-zero value. The kernel of the first time step and output is also the same as weight-tied RNN. For the gradients we have:\nδ(T )(x) = σvv φ′(g(T )(x))\nδ(t)(x) = σw√ n\n(W(t+1))> ( φ′(g(t)(x)) δ(t+1)(x) ) t ∈ [T − 1]\nFor t′ = t we have: Π(t,t)(x,x′) = σ2wVφ′ [ K(t+1,t+1)(x,x′) ] Π(t+1,t+1)(x,x′)\nΠ(t,t)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] .\nDue to using different weights for t 6= t′, we can immediately conclude that Π(t,t′)(x,x′) = 0. This set of calculation is exactly the same as the weight-tied case when τ = T − T = 0. Finally, with θ = Vect [ {{W(t),U(t),b(t)}Tt=1,v} ] we have\n〈∇θfθ(x),∇θfθ(x′)〉 = T∑ t=1 〈 ∂fθ(x) ∂W(t) , ∂fθ(x ′) ∂W(t) 〉 + T∑ t=1 〈 ∂fθ(x) ∂U(t) , ∂fθ(x ′) ∂U(t) 〉\n+ T∑ t=1 〈 ∂fθ(x) ∂b(t) , ∂fθ(x ′) ∂b(t) 〉 + 〈 ∂fθ(x) ∂v , ∂fθ(x ′) ∂v , 〉\nwith 〈 ∂fθ(x)\n∂W(t) , ∂fθ(x\n′)\n∂W(t)\n〉 = ( 1\nn\n〈 δ(t)(x), δ(t)(x′) 〉) · ( σ2w n 〈 h(t−1)(x),h(t−1)(x′) 〉) 〈 ∂fθ(x)\n∂U(t) , ∂fθ(x\n′)\n∂U(t)\n〉 = ( 1\nn\n〈 δ(t)(x), δ(t)(x′) 〉) · ( σ2u m 〈xt,x′t〉 ) 〈 ∂fθ(x)\n∂b(t) , ∂fθ(x\n′)\n∂b(t)\n〉 = ( 1\nn\n〈 δ(t)(x), δ(t)(x′) 〉) · σ2b〈\n∂fθ(x) ∂v , ∂fθ(x\n′)\n∂v\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉) .\nAs a result we obtain\n〈∇θfθ(x),∇θfθ(x′)〉 = ( T∑ t=1 Π(t,t)(x,x′) · Σ(t,t)(x′,x′) ) +K(x,x′),\nsame as the weight-tied RNN when τ = 0. This concludes the proof for Theorem 3.\nB.5 ANALYTICAL FORMULA FOR Vφ[K] For any positive definite matrixK = [ K1 K3 K3 K2 ] we have:\n• φ = ReLU (Cho & Saul, 2009)\nVφ[K] = 1\n2π\n( c(π − arccos(c)) + √ 1− c2) )√ K1K2,\nVφ′ [K] = 1\n2π (π − arccos(c)).\nwhere c = K3/ √ K1K2\n• φ = erf (Neal, 1995)\nVφ[K] = 2\nπ arcsin\n( 2K3√\n(1 + 2K1)(1 + 2K3)\n) ,\nVφ′ [K] = 4 π √ (1 + 2K1)(1 + 2K2)− 4K23 ."},{"heading":"C PROOF FOR THEOREM 2: RNTK CONVERGENCE AFTER TRAINING","text":"To prove theorem 2, we use the strategy used in (Lee et al., 2019) which relies on the the local lipschitzness of the network Jacobian J(θ,X ) = ∇θfθ(x) ∈ R|X |d×|θ| at initialization.\nDefinition 1 The Jacobian of a neural network is local lipschitz at NTK initialization (θ0 ∼ N (0, 1)) if there is constant K > 0 for every C such that{\n‖J(θ,X )‖F < K ‖J(θ,X )− J(θ̃,X )‖F < K‖θ − θ̃‖ , ∀ θ, θ̃ ∈ B(θ0, R)\nwhere B(θ,R) := {θ : ‖θ0 − θ‖ < R}.\nTheorem 4 Assume that the network Jacobian is local lipschitz with high probability and the empirical NTK of the network converges in probability at initialization and it is positive definite over the input set. For > 0, there exists N such that for n > N when applying gradient flow with η < 2 (λmin(Θ(X ,X )) + λmax(Θ(X ,X ))−1 with probability at least (1− ) we have:\nsup s ‖θs − θ0‖2√ n , sup s ‖Θ̂s(X ,X )− Θ̂0(X ,X )‖ = O ( 1√ n ) .\nProof: See (Lee et al., 2019)\nTheorem 4 holds for any network architecture and any cost function and it was used in (Lee et al., 2019) to show the stability of NTK for MLP during training.\nHere we extend the results for RNTK by proving that the Jacobian of a multi-layer RNN under NTK initialization is local lipschitz with high probability.\nTo prove it, first, we prove that for any two points θ, θ̃ ∈ B(θ0, R) there exists constant K1 such that\n‖g(`,t)(x)‖2, ‖δ(`,t)(x)‖2 ≤ K1 √ n (35)\n‖g(`,t)(x)− g̃(`,t)(x)‖2, ‖δ(`,t)(x)− δ̃(`,t)(x)‖2 ≤ ‖θ̄ − θ̃‖ ≤ K1 √ n‖θ − θ̃‖. (36)\nTo prove (35) and (36) we use the following lemmas.3\nLemma 1 Let A ∈ Rn×m be a random matrix whose entries are independent standard normal random variables. Then for every t ≥ 0, with probability at least 1− e(−ct2) for some constant c we have:\n‖A‖2 ≤ √ m+ √ n+ t.\nLemma 2 Let a ∈ Rn be a random vector whose entries are independent standard normal random variables. Then for every t ≥ 0, with probability at least 1− e(−ct2) for some constant c we have:\n‖a‖2 ≤ √ n+ √ t.\nSetting t = √ n for any θ ∈ R(θ0, R). With high probability, we get:\n‖W(`)‖2, ‖U(`)‖2 ≤ 3 √ n, ‖b`‖2 ≤ 2 √ n, ‖h(`,0)(x)‖2 ≤ 2σh √ n.\nWe also assume that there exists some finite constant C such that\n|φ(x)| < C|x|, |φ(x)− φ(x′)| < C|x− x′|, |φ′(x)| < C, , |φ′(x)− φ′(x′)| < C|x− x′|.\nThe proof is obtained by induction. From now on assume that all inequalities in (35) and (36) holds with some k for the previous layers. We have\n‖g(`,t)(x)‖2 = ‖ σ`w√ n W(`)h(`,t−1)(x) + σ`u√ n U(`) h(`−1,t)(x) + σ`bb (`)‖2\n≤ σ ` w√ n ‖W(`)‖2‖φ\n( g(`,t−1)(x) ) ‖2 +\nσ`u√ n ‖U(`)‖2‖φ\n( g(`−1,t)(x) ) ‖2 + σ`b‖b(`)‖2\n≤ ( 3σ`wCk + 3σ ` uCk + 2σb )√ n.\nAnd the proof for (35) and (36) is completed by showing that the first layer is bounded\n‖g(1,1)(x)‖2 = ‖ σ1w√ n W(`)h(1,0)(x) + σ1u√ m U(`) x1 + σ 1 bb (1)‖2\n≤ (3σ1wσh + 3σu√ m ‖x1‖2 + 2σb) √ n.\nFor the gradient of first layer we have\n‖δ(L,T )(x)‖2 = ‖σvv φ′(g(L,T )(x))‖2 ≤ σv‖v‖2‖φ′(g(L,T )(x))‖∞ = 2σvC √ n.\nAnd similarly we have\n‖δ(`,t)(x)‖ ≤ (3σwCk′ + 3σuCk′) √ n.\n3See math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf for proofs\nFor θ, θ̃ ∈ B(θ0, R) we have\n‖g(1,1)(x)− g̃(1,1)(x)‖2 = ‖ σ1w√ n (W(1) − W̃(1))h(1,0)(x) + σ 1 u√ m (U(1) − Ũ(1))h(1,0)(x)‖2\n≤ (\n3σ1wσh + 3σ1u m ‖x1‖2\n) ‖θ − θ̃‖2 √ n.\n‖g(`,t)(x)− g̃(`,t)(x)‖2 ≤ ‖φ(g(`,t−1)(x))‖2‖ σ`w√ n (W(`) − W̃(`))‖2\n+ ‖ σ ` w√ n W̃(`)‖2‖φ(g(`,t−1)(x))− φ(g̃(`,t−1)(x))‖2\n+ ‖φ(g(`−1,t)(x))‖2‖ σ`u√ n (U(`) − Ũ(`))‖2\n+ ‖ σ ` u√ n Ũ(`)‖2‖φ(g(`−1,t)(x))− φ(g̃(`−1,t)(x))‖2 + σb‖b(`) − b̃(`)‖ ≤ (kσ`w + 3σ`wCk + kσ`u + 3σ`uCk + σb)‖θ − θ̃‖2 √ n.\nFor gradients we have\n‖δ(L,T )(x)− δ̃(L,T )(x)‖2 ≤ σv‖φ′(g(L,T ))‖∞‖(v − ṽ)‖2 + σv‖v‖2‖φ′(g(L,T )(x))− φ′(g(L,T )(x))‖2 ≤ (σvC + 2σvCk)‖θ − θ̃‖2 √ n.\nAnd similarly using same techniques we have\n‖δ(`,t)(x)− δ̃(`,t)(x)‖2 ≤ (σwC + 3σwCk + σuC + 3σuCk)‖θ − θ̃‖2 √ n.\nAs a result, there exists K1 that is a function of σw, σu, σb, L, T and the norm of the inputs.\nNow we prove the local Lipchitzness of the Jacobian\n‖J(θ,x)‖F ≤ L∑ `=2 T∑ t=1 ( 1 n ∥∥∥∥δ(`,t)(x)(σ`wh(`,t−1)(x))>∥∥∥∥ F\n+ 1\nn ∥∥∥∥δ(`,t)(x)(σ`uh(`,t−1)(x))>∥∥∥∥ F + 1√ n ∥∥∥δ(`,t)(x) · σ`b∥∥∥ F ) +\nT∑ t=1 ( 1 n ∥∥∥∥δ(1,t)(x)(σ1wh(1,t−1)(x))>∥∥∥∥ F\n+ 1√ nm ∥∥∥δ(1,t−1)(x) (σ1uxt)>∥∥∥ F + 1√ n ∥∥∥δ(1,t)(x) · σ1b∥∥∥ F ) + σv√ n ‖h(L,T )(x)‖F\n≤ ( L∑ `=2 T∑ t=1 (K21Cσ ` w +K 2 1Cσ ` u + σ ` bK1)\n+ T∑ t=1 (K21Cσ 1 w + K1σ 1 u√ m ‖xt‖2 + σ1bK1) + σvCK1 ) .\nAnd for θ, θ̃ ∈ B(θ0, R) we have ‖J(θ,x)− J̃(θ,x)‖F ≤ L∑ `=2 T∑ t=1 ( 1 n ∥∥∥∥δ(`,t)(x)(σ`wh(`,t−1)(x))> − δ̃(`,t)(x)(σ`wh̃(`,t−1)(x))>∥∥∥∥ F\n+ 1\nn ∥∥∥∥δ(`,t)(x)(σ`uh(`,t−1)(x))> − δ̃(`,t)(x)(σ`uh̃(`,t−1)(x))>∥∥∥∥ F\n+ 1√ n ∥∥∥δ(`,t)(x) · σ`b − δ̃(`,t)(x) · σ`b∥∥∥ F\n+ T∑ t=1 ( 1 n ∥∥∥∥δ(1,t)(x)(σ1wh(1,t−1)(x))> − δ̃(1,t)(x)(σ1w ˜h(1,t−1)(x))>∥∥∥∥ F\n+ 1√ nm ∥∥∥δ(1,t−1)(x) (σ1uxt)> − δ̃(1,t−1)(x) (σ1uxt)>∥∥∥ F\n+ 1√ n ∥∥∥δ(`,t)(x) · σ`b − δ̃(`,t)(x) · σ`b∥∥∥)+ σv√n‖h(L,T )(x)− ˜h(L,T )(x)‖F ≤ ( L∑ `=2 T∑ t=1 (4K21Cσ ` w + 4K 2 1Cσ ` u + σ ` bK1)\n+ T∑ t=1 (4K21Cσ 1 w + K1σ 1 u√ m ‖xt‖2 + σ1bK1) + σvCK1 ) ‖θ − θ̃‖2.\nThe above proof can be generalized to the entire dataset by a straightforward application of the union bound. This concludes the proof for Theorem 2."}],"string":"[\n {\n \"heading\": \"1 INTRODUCTION\",\n \"text\": \"The overparameterization of modern deep neural networks (DNNs) has resulted in not only remarkably good generalization performance on unseen data (Novak et al., 2018; Neyshabur et al., 2019; Belkin et al., 2019) but also guarantees that gradient descent learning can find the global minimum of their highly nonconvex loss functions (Du et al., 2019b; Allen-Zhu et al., 2019b;a; Zou et al., 2018; Arora et al., 2019b). From these successes, a natural question arises: What happens when we take overparameterization to the limit by allowing the width of a DNN’s hidden layers to go to infinity? Surprisingly, the analysis of such an (impractical) DNN becomes analytically tractable. Indeed, recent work has shown that the training dynamics of (infinite-width) DNNs under gradient flow is captured by a constant kernel called the Neural Tangent Kernel (NTK) that evolves according to a linear ordinary differential equation (ODE) (Jacot et al., 2018; Lee et al., 2019; Arora et al., 2019a).\\nEvery DNN architecture and parameter initialization produces a distinct NTK. The original NTK was derived from the Multilayer Perceptron (MLP)(Jacot et al., 2018) and was soon followed by kernels derived from Convolutional Neural Networks (CNTK) (Arora et al., 2019a; Yang, 2019a), Residual DNNs (Huang et al., 2020), and Graph Convolutional Neural Networks (GNTK) (Du et al., 2019a). In (Yang, 2020a), a general strategy to obtain the NTK of any architecture is provided.\\nIn this paper, we extend the NTK concept to the important class of overparametrized Recurrent Neural Networks (RNNs), a fundamental DNN architecture for processing sequential data. We show that RNN in its infinite-width limit converges to a kernel that we dub the Recurrent Neural Tangent Kernel (RNTK). The RNTK provides high performance for various machine learning tasks, and an analysis of the properties of the kernel provides useful insights into the behavior of RNNs in the following overparametrized regime. In particular, we derive and study the RNTK to answer the following theoretical questions:\\nQ: Can the RNTK extract long-term dependencies between two data sequences? RNNs are known to underperform at learning long-term dependencies due to the gradient vanishing or exploding (Bengio et al., 1994). Attempted ameliorations have included orthogonal weights (Arjovsky et al., 2016; Jing et al., 2017; Henaff et al., 2016) and gating such as in Long Short-Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Unit (GRU) (Cho et al., 2014) RNNs. We demonstrate that the RNTK can detect long-term dependencies with proper initialization of the hyperparameters, and moreover, we show how the dependencies are extracted through time via different hyperparameter choices.\\nQ: Do the recurrent weights of the RNTK reduce its representation power compared to other NTKs? An attractive property of an RNN that is shared by the RNTK is that it can deal with sequences of different lengths via weight sharing through time. This enables the reduction of the number of learnable parameters and thus more stable training at the cost of reduced representation power. We prove the surprising fact that employing tied vs. untied weights in an RNN does not impact the analytical form of the RNTK.\\nQ: Does the RNTK generalize well? A recent study has revealed that the use of an SVM classifier with the NTK, CNTK, and GNTK kernels outperforms other classical kernel-based classifiers and trained finite DNNs on small data sets (typically fewer than 5000 training samples) (Lee et al., 2020; Arora et al., 2019a; 2020; Du et al., 2019a). We extend these results to RNTKs to demonstrate that the RNTK outperforms a variety of classic kernels, NTKs and finite RNNs for time series data sets in both classification and regression tasks. Carefully designed experiments with data of varying lengths demonstrate that the RNTK’s performance accelerates beyond other techniques as the difference in lengths increases. Those results extend the empirical observations from (Arora et al., 2019a; 2020; Du et al., 2019a; Lee et al., 2020) into finite DNNs, NTK, CNTK, and GNTK comparisons by observing that their performance-wise ranking depends on the employed DNN architecture.\\nWe summarize our contributions as follows:\\n[C1] We derive the analytical form for the RNTK of an overparametrized RNN at initialization using rectified linear unit (ReLU) and error function (erf) nonlinearities for arbitrary data lengths and number of layers (Section 3.1).\\n[C2] We prove that the RNTK remains constant during (overparametrized) RNN training and that the dynamics of training are simplified to a set of ordinary differential equations (ODEs) (Section 3.2).\\n[C3] When the input data sequences are of equal length, we show that the RNTKs of weight-tied and weight-untied RNNs converge to the same RNTK (Section 3.3).\\n[C4] Leveraging our analytical formulation of the RNTK, we empirically demonstrate how correlations between data at different times are weighted by the function learned by an RNN for different sets of hyperparameters. We also offer practical suggestions for choosing the RNN hyperparameters for deep information propagation through time (Section 3.4).\\n[C5] We demonstrate that the RNTK is eminently practical by showing its superiority over classical kernels, NTKs, and finite RNNs in exhaustive experiments on time-series classification and regression with both synthetic and 56 real-world data sets (Section 4).\"\n },\n {\n \"heading\": \"2 BACKGROUND AND RELATED WORK\",\n \"text\": \"Notation. We denote [n] = {1, . . . , n}, and Id the identity matrix of size d. [A]i,j represents the (i, j)-th entry of a matrix, and similarly [a]i represents the i-th entry of a vector. We use φ(·) : R → R to represent the activation function that acts coordinate wise on a vector and φ′ to denote its derivative. We will often use the rectified linear unit (ReLU) φ(x) = max(0, x) and error function (erf) φ(x) = 2√\\nπ ∫ x 0 e−z 2\\ndz activation functions. N (µ,Σ) represents the multidimensional Gaussian distribution with the mean vector µ and the covariance matrix Σ.\\nRecurrent Neural Networks (RNNs). Given an input sequence data x = {xt}Tt=1 of length T with data at time t, xt ∈ Rm, a simple RNN (Elman, 1990) performs the following recursive computation at each layer ` and each time step t\\ng(`,t)(x) = W (`)h(`,t−1)(x) +U (`)h(`−1,t)(x) + b(`), h(`,t)(x) = φ ( g(`,t)(x) ) ,\\nwhere W (`) ∈ Rn×n, b(`) ∈ Rn for ` ∈ [L], U (1) ∈ Rn×m and U (`) ∈ Rn×n for ` ≥ 2 are the RNN parameters. g(`,t)(x) is the pre-activation vector at layer ` and time step t, and h(`,t)(x) is the after-activation (hidden state). For the input layer ` = 0, we define h(0,t)(x) := xt. h(`,0)(x) as the initial hidden state at layer ` that must be initialized to start the RNN recursive computation.\\nThe output of an L-hidden layer RNN with linear read out layer is achieved via\\nfθ(x) = V h (L,T )(x),\\nwhere V ∈ Rd×n. Figure 1 visualizes an RNN unrolled through time.\\nh(2,2)h (2,1)(x) h(2,3)(x)\\nh(1,2)(x)h(1,1)(x) h(1,3)(x)\\nW (2) W (2)\\nW (1) W (1) U (2) U (2) U (2)\\nh(2,0)(x)\\nh(1,0)(x)\\nW (2)\\nW (1)\\nx1 x2 x3\\nU (1) U (1) U (1)\\nNeural Tangent Kernel (NTK). Let fθ(x) ∈ Rd be the output of a DNN with parameters θ. For two input data sequences x and x′, the NTK is defined as (Jacot et al., 2018)\\nΘ̂s(x,x ′) = 〈∇θsfθs(x),∇θsfθs(x′)〉,\\nwhere fθs and θs are the network output and parameters during training at time s. 1 Let X and Y be the set of training inputs and targets, `(ŷ, y) : Rd × Rd → R+ be the loss function, and L = 1|X | ∑ (x,y)∈X×Y `(fθs(x),y) be the the empirical loss. The evolution of the parameters θs and output of the network fθs on a test input using gradient descent with infinitesimal step size (a.k.a gradient flow) with learning rate η is given by\\n∂θs ∂s = −η∇θsfθs(X )T∇fθs (X )L (1)\\n∂fθs(x)\\n∂s = −η∇θsfθs(x)∇θsfθs(X )T∇fθs (X )L = −ηΘ̂s(x,X )∇fθs (X )L. (2)\\nGenerally, Θ̂s(x,x′), hereafter referred to as the empirical NTK, changes over time during training, making the analysis of the training dynamics difficult. When fθs corresponds to an infinite-width MLP, (Jacot et al., 2018) showed that Θ̂s(x,x′) converges to a limiting kernel at initialization and stays constant during training, i.e.,\\nlim n→∞\\nΘ̂s(x,x ′) = lim\\nn→∞ Θ̂0(x,x\\n′) := Θ(x,x′) ∀s ,\\nwhich is equivalent to replacing the outputs of the DNN by their first-order Taylor expansion in the parameter space (Lee et al., 2019). With a mean-square error (MSE) loss function, the training dynamics in (1) and (2) simplify to a set of linear ODEs, which coincides with the training dynamics of kernel ridge regression with respect to the NTK when the ridge term goes to zero. A nonzero ridge regularization can be conjured up by adding a regularization term λ 2\\n2 ‖θs − θ0‖ 2 2 to the empirical loss\\n(Hu et al., 2020).\"\n },\n {\n \"heading\": \"3 THE RECURRENT NEURAL TANGENT KERNEL\",\n \"text\": \"We are now ready to derive the RNTK. We first prove the convergence of an RNN at initialization to the RNTK in the infinite-width limit and discuss various insights it provides. We then derive the convergence of an RNN after training to the RNTK. Finally, we analyze the effects of various hyperparameter choices on the RNTK. Proofs of all of our results are provided in the Appendices.\"\n },\n {\n \"heading\": \"3.1 RNTK FOR AN INFINITE-WIDTH RNN AT INITIALIZATION\",\n \"text\": \"First we specify the following parameter initialization scheme that follows previous work on NTKs (Jacot et al., 2018), which is crucial to our convergence results:\\nW (`) = σ`w√ n W(`), U (1) = σ1u√ m U(1), U (`) = σ`u√ n U(`)(`≥2), V = σv√ n V, b(`) =σbb (`) , (3)\\nwhere\\n[W`]i,j , [U (`)]i,j , [V]i,j , [b (`)]i ∼ N (0, 1) . (4)\\nWe will refer to (3) and (4) as the NTK initialization. The choices of the hyperparameters σw, σu, σv and σb can significantly impact RNN performance, and we discuss them in detail in Section\\n1We use s to denote time here, since t is used to index the time steps of the RNN inputs.\\n3.4. For the initial (at time t = 0) hidden state at each layer `, we set h(`,0)(x) to an i.i.d. copy of N (0, σh) (Wang et al., 2018) . For convenience, we collect all of the learnable parameters of the RNN into θ = vect [ {{W(`),U(`),b(`)}L`=1,V} ] .\\nThe derivation of the RNTK at initialization is based on the correspondence between Gaussian initialized, infinite-width DNNs and Gaussian Processes (GPs), known as the DNN-GP. In this setting every coordinate of the DNN output tends to a GP as the number of units/neurons in the hidden layer (its width) goes to infinity. The corresponding DNN-GP kernel is computed as\\nK(x,x′) = E θ∼N\\n[ [fθ(x)]i · [fθ(x′)]i ] , ∀i ∈ [d]. (5)\\nFirst introduced for a single-layer, fully-connected neural network by (Neal, 1995), recent works on NTKs have extended the results for various DNN architectures (Lee et al., 2018; Duvenaud et al., 2014; Novak et al., 2019; Garriga-Alonso et al., 2019; Yang, 2019b), where in addition to the output, all pre-activation layers of the DNN tends to a GPs in the infinite-width limit. In the case of RNNs, each coordinate of the RNN pre-activation g(`,t)(x) converges to a centered GP depending on the inputs with kernel\\nΣ(`,t,t ′)(x,x′) = E\\nθ∼N\\n[ [g(`,t)(x)]i · [g(`,t ′)(x′)]i ] ∀i ∈ [n]. (6)\\nAs per (Yang, 2019a), the gradients of random infinite-width DNNs computed during backpropagation are also Gaussian distributed. In the case of RNNs, every coordinate of the vector δ(`,t)(x) :=√ n ( ∇g(`,t)(x)fθ(x) ) converges to a GP with kernel\\nΠ(`,t,t ′)(x,x′) = E\\nθ∼N\\n[ [δ(`,t)(x)]i · [δ(`,t ′)(x′)]i ] ∀i ∈ [n]. (7)\\nBoth convergences occur independently of the coordinate index i and for inputs of possibly different lengths, i.e., T 6= T ′. With (6) and (7), we now prove that an infinite-width RNN at initialization converges to the limiting RNTK.\\nTheorem 1 Let x and x′ be two data sequences of potentially different lengths T and T ′, respectively. Without loss of generality, assume that T ≤ T ′, and let τ := T ′ − T . Let n be the number of units in the hidden layers, the empirical RNTK for an L-layer RNN with NTK initialization converges to the following limiting kernel as n→∞\\nlim n→∞\\nΘ̂0(x,x ′) = Θ(x,x′) = Θ(L,T,T ′)(x,x′)⊗ Id , (8)\\nwhere\\nΘ(L,T,T ′)(x,x′) = ( L∑ `=1 T∑ t=1 ( Π(`,t,t+τ)(x,x′) · Σ(`,t,t+τ)(x,x′) )) +K(x,x′) , (9)\\nwith K(x,x′), Σ(`,t,t+τ)(x,x′), and Π(`,t,t+τ)(x,x′) defined in (5)–(7).\\nRemarks. Theorem 1 holds generally for any two data sequences, including different lengths ones. This highlights the RNTK’s ability to produce a similarity measure Θ(x,x′) even if the inputs are of different lengths, without resorting to heuristics such as zero padding the inputs to the to the max length of both sequences. Dealing with data of different length is in sharp contrast to common kernels such as the classical radial basis functions, polynomial kernels, and current NTKs. We showcase this capability below in Section 4.\\nTo visualize Theorem 1, we plot in the left plot in Figure 2 the convergence of a single layer, sufficiently wide RNN to its RNTK with the two simple inputs x = {1,−1, 1} of length 3 and x′ = {cos(α), sin(α)} of length 2, where α = [0, 2π]. For an RNN with a sufficiently large hidden state (n = 1000), we see clearly that it converges to the RNTK (n =∞). RNTK Example for a Single-Layer RNN. We present a concrete example of Theorem 1 by showing how to recursively compute the RNTK for a single-layer RNN; thus we drop the layer index for notational simplicity. We compute and display the RNTK for the general case of a multilayer RNN in Appendix B.3. To compute the RNTK Θ(T,T ′)(x,x′), we need to compute the GP\\nkernels Σ(t,t+τ)(x,x′) and Π(t,t+τ)(x,x′). We first define the operator Vφ [ K ]\\nthat depends on the nonlinearity φ(·) and a positive semi-definite matrixK ∈ R2×2\\nVφ [ K ]\\n= E[φ(z1) · φ(z2)], (z1, z2) ∼ N (0,K) . (10) Following (Yang, 2019a), we obtain the analytical recursive formula for the GP kernel Σ(t,t+τ)(x,x′) for a single layer RNN as\\nΣ(1,1)(x,x′) = σ2wσ 2 h1(x=x′) + σ2u m 〈x1,x′1〉+ σ2b (11) Σ(t,1)(x,x′) = σ2u m 〈xt,x′1〉+ σ2b t > 1 (12)\\nΣ(1,t ′)(x,x′) = σ2u m 〈x1,x′t′〉+ σ2b t′ > 1 (13)\\nΣ(t,t ′)(x,x′) = σ2wVφ [ K(t,t ′)(x,x′) ]\\n+ σ2u m 〈xt,x′t′〉+ σ2b t, t′ > 1 (14)\\nK(x,x′) = σ2vVφ [ K(T+1,T ′+1)(x,x′) ] , (15)\\nwhere\\nK(t,t ′)(x,x′) =\\n[ Σ(t−1,t−1)(x,x) Σ(t−1,t ′−1)(x,x′)\\nΣ(t−1,t ′−1)(x,x′) Σ(t ′−1,t′−1)(x′,x′)\\n] . (16)\\nSimilarly, we obtain the analytical recursive formula for the GP kernel Π(t,t+τ)(x,x′) as\\nΠ(T,T ′)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] (17)\\nΠ(t,t+τ)(x,x′) = σ2wVφ′ [ K(t+1,t+τ+1)(x,x′) ] Π(t+1,t+1+τ)(x,x′) t ∈ [T − 1] (18)\\nΠ(t,t ′)(x,x′) = 0 t′ − t 6= τ. (19) For φ = ReLU and φ = erf , we provide analytical expressions for Vφ [ K ] and Vφ′ [ K ]\\nin Appendix B.5. These yield an explicit formula for the RNTK that enables fast and point-wise kernel evaluations. For other activation functions, one can apply the Monte Carlo approximation to obtain Vφ [ K ]\\nand Vφ′ [ K ] (Novak et al., 2019).\"\n },\n {\n \"heading\": \"3.2 RNTK FOR AN INFINITE-WIDTH RNN DURING TRAINING\",\n \"text\": \"We prove that an infinitely-wide RNN, not only at initialization but also during gradient descent training, converges to the limiting RNTK at initialization.\\nTheorem 2 Let n be the number of units of each RNN’s layer. Assume that Θ(X ,X ) is positive definite on X such that λmin(Θ(X ,X )) > 0. Let η∗ := 2 ( λmin(Θ(X ,X )) + λmax(Θ(X ,X ) ))−1 . For an L-layer RNN with NTK initialization as in (3), (4) trained under gradient flow (recall (1) and (2)) with η < η∗, we have with high probability\\nsup s ‖θs − θ0‖2√ n , sup s ‖Θ̂s(X ,X )− Θ̂0(X ,X )‖2 = O ( 1√ n ) .\\nRemarks. Theorem 2 states that the training dynamics of an RNN in the infinite-width limit as in (1), (2) are governed by the RNTK derived from the RNN at its initialization. Intuitively, this is due to the NTK initialization (3), (4) which positions the parameters near a local minima, thus minimizing the amount of update that needs to be applied to the weights to obtain the final parameters.\"\n },\n {\n \"heading\": \"3.3 RNTK FOR AN INFINITE-WIDTH RNN WITHOUT WEIGHT SHARING\",\n \"text\": \"We prove that, in the infinite-width limit, an RNN without weight sharing (untied weights), i.e., using independent new weights W(`,t), U(`,t) and b(`,t) at each time step t, converges to the same RNTK as an RNN with weight sharing (tied weights). First, recall that it is a common practice to use weight-tied RNNs, i.e., in layer `, the weights W(`), U(`) and b(`) are the same across all time steps t. This practice conserves memory and reduces the number of learnable parameters. We demonstrate that, when using untied-weights, the RNTK formula remains unchanged.\\nTheorem 3 For inputs of the same length, an RNN with untied weights converges to the same RNTK as an RNN with tied weights in the infinite-width (n→∞) regime.\\nRemarks. Theorem 3 implies that weight-tied and weight-untied RNNs have similar behaviors in the infinite-width limit. It also suggests that existing results on the simpler, weight-untied RNN setting may be applicable for the more general, weight-tied RNN. The plot on the right side of Figure 2 empirically demonstrates the convergence of both the weight-tied and weight-untied RNNs to the RNTK with increasing hidden layer size n; moreover, the convergence rates are similar.\"\n },\n {\n \"heading\": \"3.4 INSIGHTS INTO THE ROLES OF THE RNTK’S HYPERPARAMETERS\",\n \"text\": \"Our analytical form for the RNTK is fully determined by a small number of hyperparameters, which contains the various weight variances collected into S = {σw, σu, σb, σh} and the activation function.2 In standard supervised-learning settings, one often performs cross-validation to select the hyperparameters. However, since kernel methods become computationally intractable for large datasets, we seek a more computationally friendly alternative to cross-validation. Here we conduct a novel exploratory analysis that provides new insights into the impact of the RNTK hyperparameters on the RNTK output and suggests a simple method to select them a priori in a deliberate manner.\\nTo visualize the role of the RNTK hyperparameters, we introduce the sensitivity s(t) of the RNTK of two input sequences x and x′ with respect to the input xt at time t\\ns(t) = ‖∇xtΘ(x,x′)‖2 . (20)\\n2From (11) to (18) we emphasize that σv merely scales the RNTK and does not change its overall behavior.\\nHere, s(t) indicates how sensitive the RNTK is to the data at time t, i.e., xt, in presence of another data sequence x′. Intuitively, large/small s(t) indicates that the RNTK is relatively sensitive/insensitive to the input xt at time t.\\nThe sensitivity is crucial to understanding to which extent the RNTK prediction is impacted by the input at each time step. In the case where some time indices have a small sensitivity, then any input variation in those corresponding times will not alter the RNTK output and thus will produce a metric that is invariant to those changes. This situation can be beneficial or detrimental based on the task at hand. Ideally, and in the absence of prior knowledge on the data, one should aim to have a roughly constant sensitivity across time in order to treat all time steps equally in the RNTK input comparison.\\nFigure 3 plots the normalized sensitivity s(t)/maxt(s(t)) for two data sequences of the same length T = 100, with s(t) computed numerically for xt,x′t ∼ N (0, 1). We repeated the experiments 10000 times; the mean of the sensitivity is shown in Figure 3. Each of the plots shows the changes of parameters SReLU = { √ 2, 1, 0, 0} for φ = ReLU and Serf = {1, 0.01, 0.05, 0} for φ = erf .\\nFrom Figure 3 we first observe that both ReLU and erf show similar per time step sensitivity measure s(t) behavior around the hyperparameters SReLU and Serf . If one varies any of the weight variance parameters, the sensitivity exhibits a wide range of behavior, and in particular with erf . We observe that σw has a major influence on s(t). For ReLU, a small decrease/increase in σw can lead to over-sensitivity of the RNTK to data at the last/first times steps, whereas for erf , any changes in σw leads to over-sensitivity to the last time steps.\\nAnother notable observation is the importance of σh, which is usually set to zero for RNNs. (Wang et al., 2018) showed that a non-zero σh acts as a regularization that improves the performance of RNNs with the ReLU nonlinearity. From the sensitivity perspective, a non-zero σh results in reducing the importance of the first time steps of the input. We also see the same behavior in erf , but with stronger changes as σh increases. Hence whenever one aims at reinforcing the input pairwise comparisons, such parameters should be favored.\\nThis sensitivity analysis provides a practical tool for RNTK hyperparameter tuning. In the absence of knowledge about the data, hyperparameters should be chosen to produce the least time varying sensitivity. If given a priori knowledge, hyperparameters can be selected that direct the RNTK to the desired time-steps.\"\n },\n {\n \"heading\": \"4 EXPERIMENTS\",\n \"text\": \"We now empirically validate the performance of the RNTK compared to classic kernels, NTKs, and trained RNNs on both classification and regression tasks using a large number of time series data sets. Of particular interest is the capability of the RNTK to offer high performance even on inputs of different lengths.\\nTime Series Classification. The first set of experiments considers time series inputs of the same lengths from 56 datasets in the UCR time-series classification data repository (Dau et al., 2019). We restrict ourselves to selected data sets with fewer than 1000 training samples and fewer than 1000 time steps (T ) as kernel methods become rapidly intractable for larger datasets. We compare the RNTK with a variety of other kernels, including the Radial Basis Kernel (RBF), polynomial kernel, and NTK (Jacot et al., 2018), as well as finite RNNs with Gaussian, identity (Le et al., 2015) initialization, and GRU (Cho et al., 2014). We use φ = ReLU for both the RNTKs and NTKs. For each kernel, we train a C-SVM (Chang & Lin, 2011) classifier, and for each finite RNN we use gradient descent training. For model hyperparameter tuning, we use 10-fold cross-validation. Details on the data sets and experimental setup are available in Appendix A.1.\\nWe summarize the classification results over all 56 datasets in Table 1; detailed results on each data set is available in Appendix A.2. We see that the RNTK outperforms not only the classical kernels but also the NTK and trained RNNs in all metrics. The results demonstrate the ability of RNTK to provide increased performances compare to various other methods (kernels and RNNs). The superior performance of RNTK compared to other kernels, including NTK, can be explained by the internal recurrent mechanism present in RNTK, allowing time-series adapted sample comparison. In addition, RNTK also outperforms RNN and GRU. As the datasets we consider are relative small in size, finite RNNs and GRUs that typically require large amount of data to succeed do not perform well in our setting. An interesting future direction would be to compare RNTK to RNN/GRU on larger datasets.\\nTime Series Regression. We now validate the performance of the RNTK on time series inputs of different lengths on both synthetic data and real data. For both scenarios, the target is to predict the next time-step observation of the randomly extracted windows of different length using kernel ridge regression.\\nWe compare the RNTK to other kernels, the RBF and polynomial kernels and the NTK. We also compare our results with a data independent predictor that requires no training, that is simply to predict the next time step with previous time step (PTS).\\nFor the synthetic data experiment, we simulate 1000 samples of one period of a sinusoid and add white Gaussian noise with default σn = 0.05. From this fixed data, we extract training set size Ntrain = 20 segments of uniform random lengths in the range of [Tfixed, Tfixed + Tvar] with Tfixed = 10. We use standard kernel ridge regression for this task. The test set is comprised of Ntest = 5000 obtained from other randomly extracted segments, again of varying lengths. For the real data, we use 975 days of the Google stock value in the years 2014–2018. As in the simulated signal setup above, we extract Ntrain segments of different lengths from the first 700 days and test on the Ntest segments from days 701 to 975. Details of the experiment are available in Appendix A.2.\\nWe report the predicted signal-to-noise ratio (SNR) for both datasets in Figures 4a and 4c for various values of Tvar. We vary the noise level and training set size for fixed Tvar = 10 in Figures 4b and 4d. As we see from Figures 4a and 4c, the RNTK offers substantial performance gains compared to the other kernels, due to its ability to naturally deal with variable length inputs. Moreover, the performance gap increases with the amount of length variation of the inputs Tvar. Figure 4d demonstrates that, unlike the other methods, the RNTK maintains its performance even when the training set is small. Finally, Figure 4c demonstrates that the impact of noise in the data on the regression performance is roughly the same for all models but becomes more important for RNTK with a large σn; this might be attributed to the recurrent structure of the model allowing for a time propagation and amplification of the noise for very low SNR. These experiments demonstrate the distinctive advantages of the RNTK over classical kernels, and NTKs for input data sequences of varying lengths.\\nIn the case of PTS, we expect the predictor to outperform kernel methods when learning from the training samples is hard, due to noise in the data or small training size which can lead to over fitting. In Figure 4a RNTK and Polynomial kernels outperforms PTS for all values of Tvar, but for larger Tvar, NTK and RBF under perform PTS due to the increasing detrimental effect of zero padding.\\nFor the Google stock value, we see a superior performance of PTS with respect to all other kernel methods due to the nature of those data heavily relying on close past data. However, RNTK is able to reduce the effect of over-fitting, and provide the closest results to PTS among all kernel methods we employed, with increasing performance as the number of training samples increase.\"\n },\n {\n \"heading\": \"5 CONCLUSIONS\",\n \"text\": \"In this paper, we have derived the RNTK based on the architecture of a simple RNN. We have proved that, at initialization, after training, and without weight sharing, any simple RNN converges to the same RNTK. This convergence provides new insights into the behavior of infinite-width RNNs, including how they process different-length inputs, their training dynamics, and the sensitivity of their output at every time step to different nonlinearities and initializations. We have highlighted the RNTK’s practical utility by demonstrating its superior performance on time series classification and regression compared to a range of classical kernels, the NTK, and trained RNNs. There are many avenues for future research, including developing RNTKs for gated RNNs such as the LSTM (Hochreiter & Schmidhuber, 1997) and investigating which of our theoretical insights extend to finite RNNs.\"\n },\n {\n \"heading\": \"ACKNOWLEDGMENTS\",\n \"text\": \"This work was supported by NSF grants CCF-1911094, IIS-1838177, and IIS-1730574; ONR grants N00014-18-12571, N00014-20-1-2787, and N00014-20-1-2534; AFOSR grant FA9550-18-1-0478; and a Vannevar Bush Faculty Fellowship, ONR grant N00014-18-1-2047.\"\n },\n {\n \"heading\": \"A EXPERIMENT DETAILS\",\n \"text\": \"\"\n },\n {\n \"heading\": \"A.1 TIME SERIES CLASSIFICATION\",\n \"text\": \"Kernel methods settings. We used RNTK, RBF, polynomial and NTK (Jacot et al., 2018). For data pre-processing, we normalized the norm of each x to 1. For training we used C-SVM in LIBSVM library (Chang & Lin, 2011) and for hyperparameter selection we performed 10-fold validation for splitting the training data into 90% training set and 10% validation test. We then choose the best performing set of hyperparameters on all the validation sets, retrain the models with the best set of hyperparameters on the entire training data and finally report the performance on the unseen test data. The performance of all kernels on each data set is shown in table 2.\\nFor C-SVM we chose the cost function value\\nC ∈ {0.01, 0.1, 1, 10, 100}\\nand for each kernel we used the following hyperparameter sets\\n• RNTK: We only used single layer RNTK, we φ = ReLU and the following hyperparameter sets for the variances:\\nσw ∈ {1.34, 1.35, 1.36, 1.37, 1.38, 1.39, 1.40, 1.41, 1.42, √\\n2, 1.43, 1.44, 1.45, 1.46, 1.47} σu = 1\\nσb ∈ {0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.7, 0.9, 1, 2} σh ∈ {0, 0.01, 0.1, 0.5, 1}\\n• NTK: The formula for NTK of L-layer MLP (Jacot et al., 2018) for x,x′ ∈ Rm is:\\nΣ(1) = σ2w m 〈x,x′〉+ σ2b\\nΣ(`)(x,x′) = σ2wVφ[K (`)(x,x′)] + σ2b ` ∈ [L] Σ̇(`)(x,x′) = σ2wVφ′ [K (`+1)(x,x′)] ` ∈ [L]\\nK(`)(x,x′) = [ Σ(`−1)(x,x) Σ(`−1)(x,x′) Σ(`−1)(x,x′) Σ(`−1)(x′,x′) ] K(x,x′) = σ2vVφ[K(L+1)(x,x′)]\\nkNTK = L∑ `=1\\n( Σ(`)(x,x′)\\nL∏ `′=` Σ̇(`)(x,x′)\\n) +K(x,x′)\\nand we used the following hyperparamters\\nL ∈ [10] σw ∈ {0.5, 1, √ 2, 2, 2.5, 3}\\nσb ∈ {0, 0.01, 0.1, 0.2, 0.5, 0.8, 1, 2, 5}\\n• RBF:\\nkRBF(x,x ′) = e(−α‖x−x ′‖22)\\nα ∈ {0.01, 0.05, 0.1, 0.2, 0.5, 0.6, 0.7, 0.8, 1, 2, 3, 4, 5, 10, 20, 30, 40, 100}\\n• Polynomial:\\nkPolynomial(x,x ′) = (r + 〈x,x′〉)d\\nd ∈ [5] r ∈ {0, 0.1, 0.2, 0.5, 1, 2}\\nFinite-width RNN settings. We used 3 different RNNs. The first is a ReLU RNN with Gaussian initialization with the same NTK initialization scheme, where parameter variances are σw = σv = √ 2,\\nσu = 1 and σb = 0. The second is a ReLU RNN with identity initialization following (Le et al., 2015). The third is a GRU (Cho et al., 2014) with uniform initialization. All models are trained with RMSProp algorithm for 200 epochs. Early stopping is implemented when the validation set accuracy does not improve for 5 consecutive epochs.\\nWe perform standard 5-fold cross validation. For each RNN architecture we used hyperparamters of number of layer, number of hidden units and learning rate as\\nL ∈ {1, 2} n ∈ {50, 100, 200, 500}\\nη ∈ {0.01, 0.001, 0.0001, 0.00001}\\nMetrics descriptions First, only in this paragraph, let i ∈ {1, 2, ..., N} index a total of N datasets and j ∈ {1, 2, ...,M} index a total of M classifiers. Let yij be the accuracy of the j-th classifer on the i-th dataset. We reported results on 4 metrics: average accuracy (Acc. mean), P90, P95, PMA and Friedman Rank. P90 and P95 is the fraction of datasets that the classifier achieves at least 90% and 95% of the maximum achievable accuracy for each dataset, i.e.,\\nP90j = 1\\nN ∑ i 1(yij ≥ 0.9(max j yij)) . (21)\\nPMA is the accuracy of the classifier on a dataset divided by the maximum achievable accuracy on that dataset, averaged over all datasets:\\nPMAj = 1\\nN ∑ i yij max j yij . (22)\\nFriedman Rank (Fernández-Delgado et al., 2019) first ranks the accuracy of each classifier on each dataset and then takes the average of the ranks for each classifier over all datasets, i.e.,\\nFRj = 1\\nN ∑ i rij , (23)\\nwhere rij is the ranking of the j-th classifier on the i-th dataset.\\nNote that a better classifier achieves a lower Friedman Rank, Higher P/90 and PMA.\\nRemark. In order to provide insight into the performance of RNTK in long time steps setting, we picked two datasets with more that 1000 times steps: SemgHandSubjectCh2 (T = 1024) and StarLightCurves (T = 1024).\"\n },\n {\n \"heading\": \"A.2 TIME SERIES REGRESSION\",\n \"text\": \"For time series regression, we used the 5-fold validation of training set and same hyperparamter sets for all kernels. For training we kernel ridge regression with ridge term chosen form\\nλ ∈ {0, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.8, 1, 2, 3, 4, 5, 6, 7, 8, 10, 100}\"\n },\n {\n \"heading\": \"B PROOFS FOR THEOREMS 1 AND 3: RNTK CONVERGENCE AT INITIALIZATION\",\n \"text\": \"\"\n },\n {\n \"heading\": \"B.1 PRELIMINARY: NETSOR PROGRAMS\",\n \"text\": \"Calculation of NTK in any architecture relies on finding the GP kernels that correspond to each pre-activation and gradient layers at initialization. For feedforward neural networks with n1, . . . , nL number of neurons (channels in CNNs) at each layer the form of this GP kernels can be calculated via taking the limit of n1, . . . , nL sequentially one by one. The proof is given by induction, where by conditioning on the previous layers, each entry of the current layer is sum of infinite i.i.d Gaussian random variables, and based on Central Limit Theorem (CLT), it becomes a Gaussian process with\\nkernel calculated based on the previous layers. Since the first layer is an affine transformation of input with Gaussian weights, it is a Gaussian process and the proof is completed. See (Lee et al., 2018; Duvenaud et al., 2014; Novak et al., 2019; Garriga-Alonso et al., 2019) for a formal treatment. However, due to weight-sharing, sequential limit is not possible and condoning on previous layers does not result in i.i.d. weights. Hence the aforementioned arguments break. To deal with it, in (Yang, 2019a) a proof using Gaussian conditioning trick (Bolthausen, 2014) is presented which allows use of recurrent weights in a network. More precisely, it has been demonstrated than neural networks (without batch normalization) can be expressed and a series of matrix multiplication and (piece wise) nonlinearity application, generally referred as Netsor programs. It has been shown that any architecture that can be expressed as Netsor programs that converge to GPs as width goes to infinity in the same rate, which a general rule to obtain the GP kernels. For completeness of this paper, we briefly restate the results from (Yang, 2019a) which we will use later for calculation derivation of RNTK.\\nThere are 3 types of variables in Netsor programs; A-vars, G-vars and H-vars. A-vars are matrices and vectors with i.i.d Gaussian entries, G-vars are vectors introduced by multiplication of a vector by an A-var and H-vars are vectors after coordinate wise nonlinearities is applied to G-vars. Generally, G-vars can be thought of as pre-activation layers which are asymptotically treated as a Gaussian distributed vectors, H-vars as after-activation layers and A-vars are the weights. Since in neural networks inputs are immediately multiplied by a weight matrix, it can be thought of as an G-var, namely gin. Generally Netsor programs supports G-vars with different dimension, however the asymptotic behavior of a neural networks described by Netsor programs does not change under this degree of freedom, as long as they go to infinity at the same rate. For simplicity, let the G-vars and H-vars have the same dimension n since the network of interest is RNN and all pre-activation layers have the same dimension. We introduce the Netsor programs under this simplification. To produce the output of a neural network, Netsor programs receive a set of G-vars and A-vars as input, and new variables are produced sequentially using the three following operators:\\n• Matmul : multiplication of an A-var: A with an H-var: h, which produce a new G-var, g. g = Ah (24)\\n• Lincomp: Linear combination of G-vars, gi, 1 ≤ i ≤ k , with coefficients ai ∈ R 1 ≤ i ≤ k which produce of new G-var:\\ng = k∑ i=1 aigi (25)\\n• Nonlin: creating a new H-var, h, by using a nonlinear function φ : Rk → R that act coordinate wise on a set of G-vars, gi, 1 ≤ i ≤ k :\\nh = ϕ(g1, . . . , gk) (26)\\nAny output of the neural network y ∈ R should be expressed as inner product of a new A-var which has not been used anywhere else in previous computations and an H-var:\\ny = v>h\\nAny other output can be produced by another v′ and h′ (possibility the same h or v).\\nIt is assumed that each entry of any A-var : A ∈ Rn×n in the netsor programs computations is drawn from N (0, σ 2 a\\nn ) and the input G-vars are Gaussian distributed. The collection of a specific entry of all G-vars of in the netsor program converges in probability to a Gaussian vector {[g1]i, . . . , [gk]i} ∼ N (µ,Σ) for all i ∈ [n] as n goes to infinity. Let µ(g) := E [ [g]i ] be the mean of aG-var and Σ(g, g′) := E [ [g]i · [g′]i ] be the covariance between any two G-vars. The general rule for µ(g) is given by the following equations:\\nµ(g) = µin(g) if g is input k∑ i=1 aiµ(gi) if g = k∑ i=1 aigi\\n0 otherwise\\n(27)\\nFor g and g′, let G = {g1, . . . , gr} be the set of G-vars that has been introduced before g and g′ with distribution N (µG ,ΣG), where ΣG ∈ R|G|×|G| containing the pairwise covariances between the G-vars. Σ(g, g′) is calculated via the following rules:\\nΣ(g, g′) = Σin(g, g′) if g and g′ are inputs k∑ i=1 aiΣ(gi, g′) if g = k∑ i=1 aigi k∑ i=1 aiΣ(g, gi) if g′ = k∑ i=1 aigi σ2A E z∼N (µ,ΣG) [ϕ(z)ϕ̄(z)] if g = Ah and g′ = Ah′\\n0 otherwise\\n(28)\\nWhere h = ϕ(g1, . . . , gr) and h′ = ϕ̄(g1, . . . , gr) are functions of G-vars in G from possibly different nonlinearities. This set of rules presents a recursive method for calculating the GP kernels in a network where the recursive formula starts from data dependent quantities Σin and µin which are given.\\nAll the above results holds when the nonlinearities are bounded uniformly by e(cx 2−α) for some α > 0 and when their derivatives exist.\\nStandard vs. NTK initialization. The common practice (which netsor programs uses) is to initialize DNNs weights [A]i,j with N (0, σa√n ) (known as standard initialization) where generally n is the number of units in the previous layer. In this paper we have used a different parameterization scheme as used in (Jacot et al., 2018) and we factor the standard deviation as shown in 3 and initialize weights with standard standard Gaussian. This approach does not change the the forward computation of DNN, but normalizes the backward computation (when computing the gradients) by factor 1n , otherwise RNTK will be scales by n. However this problem can be solved by scaling the step size by 1 n and there is no difference between NTK and standard initialization (Lee et al., 2019).\"\n },\n {\n \"heading\": \"B.2 PROOF FOR THEOREM 1: SINGLE LAYER CASE\",\n \"text\": \"We first derive the RNTK in a simpler setting, i.e., a single layer and single output RNN. We then generalize the results to multi-layer and multi-output RNNs. We drop the layer index ` to simplify notation. From 3 and 4, the forward pass for computing the output under NTK initialization for each input x = {xt}Tt=1 is given by:\\ng(t)(x) = σw√ m Wh(t−1)(x) + σu√ n Uxt + σbb (29)\\nh(t)(x) = φ ( g(t)(x) ) (30)\\nfθ(x) = σv√ n v>h(T )(x) (31)\\nNote that (29), (30) and (31) use all the introduced operators introduced in 24, 25 and 26 given input variables W, {Uxt}Tt=1,b,v and h(0)(x).\\nFirst, we compute the kernels of forward pass Σ(t,t ′)(x,x′) and backward pass Π(t,t ′)(x,x′) introduced in (6) and (7) for two input x and x′. Note that based on (27) the mean of all variables is zero since the inputs are all zero mean. In the forward pass for the intermediate layers we have:\\nΣ(t,t ′)(x,x′) = Σ(g(t)(x), g(t ′)(x′))\\n= Σ ( σw√ n Wh(t−1)(x) + σu√ m Uxt + σbb, σw√ n Wh(t ′−1)(x′) + σu√ m Ux′t′ + σbb ) = Σ ( σw√ n Wh(t−1)(x), σw√ n Wh(t ′−1)(x′) ) + Σin ( σu√ m Uxt, σu√ m Ux′t′ ) + Σin (σbb, σbb) .\\nWe have used the second and third rule in (28) to expand the formula, We have also used the first and fifth rule to set the cross term to zero, i.e.,\\nΣ ( σw√ n Wh(t−1)(x), σu√ n Ux′t′ ) = 0\\nΣ ( σw√ n Wh(t−1)(x), σbb ) = 0\\nΣ ( σu√ m Uxt, σw√ n Wh(t ′−1)(x′) ) = 0\\nΣ ( σbb,\\nσw√ n Wh(t ′−1)(x′)\\n) = 0\\nΣin ( σu√ m Uxt, σbb ) = 0\\nΣin ( σbb,\\nσu√ m Ux′t′\\n) = 0.\\nFor the non-zero terms we have\\nΣin (σbb, σbb) = σ 2 b Σin ( σu√ m Uxt, σu√ m Ux′t′ ) = σ2u m 〈xt,x′t′〉,\\nwhich can be achieved by straight forward computation. If t 6= 1 and t′ 6= 1, by using the forth rule in (28) we have\\nΣ ( σw√ n Wh(t−1)(x), σw√ n Wh(t ′−1)(x′) ) = σ2w E z∼N (0,K(t,t′)(x,x′)) [φ(z1)φ(z2)] = Vφ [ K(t,t ′)(x,x′) ] .\\nWithK(t,t ′)(x,x′) defined in (16). Otherwise, it will be zero by the fifth rule (if t or t = 1) . Here the set of previously introduced G-vars is G ={ {g(α)(x)},Uxα}t−1α=1, {g(α ′)(x′),Ux′α′} t′−1 α′=1,h (0)(x),h(0)(x′) }\\n, but the dependency is only on the last layer G-vars, ϕ({g : g ∈ G}) = φ(g(t−1)(x)), ϕ̄(({g : g ∈ G})) = φ(g(t′−1)(x′)), leading the calculation to the operator defined in (10). As a result\\nΣ(t,t ′)(x,x′) = σ2wVφ [ K(t,t ′)(x,x′) ]\\n+ σ2u m 〈xt,x′t′〉+ σ2b .\\nTo complete the recursive formula, using the same procedure for the first layers we have\\nΣ(1,1)(x,x′) = σ2wσ 2 h1(x=x′) + σ2u m 〈x1,x′1〉+ σ2b ,\\nΣ(1,t ′)(x,x′) = σ2u m 〈x1,x′t′〉+ σ2b ,\\nΣ(t,1)(x,x′) = σ2u m 〈xt,x′1〉+ σ2b .\\nThe output GP kernel is calculated via K(x,x′) = σ2vVφ [ K(T+1,T ′+1)(x,x′) ] The calculation of the gradient vectors δ(t)(x) = √ n ( ∇g(t)(x)fθ(x) ) in the backward pass is given by\\nδ(T )(x) = σvv φ′(g(T )(x))\\nδ(t)(x) = σw√ n\\nW> ( φ′(g(t)(x)) δ(t+1)(x) ) t ∈ [T − 1]\\nTo calculate the backward pass kernels, we rely on the following Corollary from (Yang, 2020b)\\nCorollary 1 In infinitely wide neural networks weights used in calculation of back propagation gradients (W>) is an i.i.d copy of weights used in forward propagation (W) as long as the last layer weight (v) is sampled independently from other parameters and has mean 0.\\nThe immediate result of Corollary 1 is that g(t)(x) and δ(t)(x) are two independent Gaussian vector as their covariance is zero based on the fifth rule in (28). Using this result, we have:\\nΠ(t,t ′)(x,x′) = Σ ( δ(t)(x), δ(t ′)(x) )\\n= E [ [δ(t)(x)]i · [δ(t ′)(x′)]i ] = σ2wE [ [φ′(g(t)(x))]i · [δ(t+1)(x)]i · [φ′(g(t ′)(x′))]i · [δ(t ′+1)(x′)]i\\n] = σ2w E\\nz∼N (0,K(t+1,t+1′)(x,x′)) [φ′(z1) · φ′(z2)] · E\\n[ [δ(t+1)(x)]i · [δ(t ′+1)(x′)]i ] = σ2wVφ′ [ K(t+1,t ′+1)(x,x′) ] Π(t+1,t ′+1)(x,x′).\\nIf T ′ − t′ = T − t, then the the formula will lead to\\nΠ(T,T ′)(x,x′) = E [ [δ(T )(x)]i, [δ (T ′)(x′)]i ] = σ2vE [ [v]i · [φ′(g(T )(x))]i · [v]i · [φ′(g(T ′)(x′))]i\\n] = E [ [φ′(g(T )(x))]i · [φ′(g(T ′)(x′))]i ] · E [[v]i [v]i]\\n= σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] .\\nOtherwise it will end to either of two cases for some t′′ < T or T ′ and by the fifth rule in (28) we have:\\nΣ ( δ(t ′′)(x), δ(T ′)(x) ) = Σ ( σw√ n W> ( φ′(g(t ′′)(x)) δ(t ′′+1)(x′) ) ,v φ′(g(T ′)(x)) ) = 0\\nΣ ( δ(T )(x), δ(t ′′)(x) ) = Σ ( v φ′(g(T )(x)), σw√\\nn W>\\n( φ′(g(t ′′)(x′)) δ(t ′′+1)(x′) )) = 0.\\nWithout loss of generality, from now on assume T ′ < T and T ′ − T = τ , the final formula for computing the backward gradients becomes:\\nΠ(T,T+τ)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] Π(t,t+τ)(x,x′) = σ2wVφ′ [ K(t+1,t+τ+1)(x,x′) ] Π(t+1,t+1+τ)(x,x′) t ∈ [T − 1]\\nΠ(t,t ′)(x,x′) = 0 t′ − t 6= τ\\nNow we have derived the single layer RNTK. Recall that θ = Vect [ {W,U,b,v} ] contains all of the network’s learnable parameters. As a result, we have:\\n∇θfθ(x) = Vect [ {∂fθ(x) ∂W , ∂fθ(x) ∂U , ∂fθ(x) ∂b , ∂fθ(x) ∂v } ] .\\nAs a result 〈∇θfθ(x),∇θfθ(x′)〉 = 〈 ∂fθ(x)\\n∂W , ∂fθ(x\\n′)\\n∂W\\n〉 + 〈 ∂fθ(x)\\n∂U , ∂fθ(x\\n′)\\n∂U\\n〉 + 〈 ∂fθ(x)\\n∂b , ∂fθ(x\\n′)\\n∂b 〉 + 〈 ∂fθ(x)\\n∂v , ∂fθ(x\\n′)\\n∂v\\n〉\\nWhere the gradients of output with respect to weights can be formulated as the following compact form:\\n∂fθ(x)\\n∂W = T∑ t=1 ( 1√ n δ(t)(x) ) · ( σw√ n h(t−1)(x) )> ∂fθ(x)\\n∂U = T∑ t=1 ( 1√ n δ(t)(x) ) · ( σu√ m xt )> ∂fθ(x)\\n∂b = T∑ t=1 ( σb√ n δ(t)(x) ) ∂fθ(x)\\n∂v = σv√ n h(T )(x).\\nAs a result we have:〈 ∂fθ(x)\\n∂W , ∂fθ(x\\n′)\\n∂W\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉) · ( σ2w n 〈 h(t−1)(x),h(t ′−1)(x′) 〉)\\n〈 ∂fθ(x)\\n∂U , ∂fθ(x\\n′)\\n∂U\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉) · ( σ2u m 〈xt,x′t′〉 ) 〈 ∂fθ(x)\\n∂b , ∂fθ(x\\n′)\\n∂b\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(t)(x), δ(t ′)(x′) 〉)\\n· σ2b〈 ∂fθ(x)\\n∂v , ∂fθ(x\\n′)\\n∂v\\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉) .\\nRemember that for any two G-var E [[g]i[g′]i] is independent of index i. Therefore,\\n1\\nn\\n〈 h(t−1)(x),h(t ′−1)(x′) 〉 → Vφ [ K(t,t ′)(x,x′) ]\\nt > 1\\n1\\nn\\n〈 h(0)(x),h(0)(x′) 〉 → σ2h.\\nHence, by summing the above terms in the infinite-width limit we get\\n〈∇θfθ(x),∇θfθ(x′)〉 → T ′∑ t′=1 T∑ t=1 Π(t,t ′)(x,x′) · Σ(t,t ′)(x′,x′) +K(x,x′). (32) Since Π(t,t ′)(x,x′) = 0 for t′ − t 6= τ it is simplified to\\n〈∇θfθ(x),∇θfθ(x′)〉 = ( T∑ t=1 Π(t,t+τ)(x,x′) · Σ(t,t+τ)(x′,x′) ) +K(x,x′).\\nMulti-dimensional output. For fθ(x) ∈ Rd, the i-th output for i ∈ [d] is obtained via\\n[fθ(x)]i = σv√ n v>i h (T )(x),\\nwhere vi is independent of vj for i 6= j. As a result, for The RNTK Θ(x,x′) ∈ Rd×d for multidimensional output we have\\n[Θ(x,x′)]i,j = 〈 ∇θ [fθ(x)]i ,∇θ [fθ(x ′)]j 〉 For i = j, the kernel is the same as computed in (32) and we denote it as\\n〈∇θ [fθ(x)]i ,∇θ [fθ(x ′)]i〉 = Θ (T,T ′)(x,x′).\\nFor i 6= j, since vi is independent of vj , Π(T,T ′)(x,x′) and all the backward pass gradients become zero, so 〈 ∇θ [fθ(x)]i ,∇θ [fθ(x ′)]j 〉 = 0 i 6= j\\nwhich gives us the following formula\\nΘ(x,x′) = Θ(T,T ′)(x,x′)⊗ Id.\\nThis concludes the proof for Theorem 1 for single-layer case.\"\n },\n {\n \"heading\": \"B.3 PROOF FOR THEOREM 1: MULTI-LAYER CASE\",\n \"text\": \"Now we drive the RNTK for multi-layer RNTK. We will only study single output case and the generalization to multi-dimensional case is identical as the single layer case. The set of equations for calculation of the output of a L-layer RNN for x = {xt}Tt=1 are\\ng(`,t)(x) = σ`w√ n W(`)h(`,t−1)(x) + σ`u√ m U(`)xt + σ ` bb (`) ` = 1\\ng(`,t)(x) = σ`w√ n W(`)h(`,t−1)(x) + σ`u√ n U(`)h(`−1,t)(x) + σ`bb (`) ` > 1\\nh(`,t)(x) = φ ( g(`,t)(x) ) fθ(x) =\\nσv√ n v>h(L,T )(x)\\nThe forward pass kernels for the first layer is the same as calculated in B.2. For ` ≥ 2 we have:\\nΣ(`,t,t ′)(x,x′) = Σ(g(`,t)(x), g(`,t ′)(x′))\\n= Σ ( σ`w√ n W(`)h(`,t−1)(x), σ`w√ n W(`)h(`,t ′−1)(x′) ) + Σ ( σ`u√ n U(`)h(`−1,t)(x), σ`u√ n U(`)h(`−1,t ′)(x′) ) + Σin ( σ`bb (`), σ`bb (`) )\\n= (σ`w) 2Vφ\\n[ K(`,t,t ′)(x,x′) ]\\n+ (σ`u) 2Vφ\\n[ K(`−1,t+1,t ′+1)(x,x′) ]\\n+ (σ`b) 2,\\nwhere\\nK(`,t,t ′)(x,x′) =\\n[ Σ(`,t−1,t−1)(x,x) Σ(`,t−1,t ′−1)(x,x′)\\nΣ(`,t−1,t ′−1)(x,x′) Σ(`,t ′−1,t′−1)(x′,x′)\\n] ,\\nand Σin is defined in (B.2). For the first first time step we have:\\nΣ(`,1,1)(x,x′) = (σ`w) 2σ2h1(x=x′) + (σ ` u) 2Vφ [ K(`,2,2)(x,x′) ] + (σ`b) 2 ,\\nΣ(`,t,1)(x,x′) = (σ`u) 2Vφ\\n[ K(`,t+1,2)(x,x′) ] + (σ`b) 2 ,\\nΣ(`,1,t ′)(x,x′) = (σ`u) 2Vφ [ K(`,2,t ′+1)(x,x′) ] + (σ`b) 2 .\\nAnd the output layer\\nK(x,x′) = σ2vVφ [ K(L,T+1,T ′+1)(x,x′) ] .\\nNote that because of using new weights at each layer we get\\nΣ(g(`,t)(x), g(` ′,t′))(x)) = 0 ` 6= `′\\nNow we calculate the backward pass kernels in multi-layer RNTK. The gradients at the last layer is calculated via\\nδ(L,T )(x) = σvv φ′(g(L,T )(x)). In the last hidden layer for different time steps we have\\nδ(L,t)(x) = σLw√ n\\n( W(L) )> ( φ′(g(L,t)(x)) δ(L,t+1)(x) ) t ∈ [T − 1]\\nIn the last time step for different hidden layers we have\\nδ(`,T )(x) = σ`+1u√ n\\n( U(`+1) )> ( φ′(g(`,T )(x)) δ(`+1,T )(x) ) ` ∈ [L− 1]\\nAt the end for the other layers we have\\nδ(`,t)(x) = σ`w√ n\\n( W(`) )> ( φ′(g(`,t)(x)) δ(`,t+1)(x) ) + σ`+1u√ n ( U(`+1) )> ( φ′(g(`,t)(x)) δ(`+1,t)(x) ) ` ∈ [L− 1], t ∈ [T − 1]\\nThe recursive formula for the Π(L,t,t ′)(x,x′) is the same as the single layer, and it is non-zero for t′ − t = T ′ − T = τ . As a result we have Π(L,T,T+τ)(x,x′) = σ2vVφ′ [ K(L,T+1,T+τ+1) ] (x,x′)\\nΠ(L,t,t+τ)(x,x′) = (σLw) 2Vφ′\\n[ K(L,t+1,t+τ+1) ] (x,x′) · Π(L,t+1,t+1+τ)(x,x′) t ∈ [T − 1]\\nΠ(L,t,t ′)(x,x′) = 0 t′ − t 6= τ\\n(33) Similarly by using the same course of arguments used in the single layer setting, for the last time step we have Π(`,T,T+τ)(x,x′) = (σ`+1u ) 2Vφ′ [ K(`,T+1,T+τ+1) ] (x,x′) · Π(`+1,T+1,T+τ+1)(x,x′) ` ∈ [L− 1]\\nFor the other layers we have\\nΠ(`,t,t ′)(x,x′) = (σ`w) 2Vφ′ [ K(`,t+1,t+τ+1) ] (x,x′) · Π(`,t+1,t+1+τ)(x,x′)\\n+ (σ`+1u ) 2Vφ′\\n[ K(`,t+1,t+τ+1) ] (x,x′) · Π(`+1,t+1,t ′+1)(x,x′).\\nFor t′ − t 6= τ the recursion continues until it reaches Π(L,T,t′′)(x,x′), t′′ < T ′ or Π(L,t ′′,T ′)(x,x′), t′′ < T and as a result based on (33) we get\\nΠ(`,t,t ′)(x,x′) = 0 t′ − t 6= τ (34)\\nFor t′ − t = τ it leads to Π(L,T,T ′)(x,x′) and has a non-zero value. Now we derive RNTK for multi-layer:\\n〈∇θfθ(x),∇θfθ(x′)〉 = L∑ `=1 〈 ∂fθ(x) ∂W(`) , ∂fθ(x ′) ∂W(`) 〉 + L∑ `=1 〈 ∂fθ(x) ∂U(`) , ∂fθ(x ′) ∂U(`) 〉\\n+ L∑ `=1 〈 ∂fθ(x) ∂b(`) , ∂fθ(x ′) ∂b(`) 〉 + 〈 ∂fθ(x) ∂v , ∂fθ(x ′) ∂v 〉 ,\\nwhere〈 ∂fθ(x)\\n∂W(`) , ∂fθ(x\\n′)\\n∂W(`)\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉) · ( (σ`w) 2 n 〈 h(`,t−1)(x),h(`,t ′−1)(x′) 〉)\\n〈 ∂fθ(x)\\n∂U(`) , ∂fθ(x\\n′)\\n∂U(`)\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉) · ( (σ`u) 2 m 〈xt,x′t′〉 ) ` = 1\\n〈 ∂fθ(x)\\n∂U(`) , ∂fθ(x\\n′)\\n∂U(`)\\n〉 = T ′∑ t′=1 T∑ t=1 [( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉)\\n· ( (σ`u) 2\\nn\\n〈 h(`−1,t)(x),h(`−1,t ′)(x′) 〉)]\\n` > 1\\n〈 ∂fθ(x)\\n∂b(`) , ∂fθ(x\\n′)\\n∂b(`)\\n〉 = T ′∑ t′=1 T∑ t=1 ( 1 n 〈 δ(`,t)(x), δ(`,t ′)(x′) 〉)\\n· (σ`b)2〈 ∂fθ(x)\\n∂v , ∂fθ(x\\n′)\\n∂v\\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉)\\nSumming up all the terms and replacing the inner product of vectors with their expectations we get\\n〈∇θfθ(x),∇θfθ(x′)〉 = Θ(L,T,T ′) = L∑ `=1 T∑ t=1 T ′∑ t′=1 Π(`,t,t ′)(x,x′) · Σ(`,t,t ′)(x,x′) +K(x,x′). By (34), we can simplify to\\nΘ(L,T,T ′) = ( L∑ `=1 T∑ t=1 Π(`,t,t ′)(x,x′) · Σ(`,t,t+τ)(x,x′) ) +K(x,x′).\\nFor multi-dimensional output it becomes\\nΘ(x,x′) = Θ(L,T,T ′)(x,x′)⊗ Id.\\nThis concludes the proof for Theorem 1 for the multi-layer case.\"\n },\n {\n \"heading\": \"B.4 PROOF FOR THEOREM 3: WEIGHT-UNTIED RNTK\",\n \"text\": \"The architecture of a weight-untied single layer RNN is\\ng(t)(x) = σw√ m W(t)h(t−1)(x) + σu√ n U(t)xt + σbb (t)\\nh(t)(x) = φ ( g(t)(x) ) fθ(x) =\\nσv√ n v>h(T )(x)\\nWhere we use new weights at each time step and we index it by time. Like previous sections, we first derive the forward pass kernels for two same length data x = {xt}Tt=1,x = {x′t′}Tt′=1\\nΣ(t,t)(x,x′) = σ2wVφ [ K(t,t)(x,x′) ] + σ2u m 〈xt,x′t〉+ σ2b .\\nΣ(t,t ′)(x,x′) = 0 t 6= t′\\nSince we are using same weight at the same time step, Σ(t,t)(x,x′) can be written as a function of the previous kernel, which is exactly as the weight-tied RNN. However for different length, it becomes zero as a consequence of using different weights, unlike weight-tied which has non-zero value. The kernel of the first time step and output is also the same as weight-tied RNN. For the gradients we have:\\nδ(T )(x) = σvv φ′(g(T )(x))\\nδ(t)(x) = σw√ n\\n(W(t+1))> ( φ′(g(t)(x)) δ(t+1)(x) ) t ∈ [T − 1]\\nFor t′ = t we have: Π(t,t)(x,x′) = σ2wVφ′ [ K(t+1,t+1)(x,x′) ] Π(t+1,t+1)(x,x′)\\nΠ(t,t)(x,x′) = σ2vVφ′ [ K(T+1,T+τ+1)(x,x′) ] .\\nDue to using different weights for t 6= t′, we can immediately conclude that Π(t,t′)(x,x′) = 0. This set of calculation is exactly the same as the weight-tied case when τ = T − T = 0. Finally, with θ = Vect [ {{W(t),U(t),b(t)}Tt=1,v} ] we have\\n〈∇θfθ(x),∇θfθ(x′)〉 = T∑ t=1 〈 ∂fθ(x) ∂W(t) , ∂fθ(x ′) ∂W(t) 〉 + T∑ t=1 〈 ∂fθ(x) ∂U(t) , ∂fθ(x ′) ∂U(t) 〉\\n+ T∑ t=1 〈 ∂fθ(x) ∂b(t) , ∂fθ(x ′) ∂b(t) 〉 + 〈 ∂fθ(x) ∂v , ∂fθ(x ′) ∂v , 〉\\nwith 〈 ∂fθ(x)\\n∂W(t) , ∂fθ(x\\n′)\\n∂W(t)\\n〉 = ( 1\\nn\\n〈 δ(t)(x), δ(t)(x′) 〉) · ( σ2w n 〈 h(t−1)(x),h(t−1)(x′) 〉) 〈 ∂fθ(x)\\n∂U(t) , ∂fθ(x\\n′)\\n∂U(t)\\n〉 = ( 1\\nn\\n〈 δ(t)(x), δ(t)(x′) 〉) · ( σ2u m 〈xt,x′t〉 ) 〈 ∂fθ(x)\\n∂b(t) , ∂fθ(x\\n′)\\n∂b(t)\\n〉 = ( 1\\nn\\n〈 δ(t)(x), δ(t)(x′) 〉) · σ2b〈\\n∂fθ(x) ∂v , ∂fθ(x\\n′)\\n∂v\\n〉 = ( σ2v n 〈 h(T )(x),h(T ′)(x′) 〉) .\\nAs a result we obtain\\n〈∇θfθ(x),∇θfθ(x′)〉 = ( T∑ t=1 Π(t,t)(x,x′) · Σ(t,t)(x′,x′) ) +K(x,x′),\\nsame as the weight-tied RNN when τ = 0. This concludes the proof for Theorem 3.\\nB.5 ANALYTICAL FORMULA FOR Vφ[K] For any positive definite matrixK = [ K1 K3 K3 K2 ] we have:\\n• φ = ReLU (Cho & Saul, 2009)\\nVφ[K] = 1\\n2π\\n( c(π − arccos(c)) + √ 1− c2) )√ K1K2,\\nVφ′ [K] = 1\\n2π (π − arccos(c)).\\nwhere c = K3/ √ K1K2\\n• φ = erf (Neal, 1995)\\nVφ[K] = 2\\nπ arcsin\\n( 2K3√\\n(1 + 2K1)(1 + 2K3)\\n) ,\\nVφ′ [K] = 4 π √ (1 + 2K1)(1 + 2K2)− 4K23 .\"\n },\n {\n \"heading\": \"C PROOF FOR THEOREM 2: RNTK CONVERGENCE AFTER TRAINING\",\n \"text\": \"To prove theorem 2, we use the strategy used in (Lee et al., 2019) which relies on the the local lipschitzness of the network Jacobian J(θ,X ) = ∇θfθ(x) ∈ R|X |d×|θ| at initialization.\\nDefinition 1 The Jacobian of a neural network is local lipschitz at NTK initialization (θ0 ∼ N (0, 1)) if there is constant K > 0 for every C such that{\\n‖J(θ,X )‖F < K ‖J(θ,X )− J(θ̃,X )‖F < K‖θ − θ̃‖ , ∀ θ, θ̃ ∈ B(θ0, R)\\nwhere B(θ,R) := {θ : ‖θ0 − θ‖ < R}.\\nTheorem 4 Assume that the network Jacobian is local lipschitz with high probability and the empirical NTK of the network converges in probability at initialization and it is positive definite over the input set. For > 0, there exists N such that for n > N when applying gradient flow with η < 2 (λmin(Θ(X ,X )) + λmax(Θ(X ,X ))−1 with probability at least (1− ) we have:\\nsup s ‖θs − θ0‖2√ n , sup s ‖Θ̂s(X ,X )− Θ̂0(X ,X )‖ = O ( 1√ n ) .\\nProof: See (Lee et al., 2019)\\nTheorem 4 holds for any network architecture and any cost function and it was used in (Lee et al., 2019) to show the stability of NTK for MLP during training.\\nHere we extend the results for RNTK by proving that the Jacobian of a multi-layer RNN under NTK initialization is local lipschitz with high probability.\\nTo prove it, first, we prove that for any two points θ, θ̃ ∈ B(θ0, R) there exists constant K1 such that\\n‖g(`,t)(x)‖2, ‖δ(`,t)(x)‖2 ≤ K1 √ n (35)\\n‖g(`,t)(x)− g̃(`,t)(x)‖2, ‖δ(`,t)(x)− δ̃(`,t)(x)‖2 ≤ ‖θ̄ − θ̃‖ ≤ K1 √ n‖θ − θ̃‖. (36)\\nTo prove (35) and (36) we use the following lemmas.3\\nLemma 1 Let A ∈ Rn×m be a random matrix whose entries are independent standard normal random variables. Then for every t ≥ 0, with probability at least 1− e(−ct2) for some constant c we have:\\n‖A‖2 ≤ √ m+ √ n+ t.\\nLemma 2 Let a ∈ Rn be a random vector whose entries are independent standard normal random variables. Then for every t ≥ 0, with probability at least 1− e(−ct2) for some constant c we have:\\n‖a‖2 ≤ √ n+ √ t.\\nSetting t = √ n for any θ ∈ R(θ0, R). With high probability, we get:\\n‖W(`)‖2, ‖U(`)‖2 ≤ 3 √ n, ‖b`‖2 ≤ 2 √ n, ‖h(`,0)(x)‖2 ≤ 2σh √ n.\\nWe also assume that there exists some finite constant C such that\\n|φ(x)| < C|x|, |φ(x)− φ(x′)| < C|x− x′|, |φ′(x)| < C, , |φ′(x)− φ′(x′)| < C|x− x′|.\\nThe proof is obtained by induction. From now on assume that all inequalities in (35) and (36) holds with some k for the previous layers. We have\\n‖g(`,t)(x)‖2 = ‖ σ`w√ n W(`)h(`,t−1)(x) + σ`u√ n U(`) h(`−1,t)(x) + σ`bb (`)‖2\\n≤ σ ` w√ n ‖W(`)‖2‖φ\\n( g(`,t−1)(x) ) ‖2 +\\nσ`u√ n ‖U(`)‖2‖φ\\n( g(`−1,t)(x) ) ‖2 + σ`b‖b(`)‖2\\n≤ ( 3σ`wCk + 3σ ` uCk + 2σb )√ n.\\nAnd the proof for (35) and (36) is completed by showing that the first layer is bounded\\n‖g(1,1)(x)‖2 = ‖ σ1w√ n W(`)h(1,0)(x) + σ1u√ m U(`) x1 + σ 1 bb (1)‖2\\n≤ (3σ1wσh + 3σu√ m ‖x1‖2 + 2σb) √ n.\\nFor the gradient of first layer we have\\n‖δ(L,T )(x)‖2 = ‖σvv φ′(g(L,T )(x))‖2 ≤ σv‖v‖2‖φ′(g(L,T )(x))‖∞ = 2σvC √ n.\\nAnd similarly we have\\n‖δ(`,t)(x)‖ ≤ (3σwCk′ + 3σuCk′) √ n.\\n3See math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf for proofs\\nFor θ, θ̃ ∈ B(θ0, R) we have\\n‖g(1,1)(x)− g̃(1,1)(x)‖2 = ‖ σ1w√ n (W(1) − W̃(1))h(1,0)(x) + σ 1 u√ m (U(1) − Ũ(1))h(1,0)(x)‖2\\n≤ (\\n3σ1wσh + 3σ1u m ‖x1‖2\\n) ‖θ − θ̃‖2 √ n.\\n‖g(`,t)(x)− g̃(`,t)(x)‖2 ≤ ‖φ(g(`,t−1)(x))‖2‖ σ`w√ n (W(`) − W̃(`))‖2\\n+ ‖ σ ` w√ n W̃(`)‖2‖φ(g(`,t−1)(x))− φ(g̃(`,t−1)(x))‖2\\n+ ‖φ(g(`−1,t)(x))‖2‖ σ`u√ n (U(`) − Ũ(`))‖2\\n+ ‖ σ ` u√ n Ũ(`)‖2‖φ(g(`−1,t)(x))− φ(g̃(`−1,t)(x))‖2 + σb‖b(`) − b̃(`)‖ ≤ (kσ`w + 3σ`wCk + kσ`u + 3σ`uCk + σb)‖θ − θ̃‖2 √ n.\\nFor gradients we have\\n‖δ(L,T )(x)− δ̃(L,T )(x)‖2 ≤ σv‖φ′(g(L,T ))‖∞‖(v − ṽ)‖2 + σv‖v‖2‖φ′(g(L,T )(x))− φ′(g(L,T )(x))‖2 ≤ (σvC + 2σvCk)‖θ − θ̃‖2 √ n.\\nAnd similarly using same techniques we have\\n‖δ(`,t)(x)− δ̃(`,t)(x)‖2 ≤ (σwC + 3σwCk + σuC + 3σuCk)‖θ − θ̃‖2 √ n.\\nAs a result, there exists K1 that is a function of σw, σu, σb, L, T and the norm of the inputs.\\nNow we prove the local Lipchitzness of the Jacobian\\n‖J(θ,x)‖F ≤ L∑ `=2 T∑ t=1 ( 1 n ∥∥∥∥δ(`,t)(x)(σ`wh(`,t−1)(x))>∥∥∥∥ F\\n+ 1\\nn ∥∥∥∥δ(`,t)(x)(σ`uh(`,t−1)(x))>∥∥∥∥ F + 1√ n ∥∥∥δ(`,t)(x) · σ`b∥∥∥ F ) +\\nT∑ t=1 ( 1 n ∥∥∥∥δ(1,t)(x)(σ1wh(1,t−1)(x))>∥∥∥∥ F\\n+ 1√ nm ∥∥∥δ(1,t−1)(x) (σ1uxt)>∥∥∥ F + 1√ n ∥∥∥δ(1,t)(x) · σ1b∥∥∥ F ) + σv√ n ‖h(L,T )(x)‖F\\n≤ ( L∑ `=2 T∑ t=1 (K21Cσ ` w +K 2 1Cσ ` u + σ ` bK1)\\n+ T∑ t=1 (K21Cσ 1 w + K1σ 1 u√ m ‖xt‖2 + σ1bK1) + σvCK1 ) .\\nAnd for θ, θ̃ ∈ B(θ0, R) we have ‖J(θ,x)− J̃(θ,x)‖F ≤ L∑ `=2 T∑ t=1 ( 1 n ∥∥∥∥δ(`,t)(x)(σ`wh(`,t−1)(x))> − δ̃(`,t)(x)(σ`wh̃(`,t−1)(x))>∥∥∥∥ F\\n+ 1\\nn ∥∥∥∥δ(`,t)(x)(σ`uh(`,t−1)(x))> − δ̃(`,t)(x)(σ`uh̃(`,t−1)(x))>∥∥∥∥ F\\n+ 1√ n ∥∥∥δ(`,t)(x) · σ`b − δ̃(`,t)(x) · σ`b∥∥∥ F\\n+ T∑ t=1 ( 1 n ∥∥∥∥δ(1,t)(x)(σ1wh(1,t−1)(x))> − δ̃(1,t)(x)(σ1w ˜h(1,t−1)(x))>∥∥∥∥ F\\n+ 1√ nm ∥∥∥δ(1,t−1)(x) (σ1uxt)> − δ̃(1,t−1)(x) (σ1uxt)>∥∥∥ F\\n+ 1√ n ∥∥∥δ(`,t)(x) · σ`b − δ̃(`,t)(x) · σ`b∥∥∥)+ σv√n‖h(L,T )(x)− ˜h(L,T )(x)‖F ≤ ( L∑ `=2 T∑ t=1 (4K21Cσ ` w + 4K 2 1Cσ ` u + σ ` bK1)\\n+ T∑ t=1 (4K21Cσ 1 w + K1σ 1 u√ m ‖xt‖2 + σ1bK1) + σvCK1 ) ‖θ − θ̃‖2.\\nThe above proof can be generalized to the entire dataset by a straightforward application of the union bound. This concludes the proof for Theorem 2.\"\n }\n]","isConversational":false},"year":{"kind":"number","value":2021,"string":"2,021"},"title":{"kind":"string","value":"THE RECURRENT NEURAL TANGENT KERNEL"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":8379,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NjgzNDA0NSwic3ViIjoiL2RhdGFzZXRzL2FsbGVuYWkvbXVwLWZ1bGwiLCJleHAiOjE3NzY4Mzc2NDUsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.cJ50LX8iKQarjYHRvAgu6H4aAsk49_HHM_eFJczNXgeWJP-hlIM5OH6J6i4R70sszLqagklWc_qBHdv1cHvLBA","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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"the paper proposes to learn parametric form of optimal quantum annealing schedule. Authors construct 2 versions of neural network parameterizations mapping problem data onto an optimal schedule. They train these networks on artifically generated sets of problem of different size and test final models on the Grover... | Adiabatic quantum computation is a form of computation that acts by slowly interpolating a quantum system between an easy to prepare initial state and a final state that represents a solution to a given computational problem. The choice of the interpolation schedule is critical to the performance: if at a certain time ... | [
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"Julia Kempe",
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"Oded Regev"
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"title": "Adiabatic quantum computation is equivalent to standard quantum computation",
"venue": "SIAM Review,",
"year": 2008
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"heading": "1 INTRODUCTION",
"text": "Many of the algorithms developed for quantum computing employ the quantum circuit model, in which a quantum state involving multiple qubits undergoes a series of invertible transformations. However, an alternative model, called Adiabatic Quantum Computation (AQC) (Far... | 2,021 | FIDELITY-BASED DEEP ADIABATIC SCHEDULING |
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"The paper builds upon previous lines of research on multi-task learning problem, such as conditional latent variable models including the Neural Process. As shown by the extensive Related Work section, this seems to be an active research direction. This makes it difficult for me to judge originality and significan... | Formulating scalable probabilistic regression models with reliable uncertainty estimates has been a long-standing challenge in machine learning research. Recently, casting probabilistic regression as a multi-task learning problem in terms of conditional latent variable (CLV) models such as the Neural Process (NP) has s... | [
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"name": "Michael Volpp"
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"name": "Fabian Flürenbrock"
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"name": "Lukas Grossberger"
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"title": "Optuna: A Next-generation Hyperparameter Optimization Framework",
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"text": "Estimating statistical relationships between physical quantities from measured data is of central importance in all branches of science and engineering and devising powerful regression models for this purpose forms a major field of study in statistics and machine lear... | 2,021 | BAYESIAN CONTEXT AGGREGATION FOR NEURAL PROCESSES |
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"Till page 3 the paper was easy to follow, i.e., the analytical expressions in eq(5), and the basic idea of Algorithm 1 (which is same as prior works by Han et al. , Wang et al., Periera et al.) are clear. However, after page 3 the paper is hard to follow. The specific points are as follows:"
] | In this paper we present a deep learning framework for solving large-scale multiagent non-cooperative stochastic games using fictitious play. The HamiltonJacobi-Bellman (HJB) PDE associated with each agent is reformulated into a set of Forward-Backward Stochastic Differential Equations (FBSDEs) and solved via forward s... | [] | [
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"venue": "Activity analysis of production and allocation,",
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"text": "Stochastic differential games represent a framework for investigating scenarios where multiple players make decisions while operating in a dynamic and stochastic environment. The theory of differential games dates back to the seminal work of Isaacs (1965) studying two... | 2,020 | MULTI-AGENT DEEP FBSDE REPRESENTATION FOR LARGE SCALE STOCHASTIC DIFFERENTIAL GAMES |
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"This paper attempts to answer the four questions raised from the mutual information estimator. To this end, this paper investigates why the MINE succeeds or fails during the optimization on a synthetic dataset. Based on the observations and discussions, the paper then proposes a novel lower bound to regularize the... | With the variational lower bound of mutual information (MI), the estimation of MI can be understood as an optimization task via stochastic gradient descent. In this work, we start by showing how Mutual Information Neural Estimator (MINE) searches for the optimal function T that maximizes the Donsker-Varadhan representa... | [] | [
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"authors": [
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"title": "Information maximization in noisy channels: A variational approach",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2004
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"authors": [
"Mohamed Ishmael Belghazi",
"Aristide... | [
{
"heading": "1 INTRODUCTION",
"text": "Identifying a relationship between two variables of interest is one of the great linchpins in mathematics, statistics, and machine learning (Goodfellow et al., 2014; Ren et al., 2015; He et al., 2016; Vaswani et al., 2017). Not surprisingly, this problem is closely ti... | 2,020 | null |
SP:b1c7e0c9656a0ec0399b6602f89f46323ff3436b | [
"The paper proposes contextual dropout as a sample-dependent dropout module, which can be applied to different models at the expense of marginal memory and computational overhead. The authors chose to focus on Visual Question Answering and Image classification tasks. The results in the paper show the contextual dr... | Dropout has been demonstrated as a simple and effective module to not only regularize the training process of deep neural networks, but also provide the uncertainty estimation for prediction. However, the quality of uncertainty estimation is highly dependent on the dropout probabilities. Most current models use the sam... | [
{
"affiliations": [],
"name": "Xinjie Fan"
},
{
"affiliations": [],
"name": "Shujian Zhang"
},
{
"affiliations": [],
"name": "Korawat Tanwisuth"
},
{
"affiliations": [],
"name": "Xiaoning Qian"
},
{
"affiliations": [],
"name": "Mingyuan Zhou"
}
] | [
{
"authors": [
"Jimmy Ba",
"Brendan Frey"
],
"title": "Adaptive dropout for training deep neural networks",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2013
},
{
"authors": [
"Emmanuel Bengio",
"Pierre-Luc Bacon",
"Joelle Pineau... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep neural networks (NNs) have become ubiquitous and achieved state-of-the-art results in a wide variety of research problems (LeCun et al., 2015). To prevent over-parameterized NNs from overfitting, we often need to appropriately regularize their training. One way t... | 2,021 | CONTEXTUAL DROPOUT: AN EFFICIENT SAMPLE- DEPENDENT DROPOUT MODULE |
SP:ee9764a48b109b9860c0a6f657a6cdd819237e7e | [
"The authors propose a end-to-end deep learning model called Net-DNF to handle tabular data. The architecture of Net-DNF has four layers: the first layer is a dense layer (learnable weights) with tanh activation eq(1). The second layer (DNNF) is formed by binary conjunctions over literals eq(2). The third layer is ... | A challenging open question in deep learning is how to handle tabular data. Unlike domains such as image and natural language processing, where deep architectures prevail, there is still no widely accepted neural architecture that dominates tabular data. As a step toward bridging this gap, we present Net-DNF a novel ge... | [
{
"affiliations": [],
"name": "Liran Katzir"
},
{
"affiliations": [],
"name": "Gal Elidan"
}
] | [
{
"authors": [
"Martin Anthony"
],
"title": "Connections between neural networks and Boolean functions",
"venue": "In Boolean Methods and Models,",
"year": 2005
},
{
"authors": [
"Sercan Ömer Arik",
"Tomas Pfister"
],
"title": "Tabnet: Attentive interpretable ta... | [
{
"heading": "1 INTRODUCTION",
"text": "A key point in successfully applying deep neural models is the construction of architecture families that contain inductive bias relevant to the application domain. Architectures such as CNNs and RNNs have become the preeminent favorites for modeling images and sequen... | 2,021 | NET-DNF: EFFECTIVE DEEP MODELING OF TABULAR DATA |
SP:9962a592fe8663bbcfe752b83aa9b666fe3a9456 | [
"The paper suggests an improvement over double-Q learning by applying the control variates technique to the target Q, in the form of $(q1 - \\beta (q2 - E(q2))$ (eqn (8)). To minimize the variance, it suggests minimizing the correlation between $q1$ and $q2$. In addition, it applies the TD3 trick. The resulting alg... | Q-learning with value function approximation may have the poor performance because of overestimation bias and imprecise estimate. Specifically, overestimation bias is from the maximum operator over noise estimate, which is exaggerated using the estimate of a subsequent state. Inspired by the recent advance of deep rein... | [] | [
{
"authors": [
"Oron Anschel",
"Nir Baram",
"Nahum Shimkin"
],
"title": "Averaged-dqn: Variance reduction and stabilization for deep reinforcement learning",
"venue": "In Proceedings of the 34th International Conference on Machine Learning, ICML 2017,",
"year": 2017
},
{
... | [
{
"heading": "1 INTRODUCTION",
"text": "Q-learning Watkins & Dayan (1992) as a model free reinforcement learning approach has gained popularity, especially under the advance of deep neural networks Mnih et al. (2013). In general, it combines the neural network approximators with the actor-critic architectur... | 2,020 | DECORRELATED DOUBLE Q-LEARNING |
SP:73f0f92f476990989fa8339f789a77fadb5c1e26 | [
"This work empirically studies the relationship between robustness and class selectivity, a measure of neuron variability between classes. Robustness to both adversarial (\"worst-case\") perturbations and corruptions (\"average-case\") are considered. This work builds off the recent work of Leavitt and Morcos (2020... | Representational sparsity is known to affect robustness to input perturbations in deep neural networks (DNNs), but less is known about how the semantic content of representations affects robustness. Class selectivity—the variability of a unit’s responses across data classes or dimensions—is one way of quantifying the s... | [] | [
{
"authors": [
"Rana Ali Amjad",
"Kairen Liu",
"Bernhard C. Geiger"
],
"title": "Understanding Individual Neuron Importance Using Information Theory. April 2018",
"venue": "URL https://arxiv.org/abs/1804.06679v3",
"year": 2018
},
{
"authors": [
"Alessio Ansuini",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Methods for understanding deep neural networks (DNNs) often attempt to find individual neurons or small sets of neurons that are representative of a network’s decision (Erhan et al., 2009; Zeiler and Fergus, 2014; Karpathy et al., 2016; Amjad et al., 2018; Lillian et ... | 2,020 | null |
SP:8fe8ad33a783b2f98816e57e88d20b67fed50e8d | [
"The authors investigate the token embedding space of a variety of contextual embedding models for natural language. Using techniques based on nearest neighbors, clustering, and PCA, they report a variety of results on local dimensionality / anisotropy / clustering / manifold structure in these embedding models whi... | The geometric properties of contextual embedding spaces for deep language models such as BERT and ERNIE, have attracted considerable attention in recent years. Investigations on the contextual embeddings demonstrate a strong anisotropic space such that most of the vectors fall within a narrow cone, leading to high cosi... | [
{
"affiliations": [],
"name": "Xingyu Cai"
},
{
"affiliations": [],
"name": "Jiaji Huang"
},
{
"affiliations": [],
"name": "Yuchen Bian"
},
{
"affiliations": [],
"name": "Kenneth Church"
}
] | [
{
"authors": [
"Laurent Amsaleg",
"Oussama Chelly",
"Teddy Furon",
"Stéphane Girard",
"Michael E Houle",
"Ken-ichi Kawarabayashi",
"Michael Nett"
],
"title": "Estimating local intrinsic dimensionality",
"venue": "In Proceedings of the 21th ACM SIGKDD Interna... | [
{
"heading": "1 INTRODUCTION",
"text": "The polysemous English word “bank” has two common senses: 1. the money sense, a place that people save or borrow money; 2. the river sense, a slope of earth that prevents the flooding. In modern usage, the two senses are very different from one another, though interes... | 2,021 | null |
SP:9e4a85fa5d76f345b5a38b6f86710a53e1d08503 | [
"This paper critically re-examines research in domain generalisation (DG), ie building models that robustly generalise to out-of-distribution data. It observes that existing methods are hard to compare, in particular due to unclear hyper-parameter and model selection criteria. It introduces a common benchmark suite... | The goal of domain generalization algorithms is to predict well on distributions different from those seen during training. While a myriad of domain generalization algorithms exist, inconsistencies in experimental conditions—datasets, network architectures, and model selection criteria—render fair comparisons difficult... | [
{
"affiliations": [],
"name": "Ishaan Gulrajani"
},
{
"affiliations": [],
"name": "David Lopez-Paz"
}
] | [
{
"authors": [
"Kartik Ahuja",
"Karthikeyan Shanmugam",
"Kush Varshney",
"Amit Dhurandhar"
],
"title": "Invariant risk minimization",
"venue": "games. arXiv,",
"year": 2020
},
{
"authors": [
"Kei Akuzawa",
"Yusuke Iwasawa",
"Yutaka Matsuo"
],... | [
{
"heading": "1 INTRODUCTION",
"text": "Machine learning systems often fail to generalize out-of-distribution, crashing in spectacular ways when tested outside the domain of training examples (Torralba and Efros, 2011). The overreliance of learning systems on the training distribution manifests widely. For ... | 2,021 | IN SEARCH OF LOST DOMAIN GENERALIZATION |
SP:04abdf6d039513f23e00e6686832cd4b950f1d75 | [
"This work proposes a specific parametrisation for the Gaussian prior and approximate posterior distribution in variational Bayesian neural networks in terms of inducing weights. The general idea is an instance of the sparse variational inference scheme for GPs proposed by Titsias back in 2009; for a given model wi... | Bayesian Neural Networks and deep ensembles represent two modern paradigms of uncertainty quantification in deep learning. Yet these approaches struggle to scale mainly due to memory inefficiency, requiring parameter storage several times that of their deterministic counterparts. To address this, we augment each weight... | [
{
"affiliations": [],
"name": "Hippolyt Ritter"
},
{
"affiliations": [],
"name": "Martin Kukla"
},
{
"affiliations": [],
"name": "Cheng Zhang"
},
{
"affiliations": [],
"name": "Yingzhen Li"
}
] | [
{
"authors": [
"F.V. Agakov",
"D. Barber"
],
"title": "An auxiliary variational method",
"venue": "In ICONIP,",
"year": 2019
},
{
"authors": [
"E. Bingham",
"J.P. Chen",
"M. Jankowiak",
"F. Obermeyer",
"N. Pradhan",
"T. Karaletsos",
"... | [
{
"heading": "1 Introduction",
"text": "Deep learning models are becoming deeper and wider than ever before. From image recognition models such as ResNet-101 (He et al., 2016a) and DenseNet (Huang et al., 2017) to BERT (Xu et al., 2019) and GPT-3 (Brown et al., 2020) for language modelling, deep neural netw... | 2,022 | Sparse Uncertainty Representation in Deep Learning with Inducing Weights |
SP:4b4f70092c9fceabdc76c6ed5c5cf83c7791e119 | [
"This paper proposes a hybrid-regressive machine translation (HRT) approach—combining autoregressive (AT) and non-autoregressive (NAT) translation paradigms: it first uses an AT model to generate a “gappy” sketch (every other token in a sentence), and then applies a NAT model to fill in the gaps with a single pass.... | Although the non-autoregressive translation model based on iterative refinement has achieved comparable performance to the autoregressive counterparts with faster decoding, we empirically found that such aggressive iterations make the acceleration rely heavily on small batch size (e.g., 1) and computing device (e.g., G... | [] | [
{
"authors": [
"Nader Akoury",
"Kalpesh Krishna",
"Mohit Iyyer"
],
"title": "Syntactically supervised transformers for faster neural machine translation",
"venue": "In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics,",
"year": 2019
},
... | [
{
"heading": "1 INTRODUCTION",
"text": "Although autoregressive translation (AT) has become the de facto standard for Neural Machine Translation (Bahdanau et al., 2015), its nature of generating target sentences sequentially (e.g., from left to right) makes it challenging to respond quickly in a production ... | 2,020 | null |
SP:41b23082a1439aa8601439e27c9abaa33e06959c | [
"This paper proposes a (decentralized) method for online adjustment of agent incentives in multi-agent learning scenarios, as a means to obtain higher outcomes for each agent and for the group as a whole. The paper uses the “price of anarchy” (the worst value of an equilibrium divided by the best value in the game)... | Even in simple multi-agent systems, fixed incentives can lead to outcomes that are poor for the group and each individual agent. We propose a method, D3C, for online adjustment of agent incentives that reduces the loss incurred at a Nash equilibrium. Agents adjust their incentives by learning to mix their incentive wit... | [] | [
{
"authors": [
"Blaise Aguera y Arcas"
],
"title": "Social intelligence",
"venue": "In Talk presented at the 33rd Conference on Neural Information Processing Systems Conference,",
"year": 2020
},
{
"authors": [
"Richard D. Alexander",
"Gerald Bargia"
],
"title":... | [
{
"heading": "1 INTRODUCTION",
"text": "We consider a setting composed of multiple interacting artificially intelligent agents. These agents will be instantiated by humans, corporations, or machines with specific individual incentives. However, it is well known that the interactions between individual agent... | 2,020 | null |
SP:87bda29654ffe25cda14e3b27a6e4b53e2a40164 | [
"The paper investigates whether languages are equally hard to Conditional-Language-Model (CLM). To do this, the authors perform controlled experiments by modeling text from parallel data from 6 typologically diverse languages. They pair the languages and perform experiments in 30 directions with Transformers, and c... | Inspired by the phenomenon of performance disparity between languages in machine translation, we investigate whether and to what extent languages are equally hard to “conditional-language-model”. Our goal is to improve our understanding and expectation of the relationship between language, data representation, size, an... | [] | [
{
"authors": [
"Mikhail Belkin",
"Daniel Hsu",
"Siyuan Ma",
"Soumik Mandal"
],
"title": "Reconciling modern machine-learning practice and the classical bias–variance trade-off",
"venue": "Proceedings of the National Academy of Sciences,",
"year": 2019
},
{
"author... | [
{
"heading": null,
"text": "Inspired by the phenomenon of performance disparity between languages in machine translation, we investigate whether and to what extent languages are equally hard to “conditional-language-model”. Our goal is to improve our understanding and expectation of the relationship between... | 2,020 | null |
SP:0cab715d71a765b97066673f3a2d0e00d22ffa3c | [
"The authors propose a neural architecture search (NAS) algorithm inspired by brain physiology. In particular, they propose a NAS algorithm based on neural dendritic branching, and apply it to three different segmentation tasks (namely cell nuclei, electron microscopy, and chest X-ray lung segmentation). The author... | Researchers manually compose most neural networks through painstaking experimentation. This process is taxing and explores only a limited subset of possible architecture. Researchers design architectures to address objectives ranging from low space complexity to high accuracy through hours of experimentation. Neural ar... | [] | [
{
"authors": [
"Md Zahangir Alom",
"Mahmudul Hasan",
"Chris Yakopcic",
"Tarek M Taha",
"Vijayan K Asari"
],
"title": "Recurrent residual convolutional neural network based on u-net (r2u-net) for medical image segmentation",
"venue": "arXiv preprint arXiv:1802.06955,",
... | [
{
"heading": null,
"text": "1 INTRODUCTION\nResearchers manually composing neural networks must juggle multiple goals for their architectures. Architectures must make good decisions; they must be fast, and they should work even with limited computational resources. These goals are challenging to achieve man... | 2,020 | null |
SP:232edf223e799126992acd9ee04d88c22ff57110 | [
"The authors propose two approaches for pruning: (a) \"Evolution-style\": start with K random masks associated with the weights, update weights on gradient descent corresponding to those active in the “fittest” mask, and overtime throw away all but one masks which are less fit. (b) \"Dissipating-gradients”: Here th... | Post-training dropout based approaches achieve high sparsity and are well established means of deciphering problems relating to computational cost and overfitting in Neural Network architectures citesrivastava2014dropout, (Pan et al., 2016), Zhu & Gupta (2017), LeCun et al. (1990). Contrastingly, pruning at initializat... | [] | [
{
"authors": [
"Simon Alford",
"Ryan Robinett",
"Lauren Milechin",
"Jeremy Kepner"
],
"title": "Training behavior of sparse neural network topologies",
"venue": "IEEE High Performance Extreme Computing Conference (HPEC),",
"year": 2019
},
{
"authors": [
"Ayd... | [
{
"heading": "1 INTRODUCTION",
"text": "Computational complexity and overfitting in neural networks is a well established problem Frankle & Carbin (2018), Han et al. (2015), LeCun et al. (1990), Denil et al. (2013). We utilize pruning approaches for the following two reasons: 1) To reduce the computational ... | 2,020 | null |
SP:bb0b99194e5d102320ca4cc7c89c4ae6ee514d83 | [
"The paper studies “butterfly networks”, where, a logarithmic number of linear layers with sparse connections resembling the butterfly structure of the FFT algorithm, along with linear layers in smaller dimensions are used to approximate linear layers in larger dimensions. In general, the paper follows the idea of ... | A butterfly network consists of logarithmically many layers, each with a linear number of non-zero weights (pre-specified). The fast Johnson-Lindenstrauss transform (FJLT) can be represented as a butterfly network followed by a projection onto a random subset of the coordinates. Moreover, a random matrix based on FJLT ... | [
{
"affiliations": [],
"name": "FIXED BUTTER"
}
] | [
{
"authors": [
"N. Ailon",
"B. Chazelle"
],
"title": "The fast johnson–lindenstrauss transform and approximate nearest neighbors",
"venue": "SIAM J. Comput.,",
"year": 2009
},
{
"authors": [
"N. Ailon",
"E. Liberty"
],
"title": "Fast dimension reduction us... | [
{
"heading": "1 INTRODUCTION",
"text": "A butterfly network (see Figure 6 in Appendix A) is a layered graph connecting a layer of n inputs to a layer of n outputs withO(log n) layers, where each layer contains 2n edges. The edges connecting adjacent layers are organized in disjoint gadgets, each gadget conn... | 2,020 | null |
SP:fb0eda1f20d9b0a63164e96a2bf9ab4bee365eea | [
"The paper considers the problem of partitioning the atoms (e.g., pixels of an image) of a reinforcement learning task to latent states (e.g., a grid that determines whether there exists furniture in each cell). The number of states grows exponentially with the number of cells of the grid. So the algorithms that ar... | We propose a novel setting for reinforcement learning that combines two common real-world difficulties: presence of observations (such as camera images) and factored states (such as location of objects). In our setting, the agent receives observations generated stochastically from a latent factored state. These observa... | [
{
"affiliations": [],
"name": "Dipendra Misra"
},
{
"affiliations": [],
"name": "Qinghua Liu"
}
] | [
{
"authors": [
"Alekh Agarwal",
"Sham Kakade",
"Akshay Krishnamurthy",
"Wen Sun"
],
"title": "Flambe: Structural complexity and representation learning of low rank mdps",
"venue": "Advances in Neural Information Processing Systems,",
"year": 2020
},
{
"authors": [... | [
{
"heading": "1 INTRODUCTION",
"text": "Most reinforcement learning (RL) algorithms scale polynomially with the size of the state space, which is inadequate for many real world applications. Consider for example a simple navigation task in a room with furniture where the set of furniture pieces and their lo... | 2,021 | PROVABLE RICH OBSERVATION REINFORCEMENT LEARNING WITH COMBINATORIAL LATENT STATES |
SP:5908636440ae0162f1bf98b6e7b8969cc163f9a6 | [
"Motivated by the observation that prevalent metrics (Inception Score, Frechet Inception Distance) used to assess the quality of samples obtained from generative models are gameable (due to either the metric not correlating well with visually assessed sample quality or the metric being susceptible to training sampl... | Many recent developments on generative models for natural images have relied on heuristically-motivated metrics that can be easily gamed by memorizing a small sample from the true distribution or training a model directly to improve the metric. In this work, we critically evaluate the gameability of such metrics by run... | [] | [
{
"authors": [
"Shane Barratt",
"Rishi Sharma"
],
"title": "A note on the inception score",
"venue": "arXiv preprint arXiv:1801.01973,",
"year": 2018
},
{
"authors": [
"Ali Borji"
],
"title": "Pros and cons of gan evaluation measures",
"venue": "Computer Vis... | [
{
"heading": null,
"text": "Many recent developments on generative models for natural images have relied on heuristically-motivated metrics that can be easily gamed by memorizing a small sample from the true distribution or training a model directly to improve the metric. In this work, we critically evaluat... | 2,020 | null |
SP:9ce7a60c5f2e40f7d59e98c90171a7b49621c67c | [
"Observing that the existed ER-based sampling method may introduce bias or redundancy in sampled transitions, the paper proposes a new sampling method in the ER learning setting. The idea is to take into consideration the context, i.e. many visited transitions, rather than a single one, based on which one can measu... | Experience replay, which enables the agents to remember and reuse experience from the past, has played a significant role in the success of off-policy reinforcement learning (RL). To utilize the experience replay efficiently, the existing sampling methods allow selecting out more meaningful experiences by imposing prio... | [
{
"affiliations": [],
"name": "REPLAY BUFFERS"
},
{
"affiliations": [],
"name": "Youngmin Oh"
},
{
"affiliations": [],
"name": "Kimin Lee"
},
{
"affiliations": [],
"name": "Jinwoo Shin"
},
{
"affiliations": [],
"name": "Eunho Yang"
},
{
"affiliations":... | [
{
"authors": [
"Marc G Bellemare",
"Yavar Naddaf",
"Joel Veness",
"Michael Bowling"
],
"title": "The arcade learning environment: An evaluation platform for general agents",
"venue": "Journal of Artificial Intelligence Research,",
"year": 2013
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Experience replay (Mnih et al., 2015), which is a memory that stores the past experiences to reuse them, has become a popular mechanism for reinforcement learning (RL), since it stabilizes training and improves the sample efficiency. The success of various off-policy ... | 2,021 | null |
SP:ca6ab92369346b3d457f575fc652333255f2dfec | [
"The paper considers the problem of slow sampling in autoregressive generative models. Sampling in such models is sequential, so its computational cost scales with the data dimensionality. Existing work speeds up autoregressive sampling by caching activations or distilling into normalizing flows with fast sampling.... | Autoregressive models are widely used for tasks such as image and audio generation. The sampling process of these models, however, does not allow interruptions and cannot adapt to real-time computational resources. This challenge impedes the deployment of powerful autoregressive models, which involve a slow sampling pr... | [
{
"affiliations": [],
"name": "ORDERED AUTOENCODING"
},
{
"affiliations": [],
"name": "Yilun Xu"
},
{
"affiliations": [],
"name": "Yang Song"
},
{
"affiliations": [],
"name": "Linyuan Gong"
}
] | [
{
"authors": [
"X. Bao",
"J. Lucas",
"S. Sachdeva",
"R.B. Grosse"
],
"title": "Regularized linear autoencoders recover the principal components, eventually",
"venue": "ArXiv, abs/2007.06731,",
"year": 2020
},
{
"authors": [
"Y. Bengio",
"N. Léonard",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Autoregressive models are a prominent approach to data generation, and have been widely used to produce high quality samples of images (Oord et al., 2016b; Salimans et al., 2017; Menick & Kalchbrenner, 2018), audio (Oord et al., 2016a), video (Kalchbrenner et al., 201... | 2,021 | null |
SP:a4cda983cb5a670c3ad7054b9cd7797107af64b1 | [
"This paper presents a one-class classification method using a fully convolutional model and directly using the output map as an explanation map. The method is dubbed FCDD for fully convolutional data descriptor. FCDD uses a hypersphere classifier combined with a pseudo-Huber loss. FCDD is trained using outliers ex... | Deep one-class classification variants for anomaly detection learn a mapping that concentrates nominal samples in feature space causing anomalies to be mapped away. Because this transformation is highly non-linear, finding interpretations poses a significant challenge. In this paper we present an explainable deep one-c... | [
{
"affiliations": [],
"name": "Philipp Liznerski"
},
{
"affiliations": [],
"name": "Lukas Ruff"
},
{
"affiliations": [],
"name": "Robert A. Vandermeulen"
},
{
"affiliations": [],
"name": "Billy Joe Franks"
},
{
"affiliations": [],
"name": "Marius Kloft"
},
... | [
{
"authors": [
"C.J. Anders",
"P. Pasliev",
"A.-K. Dombrowski",
"K.-R. Müller",
"P. Kessel"
],
"title": "Fairwashing explanations with off-manifold detergent",
"venue": "In ICML,",
"year": 2020
},
{
"authors": [
"V. Barnett",
"T. Lewis"
],
... | [
{
"heading": "1 INTRODUCTION",
"text": "Anomaly detection (AD) is the task of identifying anomalies in a corpus of data (Edgeworth, 1887; Barnett and Lewis, 1994; Chandola et al., 2009; Ruff et al., 2021). Powerful new anomaly detectors based on deep learning have made AD more effective and scalable to larg... | 2,021 | EXPLAINABLE DEEP ONE-CLASS CLASSIFICATION |
SP:1d4d75e1bbb4e58273bc027f004aa986a587a6dd | [
"This paper proposes an approach to training deep latent variable models on data that is missing not at random. To learn the parameters of deep latent variable models, the paper adopts importance-weighted variational inference techniques. Experiments on a variety of datasets show that the proposed approach is effec... | When a missing process depends on the missing values themselves, it needs to be explicitly modelled and taken into account while doing likelihood-based inference. We present an approach for building and fitting deep latent variable models (DLVMs) in cases where the missing process is dependent on the missing data. Spec... | [
{
"affiliations": [],
"name": "Niels Bruun Ipsen"
},
{
"affiliations": [],
"name": "Pierre-Alexandre Mattei"
},
{
"affiliations": [],
"name": "Jes Frellsen"
}
] | [
{
"authors": [
"Alberto Bietti",
"Julien Mairal"
],
"title": "Invariance and stability of deep convolutional representations",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2017
},
{
"authors": [
"Benjamin Bloem-Reddy",
"Yee Whye Teh"
... | [
{
"heading": null,
"text": "1 INTRODUCTION\nz\nx\ns\nθ\nφ\nγ\nN\n(a)\nPPCA not-MIWAE PPCA\n(b)\nFigure 1: (a) Graphical model of the not-MIWAE. (b) Gaussian data with MNAR values. Dots are fully observed, partially observed data are displayed as black crosses. A contour of the true distribution is shown tog... | 2,021 | null |
SP:da630280f443afedfacaf7ad1abe20d97ebb60f2 | [
"In this work generative models using a GP as prior and a deep network as likelihood (GP-DGMs) are considered. In the VAE formalism for inference, the novelty of this paper is located in the encoder: It is sparse and the posterior can be computed even when part of the observations are missing. Sparsity is obtained ... | Large, multi-dimensional spatio-temporal datasets are omnipresent in modern science and engineering. An effective framework for handling such data are Gaussian process deep generative models (GP-DGMs), which employ GP priors over the latent variables of DGMs. Existing approaches for performing inference in GP-DGMs do n... | [] | [
{
"authors": [
"Mauricio A Álvarez",
"Neil D Lawrence"
],
"title": "Computationally efficient convolved multiple output Gaussian processes",
"venue": "The Journal of Machine Learning Research,",
"year": 2011
},
{
"authors": [
"Gowtham Atluri",
"Anuj Karpatne",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Increasing amounts of large, multi-dimensional datasets that exhibit strong spatio-temporal dependencies are arising from a wealth of domains, including earth, social and environmental sciences (Atluri et al., 2018). For example, consider modelling daily atmospheric m... | 2,020 | null |
SP:30ceb5d450760e9954ac86f091fb97cb14a2d092 | [
"The paper considers the problem of creating spatial memory representations, which play important roles in robotics and are crucial for real-world applications of intelligent agents. The paper proposes an ego-centric representation that stores depth values and features at each pixel in a panorama. Given the relativ... | Spatial memory, or the ability to remember and recall specific locations and objects, is central to autonomous agents’ ability to carry out tasks in real environments. However, most existing artificial memory modules are not very adept at storing spatial information. We propose a parameter-free module, Egospheric Spati... | [
{
"affiliations": [],
"name": "Daniel Lenton"
},
{
"affiliations": [],
"name": "Stephen James"
},
{
"affiliations": [],
"name": "Ronald Clark"
},
{
"affiliations": [],
"name": "Andrew J. Davison"
}
] | [
{
"authors": [
"Michael Bloesch",
"Jan Czarnowski",
"Ronald Clark",
"Stefan Leutenegger",
"Andrew J Davison"
],
"title": "Codeslam—learning a compact, optimisable representation for dense visual slam",
"venue": "In Proceedings of the IEEE conference on computer vision a... | [
{
"heading": "1 INTRODUCTION",
"text": "Egocentric spatial memory is central to our understanding of spatial reasoning in biology (Klatzky, 1998; Burgess, 2006), where an embodied agent constantly carries with it a local map of its surrounding geometry. Such representations have particular significance for ... | 2,021 | END-TO-END EGOSPHERIC SPATIAL MEMORY |
SP:0cde0537137f3eef6c9c0d6d580a610a07112a39 | [
"This paper introduces an algorithm for training neural networks in a way that parameters preserve a given property. The optimization is based on using a transformation R that perturbs parameters in a way that the desired property is preserved. Instead of directly optimizing the parameters of the network, the opti... | Many types of neural network layers rely on matrix properties such as invertibility or orthogonality. Retaining such properties during optimization with gradientbased stochastic optimizers is a challenging task, which is usually addressed by either reparameterization of the affected parameters or by directly optimizing... | [] | [
{
"authors": [
"P.A. Absil",
"R. Mahony",
"R. Sepulchre"
],
"title": "Optimization algorithms on matrix manifolds",
"venue": "ISBN 9780691132983",
"year": 2009
},
{
"authors": [
"Jens Behrmann",
"Will Grathwohl",
"Ricky T.Q. Chen",
"David Duvenau... | [
{
"heading": null,
"text": "1 INTRODUCTION\nMany deep learning applications depend critically on the neural network parameters having a certain mathematical structure. As an important example, reversible generative models rely on invertibility and, in the case of normalizing flows, efficient inversion and c... | 2,020 | null |
SP:6ba57dba7e320797ca311e5c7d6e55e130384df2 | [
"To summarize, this paper proposed a new noise injection method that is easy to implement and is able to replace the original noise injection method in StyleGAN 2. The approach is supported by detailed theoretical analysis and impactful performance improvement on GAN training and inversion. The results show that th... | Noise injection is an effective way of circumventing overfitting and enhancing generalization in machine learning, the rationale of which has been validated in deep learning as well. Recently, noise injection exhibits surprising effectiveness when generating high-fidelity images in Generative Adversarial Networks (e.g.... | [] | [
{
"authors": [
"Rameen Abdal",
"Yipeng Qin",
"Peter Wonka"
],
"title": "Image2StyleGAN: How to embed images into the StyleGAN latent space",
"venue": "In Proceedings of the IEEE International Conference on Computer Vision,",
"year": 2019
},
{
"authors": [
"Guozhon... | [
{
"heading": "1 INTRODUCTION",
"text": "Noise injection is usually applied as regularization to cope with overfitting or facilitate generalization in neural networks (Bishop, 1995; An, 1996). The effectiveness of this simple technique has also been proved in various tasks in deep learning, such as learning ... | 2,020 | null |
SP:bdbb12951868ea0864f926192fdbe2e62ecdb0e3 | [
"The authors proposed in this paper a supervised approach relying on given relative and quantitative attribute discrepancies. A UNet-like generator learns adversarially tends to generate realistic images while a \"ranker\" tends to predict the magnitude of the input parameter used to control the image manipulation.... | We propose a new model to refine image-to-image translation via an adversarial ranking process. In particular, we simultaneously train two modules: a generator that translates an input image to the desired image with smooth subtle changes with respect to some specific attributes; and a ranker that ranks rival preferenc... | [] | [
{
"authors": [
"Yazeed Alharbi",
"Peter Wonka"
],
"title": "Disentangled image generation through structured noise injection",
"venue": "In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition,",
"year": 2020
},
{
"authors": [
"Yunbo Cao",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Image-to-image (I2I) translation (Isola et al., 2017) aims to translate an input image into the desired ones with changes in some specific attributes. Current literature can be classified into two categories: binary translation (Zhu et al., 2017; Kim et al., 2017), e.... | 2,020 | TRIP: REFINING IMAGE-TO-IMAGE TRANSLATION |
SP:878a518cb77731b8b376d5fd82542670e195f0d6 | [
"This paper aims to develop a transformer-based pre-trained model for multivariate time series representation learning. Specifically, the transformer’s encoder is only used and a time-series imputation task is constructed as their unsupervised learning objective. This is a bit similar to the BERT model in NLP. But ... | In this work we propose for the first time a transformer-based framework for unsupervised representation learning of multivariate time series. Pre-trained models can be potentially used for downstream tasks such as regression and classification, forecasting and missing value imputation. We evaluate our models on severa... | [] | [
{
"authors": [
"A. Bagnall",
"J. Lines",
"A. Bostrom",
"J. Large",
"E. Keogh"
],
"title": "The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances",
"venue": "Data Mining and Knowledge Discovery,",
"year": 2... | [
{
"heading": "1 INTRODUCTION",
"text": "Multivariate time series (MTS) are an important type of data that is ubiquitous in a wide variety of domains, including science, medicine, finance, engineering and industrial applications. Despite the recent abundance of MTS data in the much touted era of “Big Data”, ... | 2,020 | null |
SP:2fe9ca0b44e57587b94159cb8fa201f79c13db50 | [
"In this paper, the authors proposed a novel reparameterization framework of the last network layer that takes semantic hierarchy into account. Specifically, the authors assume a predefined hierarchy graph, and model the classifier of child classes as a parent classifier plus offsets $\\delta$ recursively. The auth... | This paper considers classification problems with hierarchically organized classes. We force the classifier (hyperplane) of each class to belong to a sphere manifold, whose center is the classifier of its super-class. Then, individual sphere manifolds are connected based on their hierarchical relations. Our technique r... | [] | [
{
"authors": [
"P.-A. Absil",
"R. Mahony",
"R. Sepulchre"
],
"title": "Optimization Algorithms on Matrix Manifolds",
"venue": null,
"year": 2007
},
{
"authors": [
"Gregor Bachmann",
"Gary Bécigneul",
"Octavian-Eugen Ganea"
],
"title": "Constant... | [
{
"heading": "1 INTRODUCTION",
"text": "Applying inductive biases or prior knowledge to inference models is a popular strategy to improve their generalization performance (Battaglia et al., 2018). For example, a hierarchical structure is found based on the similarity or shared characteristics between sample... | 2,020 | CONNECTING SPHERE MANIFOLDS HIERARCHICALLY |
SP:cb6afa05735201fecf8106b77c2d0a883d5cd996 | [
"This paper investigates the role of pre-training as an initialization for meta-learning for few-shot classification. In particular, they look at the extent to which the pre-trained representations are disentangled with respect to the class labels. They hypothesize that this disentanglement property of those repres... | Few-shot learning aims to classify unknown classes of examples with a few new examples per class. There are two key routes for few-shot learning. One is to (pre)train a classifier with examples from known classes, and then transfer the pretrained classifier to unknown classes using the new examples. The other, called m... | [] | [
{
"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,",
"year": 2019
},
{
"authors": [
"W... | [
{
"heading": "1 INTRODUCTION",
"text": "In recent years, deep learning methods have outperformed most of the traditional methods in supervised learning, especially in image classification. However, deep learning methods generally require lots of labeled data to achieve decent performance. Some applications,... | 2,020 | null |
SP:2cfe676c21709d69aa3bab1480440fda0a365c3f | [
"The paper proposes a method, named as RG-flow, which combines the ideas of Renormalization group (RG) and flow-based models. The RG is applied to separate signal statistics of different scales in the input distribution and flow-based idea represents each scale information in its latent variables with sparse prior ... | Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key idea of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, called RG-Flow, which can separate information at different scales ... | [] | [
{
"authors": [
"Samuel K Ainsworth",
"Nicholas J Foti",
"Adrian KC Lee",
"Emily B Fox"
],
"title": "oi-vae: Output interpretable vaes for nonlinear group factor analysis",
"venue": "In International Conference on Machine Learning,",
"year": 2018
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "One of the most important unsupervised learning tasks is to learn the data distribution and build generative models. Over the past few years, various types of generative models have been proposed. Flow-based generative models are a particular family of generative mode... | 2,020 | RG-FLOW: A HIERARCHICAL AND EXPLAINABLE FLOW MODEL BASED ON RENORMALIZATION GROUP AND SPARSE PRIOR |
SP:2f7f3a043edf8bbe4164dc748c7fbfc7c7338a02 | [
"The authors propose a discriminator-based approach to inverse reinforcement learning (IRL). The discriminator function is trained to attain large values (\"energy\") on trajectories from the current policy and small values on trajectories from an expert policy. The current policy is then improved by using the nega... | Traditional reinforcement learning methods usually deal with the tasks with explicit reward signals. However, for vast majority of cases, the environment wouldn’t feedback a reward signal immediately. It turns out to be a bottleneck for modern reinforcement learning approaches to be applied into more realistic scenario... | [] | [
{
"authors": [
"Pieter Abbeel",
"Varun Ganapathi",
"Andrew Y. Ng"
],
"title": "A learning vehicular dynamics, with application to modeling helicopters",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2006
},
{
"authors": [
"Pieter Abbeel... | [
{
"heading": "1 INTRODUCTION",
"text": "Motivated by applying reinforcement learning algorithms into more realistic tasks, we find that most realistic environments cannot feed an explicit reward signal back to the agent immediately. It becomes a bottleneck for traditional reinforcement learning methods to b... | 2,020 | null |
SP:6b06c93bb2394dae7e4d6e76a8c134b6808a46e9 | [
"This paper considers solving rank-constrained convex optimization. This is a fairly general problem that contains several special cases such as matrix completion and robust PCA. This paper presents a local search approach along with an interesting theoretical analysis of their approach. Furthermore, this paper pro... | We propose greedy and local search algorithms for rank-constrained convex optimization, namely solving min rank(A)≤r∗ R(A) given a convex function R : Rm×n → R and a parameter r∗. These algorithms consist of repeating two steps: (a) adding a new rank-1 matrix to A and (b) enforcing the rank constraint on A. We refine a... | [
{
"affiliations": [],
"name": "Kyriakos Axiotis"
},
{
"affiliations": [],
"name": "Maxim Sviridenko"
}
] | [
{
"authors": [
"Kyriakos Axiotis",
"Maxim Sviridenko"
],
"title": "Sparse convex optimization via adaptively regularized hard thresholding",
"venue": "arXiv preprint arXiv:2006.14571,",
"year": 2020
},
{
"authors": [
"Thierry Bouwmans",
"Sajid Javed",
"Hongy... | [
{
"heading": null,
"text": "rank(A)≤r∗ R(A) given a convex function R : Rm×n →\nR and a parameter r∗. These algorithms consist of repeating two steps: (a) adding a new rank-1 matrix to A and (b) enforcing the rank constraint on A. We refine and improve the theoretical analysis of Shalev-Shwartz et al. (2011... | 2,021 | LOCAL SEARCH ALGORITHMS FOR RANK- CONSTRAINED CONVEX OPTIMIZATION |
SP:9eeb3b40542889b8a8e196f126a11f80e177f031 | [
"The paper uses selective training with pseudo labels. Specifically, the method selects the pseudo-labeled data associated with small loss after performing the data augmentation, and then uses the selected data for training the model. Here, the model computes the confidence of the pseudo labels and then puts a t... | We propose a novel semi-supervised learning (SSL) method that adopts selective training with pseudo labels. In our method, we generate hard pseudo-labels and also estimate their confidence, which represents how likely each pseudo-label is to be correct. Then, we explicitly select which pseudo-labeled data should be use... | [] | [
{
"authors": [
"David Berthelot",
"Nicholas Carlini",
"Ian Goodfellow",
"Nicolas Papernot",
"Avital Oliver",
"Colin A Raffel"
],
"title": "Mixmatch: A holistic approach to semi-supervised learning",
"venue": "In Advances in Neural Information Processing Systems,",... | [
{
"heading": "1 INTRODUCTION",
"text": "Semi-supervised learning (SSL) is a powerful technique to deliver a full potential of complex models, such as deep neural networks, by utilizing unlabeled data as well as labeled data to train the model. It is especially useful in some practical situations where obtai... | 2,020 | SEMI-SUPERVISED LEARNING BY SELECTIVE TRAINING WITH PSEUDO LABELS VIA CONFIDENCE ESTIMATION |
SP:d818bed28daccbda111c39cdc9d097b5755b3d89 | [
"Paper provides an evaluation of the reliability of confidence levels of well known uncertainty quantification techniques in deep learning on classification and regression tasks. The question that the authors are trying to answer empirically is: when a model claims accuracy at a confidence level within a certain i... | Uncertainty quantification for complex deep learning models is increasingly important as these techniques see growing use in high-stakes, real-world settings. Currently, the quality of a model’s uncertainty is evaluated using point-prediction metrics such as negative log-likelihood or the Brier score on heldout data. I... | [] | [
{
"authors": [
"Rina Foygel Barber",
"Emmanuel J Candès",
"Aaditya Ramdas",
"Ryan J Tibshirani"
],
"title": "The limits of distribution-free conditional predictive inference. March 2019",
"venue": "URL http://arxiv.org/ abs/1903.04684",
"year": 1903
},
{
"authors"... | [
{
"heading": null,
"text": "Uncertainty quantification for complex deep learning models is increasingly important as these techniques see growing use in high-stakes, real-world settings. Currently, the quality of a model’s uncertainty is evaluated using point-prediction metrics such as negative log-likeliho... | 2,020 | DEEP LEARNING UNCERTAINTY QUANTIFICATION PROCEDURES |
SP:74f12645ba675ccd4217ebfc0579cb4232406009 | [
"This paper proposes a general framework for boosting CNNs performance on different tasks by using'commentary' to learn meta-information. The obtained meta-information can also be used for other purposes such as the mask of objects within spurious background and the similarities among classes. The commentary module... | Effective training of deep neural networks can be challenging, and there remain many open questions on how to best learn these models. Recently developed methods to improve neural network training examine teaching: providing learned information during the training process to improve downstream model performance. In thi... | [
{
"affiliations": [],
"name": "Aniruddh Raghu"
},
{
"affiliations": [],
"name": "Maithra Raghu"
},
{
"affiliations": [],
"name": "Simon Kornblith"
}
] | [
{
"authors": [
"M.A. Badgeley",
"J.R. Zech",
"L. Oakden-Rayner",
"B.S. Glicksberg",
"M. Liu",
"W. Gale",
"M.V. McConnell",
"B. Percha",
"T.M. Snyder",
"J.T. Dudley"
],
"title": "Deep learning predicts hip fracture using confounding patient and ... | [
{
"heading": "1 INTRODUCTION",
"text": "Training, regularising, and understanding complex neural network models is challenging. There remain central open questions on making training faster and more data-efficient (Kornblith et al., 2019; Raghu et al., 2019a;b), ensuring better generalisation (Zhang et al.,... | 2,021 | null |
SP:19e2493d7bdb4be73c3b834affdb925201243aef | [
"It is well-known that neural networks (NN) perform very well in various areas and in particular if one looks at computer vision convolutional neural networks perform very well. Although convolutional neural networks (CNN) are limited in their architecture (since they only allow nearest-neighbour connections) compa... | Convolutional neural networks (CNN) exhibit unmatched performance in a multitude of computer vision tasks. However, the advantage of using convolutional networks over fully-connected networks is not understood from a theoretical perspective. In this work, we show how convolutional networks can leverage locality in the ... | [
{
"affiliations": [],
"name": "FULLY-CONNECTED NETWORKS"
},
{
"affiliations": [],
"name": "Eran Malach"
},
{
"affiliations": [],
"name": "Shai Shalev-Shwartz"
}
] | [
{
"authors": [
"Emmanuel Abbe",
"Colin Sandon"
],
"title": "Provable limitations of deep learning",
"venue": "arXiv preprint arXiv:1812.06369,",
"year": 2018
},
{
"authors": [
"Peter L Bartlett",
"Nick Harvey",
"Christopher Liaw",
"Abbas Mehrabian"
... | [
{
"heading": "1 INTRODUCTION",
"text": "Convolutional neural networks (LeCun et al., 1998; Krizhevsky et al., 2012) achieve state-of-the-art performance on every possible task in computer vision. However, while the empirical success of convolutional networks is indisputable, the advantage of using them is n... | 2,021 | null |
SP:b7b4e29defc84ee37a5a4dcaf2d393363c153b52 | [
"This paper studies short, chaotic time series and uses the Taken's theorem to discover the causality between two time series. The main challenge is that for short time series, the delay embedding is not possible. Thus, the authors propose to fit a latent neural ODE and theoretically argue that they can use the Neu... | Discovering causal structures of temporal processes is a major tool of scientific inquiry because it helps us better understand and explain the mechanisms driving a phenomenon of interest, thereby facilitating analysis, reasoning, and synthesis for such systems. However, accurately inferring causal structures within a ... | [
{
"affiliations": [],
"name": "Edward De Brouwer"
},
{
"affiliations": [],
"name": "Adam Arany"
},
{
"affiliations": [],
"name": "Yves Moreau"
}
] | [
{
"authors": [
"Mohammad Taha Bahadori",
"Yan Liu"
],
"title": "Granger causality analysis in irregular time series",
"venue": "In Proceedings of the 2012 SIAM International Conference on Data Mining,",
"year": 2012
},
{
"authors": [
"Zsigmond Benkő",
"Adám Zlatni... | [
{
"heading": "1 INTRODUCTION",
"text": "Inferring a right causal model of a physical phenomenon is at the heart of scientific inquiry. It is fundamental to how we understand the world around us and to predict the impact of future interventions (Pearl, 2009). Correctly inferring causal pathways helps us reas... | 2,021 | LATENT CONVERGENT CROSS MAPPING |
SP:474e2b9be8a3ec69a48c4ccd04a7e390ebb96347 | [
"There have been multiple attempts to use self-attention in computer vision backbones for image classification and object detection. Most of these approaches either tried to combine convolution with global self-attention, or replace it completely with local self-attention operation. The proposed approach naturally ... | Recently, a series of works in computer vision have shown promising results on various image and video understanding tasks using self-attention. However, due to the quadratic computational and memory complexities of self-attention, these works either apply attention only to low-resolution feature maps in later stages o... | [] | [
{
"authors": [
"cent 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": "Self-attention is a mechanism in neural networks that focuses on modeling long-range dependencies. Its advantage in terms of establishing global dependencies over other mechanisms, e.g., convolution and recurrence, has made it prevalent in modern deep learning. In com... | 2,020 | null |
SP:bf70c9e16933774746d621a5b8475843e723ac24 | [
"In the context of deep learning, back-propagation is stochastic in the sample level to attain bette efficiency than full-dataset gradient descent. The authors asked that, can we further randomize the gradient compute within each single minibatch / sample with the goal to achieve strong model accuracy. In modern de... | The successes of deep learning, variational inference, and many other fields have been aided by specialized implementations of reverse-mode automatic differentiation (AD) to compute gradients of mega-dimensional objectives. The AD techniques underlying these tools were designed to compute exact gradients to numerical p... | [
{
"affiliations": [],
"name": "Deniz Oktay"
},
{
"affiliations": [],
"name": "Nick McGreivy"
},
{
"affiliations": [],
"name": "Joshua Aduol"
},
{
"affiliations": [],
"name": "Alex Beatson"
},
{
"affiliations": [],
"name": "Ryan P. Adams"
}
] | [
{
"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": "1 INTRODUCTION",
"text": "Deep neural networks have taken center stage as a powerful way to construct and train massivelyparametric machine learning (ML) models for supervised, unsupervised, and reinforcement learning tasks. There are many reasons for the resurgence of neural networks—large da... | 2,021 | RANDOMIZED AUTOMATIC DIFFERENTIATION |
SP:5b707bffe506d9556ffedbe49425c57d0e21c9fa | [
"This paper studies the multi-source domain adaptation problem. The authors examine the existing MDA solutions, i.e. using a domain discriminator for each source-target pair, and argue that the existing ones are likely to distribute the domain-discriminative information across multiple discriminators. By theoretica... | Adversarial learning strategy has demonstrated remarkable performance in dealing with single-source unsupervised Domain Adaptation (DA) problems, and it has recently been applied to multi-source DA problems. Although most existing DA methods use multiple domain discriminators, the effect of using multiple discriminator... | [] | [
{
"authors": [
"Alexander A Alemi",
"Ian Fischer",
"Joshua V Dillon",
"Kevin Murphy"
],
"title": "Deep variational information bottleneck",
"venue": "arXiv preprint arXiv:1612.00410,",
"year": 2016
},
{
"authors": [
"Orly Alter",
"Patrick O Brown",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Although a large number of studies have demonstrated the ability of deep neural networks to solve challenging tasks, the tasks solved by networks are mostly confined to a similar type or a single domain. One remaining challenge is the problem known as domain shift (Gr... | 2,020 | null |
SP:825132782872f2167abd5e45773bfdef83e4bb2e | [
"This paper tackles the problem of geometrical and topological 3D reconstruction of a (botanical) tree using a drone-mounted stereo vision system and deep learning-based/aided tree branch image annotation procedures. This is an interesting computer vision 3D reconstruction task, which has important practical appl... | We tackle the challenging problem of creating full and accurate three dimensional reconstructions of botanical trees with the topological and geometric accuracy required for subsequent physical simulation, e.g. in response to wind forces. Although certain aspects of our approach would benefit from various improvements,... | [
{
"affiliations": [],
"name": "SIMULATABLE GEOMETRY"
}
] | [
{
"authors": [
"Sameer Agarwal",
"Noah Snavely",
"Ian Simon",
"Steven M Seitz",
"Richard Szeliski"
],
"title": "Building rome in a day",
"venue": "In Computer Vision,",
"year": 2009
},
{
"authors": [
"Iro Armeni",
"Ozan Sener",
"Amir R Zami... | [
{
"heading": "1 INTRODUCTION",
"text": "Human-inhabited outdoor environments typically contain ground surfaces such as grass and roads, transportation vehicles such as cars and bikes, buildings and structures, and humans themselves, but are also typically intentionally populated by a large number of trees a... | 2,020 | null |
SP:8e3a07ed19e7b0c677aae1106da801d246f5aa0c | [
"This paper addresses the task of adversarial defense, particularly against untargeted attack. It starts from the observation that these attacks mostly minimize the perturbation and the classification loss, and proposes a new training strategy named Target Training. The method duplicate training examples with a spe... | Recent adversarial defense approaches have failed. Untargeted gradient-based attacks cause classifiers to choose any wrong class. Our novel white-box defense tricks untargeted attacks into becoming attacks targeted at designated target classes. From these target classes, we derive the real classes. The Target Training ... | [] | [
{
"authors": [
"Dario Amodei",
"Chris Olah",
"Jacob Steinhardt",
"Paul F. Christiano",
"John Schulman",
"Dan Mané"
],
"title": "Concrete problems in AI safety",
"venue": "CoRR, abs/1606.06565,",
"year": 2016
},
{
"authors": [
"Anish Athalye",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Neural network classifiers are vulnerable to malicious adversarial samples that appear indistinguishable from original samples (Szegedy et al., 2013), for example, an adversarial attack can make a traffic stop sign appear like a speed limit sign (Eykholt et al., 2018)... | 2,020 | TARGET TRAINING: TRICKING ADVERSARIAL ATTACKS TO FAIL |
SP:8e2ac7405015f9d2d59c4a511df83d796ac00a9e | [
"This paper proposes the signal propagation plot (spp) which is a tool for analyzing residual networks and analyzes ResNet with/without BN. Based on the investigation, the authors first provide ResNet results without normalization with the proposed scaled weight standardization. Furthermore, the authors provide a b... | Batch Normalization is a key component in almost all state-of-the-art image classifiers, but it also introduces practical challenges: it breaks the independence between training examples within a batch, can incur compute and memory overhead, and often results in unexpected bugs. Building on recent theoretical analyses ... | [
{
"affiliations": [],
"name": "Andrew Brock"
},
{
"affiliations": [],
"name": "Soham De"
},
{
"affiliations": [],
"name": "Samuel L. Smith"
}
] | [
{
"authors": [
"Devansh Arpit",
"Yingbo Zhou",
"Bhargava Kota",
"Venu Govindaraju"
],
"title": "Normalization propagation: A parametric technique for removing internal covariate shift in deep networks",
"venue": "In International Conference on Machine Learning,",
"year": ... | [
{
"heading": null,
"text": "Batch Normalization is a key component in almost all state-of-the-art image classifiers, but it also introduces practical challenges: it breaks the independence between training examples within a batch, can incur compute and memory overhead, and often results in unexpected bugs. ... | 2,021 | CHARACTERIZING SIGNAL PROPAGATION TO CLOSE THE PERFORMANCE GAP IN UNNORMALIZED RESNETS |
SP:206600e5bfcc9ccd494b82995a7898ae81a4e0bf | [
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"affiliations": [],
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"name": "Jianing Zhu"
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"affiliations": [],
"name": "Gang Niu"
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"affiliations": [],
"name": "Bo Han"
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{
"affiliations": ... | [
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"authors": [
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"Martin Vechev"
],
"title": "Adversarial training and provable defenses: Bridging the gap",
"venue": "In ICLR,",
"year": 2020
},
{
"authors": [
"Denni D. Boos",
"L.A. Stefanski"
],
"title": "M-Estimation (Estimating Equat... | [
{
"heading": "1 INTRODUCTION",
"text": "Crafted adversarial data can easily fool the standard-trained deep models by adding humanimperceptible noise to the natural data, which leads to the security issue in applications such as medicine, finance, and autonomous driving (Szegedy et al., 2014; Nguyen et al., ... | 2,021 | GEOMETRY-AWARE INSTANCE-REWEIGHTED ADVER- |
SP:d729aacc2cd3f97011a04360a252ca7cb0489354 | [
"This paper considers adopting continual learning on the problem of causal effect estimation. The paper combines methods and algorithms for storing feature representation and representative samples (herding algorithm), avoiding drifting feature representation when new data is learned (feature representation distill... | The era of real world evidence has witnessed an increasing availability of observational data, which much facilitates the development of causal effect inference. Although significant advances have been made to overcome the challenges in causal effect estimation, such as missing counterfactual outcomes and selection bia... | [] | [
{
"authors": [
"Ahmed M Alaa",
"Mihaela van der Schaar"
],
"title": "Bayesian inference of individualized treatment effects using multi-task gaussian processes",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2017
},
{
"authors": [
"Hugh A Chi... | [
{
"heading": "1 INTRODUCTION",
"text": "Causal effect inference is a critical research topic across many domains, such as statistics, computer science, public policy, and economics. Randomized controlled trials (RCT) are usually considered as the gold-standard for causal effect inference, which randomly ass... | 2,020 | CONTINUAL LIFELONG CAUSAL EFFECT INFERENCE |
SP:864d98472c237daf2b227692c4765af9a89886cd | [
"In this paper, the authors study the problem of GCN for disassortative graphs. The authors proposed the GNAN method to allow attention on distant nodes indeed of limiting to local neighbors. The authors generalized the idea of graph wavelet with MLP to generate the attention score and utilized it to generate multi... | Graph neural networks (GNNs) have been extensively studied for prediction tasks on graphs. Most GNNs assume local homophily, i.e., strong similarities in local neighborhoods. This assumption limits the generalizability of GNNs, which has been demonstrated by recent work on disassortative graphs with weak local homophil... | [] | [
{
"authors": [
"Filippo Maria Bianchi",
"Daniele Grattarola",
"Lorenzo Livi",
"Cesare Alippi"
],
"title": "Graph neural networks with convolutional ARMA filters",
"venue": null,
"year": 1901
},
{
"authors": [
"Heng Chang",
"Yu Rong",
"Tingyang Xu... | [
{
"heading": "1 INTRODUCTION",
"text": "Graph neural networks (GNNs) have recently demonstrated great power in graph-related learning tasks, such as node classification (Kipf & Welling, 2017), link prediction (Zhang & Chen, 2018) and graph classification (Lee et al., 2018). Most GNNs follow a message-passin... | 2,020 | null |
SP:28a5570540fa769396ee73c14c25ada9669dd95f | [
"The paper presents a post-hoc calibration method for deep neural net classification. The method proposes to first reduces the well-known ECE score to a special case of the Kolmogorov-Smirnov (KS) test, and this way solves the dependency of ECE on the limiting binning assumption. The method proposes next to recalib... | Calibrating neural networks is of utmost importance when employing them in safety-critical applications where the downstream decision making depends on the predicted probabilities. Measuring calibration error amounts to comparing two empirical distributions. In this work, we introduce a binning-free calibration measure... | [
{
"affiliations": [],
"name": "Kartik Gupta"
},
{
"affiliations": [],
"name": "Amir Rahimi"
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{
"affiliations": [],
"name": "Thalaiyasingam Ajanthan"
},
{
"affiliations": [],
"name": "Thomas Mensink"
},
{
"affiliations": [],
"name": "Cristian Sminchisescu"
... | [
{
"authors": [
"Glenn W Brier"
],
"title": "Verification of forecasts expressed in terms of probability",
"venue": "Monthly weather review,",
"year": 1950
},
{
"authors": [
"Jia Deng",
"Wei Dong",
"Richard Socher",
"Li-Jia Li",
"Kai Li",
"Li Fei-... | [
{
"heading": "1 INTRODUCTION",
"text": "Despite the success of modern neural networks they are shown to be poorly calibrated (Guo et al. (2017)), which has led to a growing interest in the calibration of neural networks over the past few years (Kull et al. (2019); Kumar et al. (2019; 2018); Müller et al. (2... | 2,021 | null |
SP:cdc407d403e1008ced29c7cda727db0d631cc966 | [
"This paper proposes ProxylessKD method from a novel perspective of knowledge distillation. Instead of minimizing the outputs of teacher and student models, ProxylessKD adopts a shared classifier for two models. The shared classifier yields better aligned embedding space, so the embeddings from teacher and student ... | Knowledge Distillation (KD) refers to transferring knowledge from a large model to a smaller one, which is widely used to enhance model performance in machine learning. It tries to align embedding spaces generated from the teacher and the student model (i.e. to make images corresponding to the same semantics share the ... | [
{
"affiliations": [],
"name": "FACE RECOGNI"
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] | [
{
"authors": [
"Umar Asif",
"Jianbin Tang",
"Stefan Harrer"
],
"title": "Ensemble knowledge distillation for learning improved and efficient networks",
"venue": "arXiv preprint arXiv:1909.08097,",
"year": 2019
},
{
"authors": [
"Guobin Chen",
"Wongun Choi",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Knowledge Distillation (KD) is a process of transferring knowledge from a large model to a smaller one. This technique is widely used to enhance model performance in many machine learning tasks such as image classification (Hinton et al., 2015), object detection (Chen... | 2,020 | PROXYLESSKD: DIRECT KNOWLEDGE DISTILLATION |
SP:a15d5230fecc1dad8998905f17c82cf8e05c98d3 | [
"This paper proposes a contrastive learning approach where one of the views, x, is converted into two subviews, x' and x'', and then separate InfoNCE style bounds constructed for each of I(x'';y) and I(x';y|x'') before being combined to form an overall training objective. Critically, the second of these is based o... | Many self-supervised representation learning methods maximize mutual information (MI) across views. In this paper, we transform each view into a set of subviews and then decompose the original MI bound into a sum of bounds involving conditional MI between the subviews. E.g., given two views x and y of the same input ex... | [] | [
{
"authors": [
"Philip Bachman",
"R Devon Hjelm",
"William Buchwalter"
],
"title": "Learning representations by maximizing mutual information across views",
"venue": "In Proc. Conf. on Neural Information Processing Systems (NeurIPS),",
"year": 2019
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "The ability to extract actionable information from data in the absence of explicit supervision seems to be a core prerequisite for building systems that can, for instance, learn from few data points or quickly make analogies and transfer to other tasks. Approaches to ... | 2,020 | null |
SP:70bed0f6f729c03edcb03678fca53e1d82fc06ab | [
"The paper proposes a continual learning framework based on Bayesian non-parametric approach. The hidden layer is modeled using Indian Buffet Process prior. The inference uses a structured mean-field approximation with a Gaussian family for the weights, and Beta-Bernoulli for the task-masks. The variational infe... | Continual Learning is a learning paradigm where learning systems are trained on a sequence of tasks. The goal here is to perform well on the current task without suffering from a performance drop on the previous tasks. Two notable directions among the recent advances in continual learning with neural networks are (1) v... | [] | [
{
"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": [
"Hongjoon Ahn",
"Sungmin Cha",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Continual learning (CL) (Ring, 1997; Parisi et al., 2019) is the learning paradigm where a single model is subjected to a sequence of tasks. At any point of time, the model is expected to (i) make predictions for the tasks it has seen so far, (ii) if subjected to trai... | 2,020 | null |
SP:d27e98774183ece8d82b87f1e7067bf2a28a4fca | [
"This paper describes a system for separating \"on-screen\" sounds from \"off-screen\" sounds in an audio-visual task, meaning sounds that are associated with objects that are visible in a video versus not. It is trained to do this using mixture invariant training to separate synthetic mixtures of mixtures. It is e... | Recent progress in deep learning has enabled many advances in sound separation and visual scene understanding. However, extracting sound sources which are apparent in natural videos remains an open problem. In this work, we present AudioScope, a novel audio-visual sound separation framework that can be trained without ... | [
{
"affiliations": [],
"name": "ON-SCREEN SOUNDS"
},
{
"affiliations": [],
"name": "Efthymios Tzinis"
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{
"affiliations": [],
"name": "Scott Wisdom"
},
{
"affiliations": [],
"name": "Aren Jansen"
},
{
"affiliations": [],
"name": "Shawn Hershey"
},
{
"a... | [
{
"authors": [
"Triantafyllos Afouras",
"Andrew Owens",
"Joon Son Chung",
"Andrew Zisserman"
],
"title": "Self-supervised learning of audio-visual objects from video",
"venue": "In Proc. European Conference on Computer Vision (ECCV),",
"year": 2020
},
{
"authors":... | [
{
"heading": "1 INTRODUCTION",
"text": "Audio-visual machine perception has been undergoing a renaissance in recent years driven by advances in large-scale deep learning. A motivating observation is the interplay in human perception between auditory and visual perception. We understand the world by parsing ... | 2,021 | null |
SP:958f2aacb0790ffe7399fd918c023c7e4e4c314c | [
"The paper is generally well presented. However, a main issue is that the optimization algorithms for the l0-norm regularized problems (Section 3.1.2 and Section 3.2) are not correctly presented. Specifically, in the algorithm development to solve the \"Fix $\\boldsymbol{R}$, optimize $\\boldsymbol{Y}$\" subproblem... | Deep models have achieved great success in many applications. However, vanilla deep models are not well-designed against the input perturbation. In this work, we take an initial step to design a simple robust layer as a lightweight plug-in for vanilla deep models. To achieve this goal, we first propose a fast sparse co... | [] | [
{
"authors": [
"Chenglong Bao",
"Jian-Feng Cai",
"Hui Ji"
],
"title": "Fast sparsity-based orthogonal dictionary learning for image restoration",
"venue": "In Proceedings of the IEEE International Conference on Computer Vision, pp",
"year": 2013
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep neural networks have obtained a great success in many applications, including computer vision, reinforcement learning (RL) and natural language processing, etc. However, vanilla deep models are not robust to noise perturbations of the input. Even a small perturba... | 2,020 | A SIMPLE SPARSE DENOISING LAYER FOR ROBUST DEEP LEARNING |
SP:33673a515722e1d8288fd3014e7db507b7250b20 | [
"The paper under review proposes a new model for multi-dimensional temporal Point processes, allowing efficient estimation of high order interactions. This new model, called additive Poisson process, relies on a log-linear structure of the intensity function that is motivated thanks to the Kolmogorov-Arnold theorem... | We present the Additive Poisson Process (APP), a novel framework that can model the higher-order interaction effects of the intensity functions in point processes using lower dimensional projections. Our model combines the techniques in information geometry to model higher-order interactions on a statistical manifold a... | [] | [
{
"authors": [
"Alan Agresti"
],
"title": "Categorical Data Analysis",
"venue": "Wiley, 3 edition,",
"year": 2012
},
{
"authors": [
"S. Amari"
],
"title": "Information geometry on hierarchy of probability distributions",
"venue": "IEEE Transactions on Information ... | [
{
"heading": null,
"text": "We present the Additive Poisson Process (APP), a novel framework that can model the higher-order interaction effects of the intensity functions in point processes using lower dimensional projections. Our model combines the techniques in information geometry to model higher-order ... | 2,020 | null |
SP:e6e46c0563e852189839b2f923788165800a0f17 | [
"This paper provides an approach for treatment effect estimation when the observational data is longitudinal (with irregular time stamps) and consists of temporal confounding variables. The proposed method can be categorized under the matching methods, in which, in order to estimate the counterfactual outcomes, a s... | Estimating causal treatment effects using observational data is a problem with few solutions when the confounder has a temporal structure, e.g. the history of disease progression might impact both treatment decisions and clinical outcomes. For such a challenging problem, it is desirable for the method to be transparent... | [] | [
{
"authors": [
"Alberto Abadie"
],
"title": "Using synthetic controls: Feasibility, data requirements, and methodological aspects",
"venue": "Journal of Economic Literature,",
"year": 2019
},
{
"authors": [
"Alberto Abadie",
"Javier Gardeazabal"
],
"title": "The... | [
{
"heading": "1 INTRODUCTION",
"text": "Estimating the causal individual treatment effect (ITE) on patient outcomes using observational data (observational studies) has become a promising alternative to clinical trials as large-scale electronic health records become increasingly available (Booth & Tannock, ... | 2,020 | SYNCTWIN: TRANSPARENT TREATMENT EFFECT ESTIMATION UNDER TEMPORAL CONFOUNDING |
SP:8997ab419d35acd51ef50ef6265e5c37c468a2ac | [
"This paper proposes a method for obtaining probably-approximately correct (PAC) predictions given a pre-trained classifier. The PAC intervals are connected to calibration, and take the form of confidence intervals given the bin a prediction falls in. They demonstrate and explore two use cases: applying this techni... | A key challenge for deploying deep neural networks (DNNs) in safety critical settings is the need to provide rigorous ways to quantify their uncertainty. In this paper, we propose a novel algorithm for constructing predicted classification confidences for DNNs that comes with provable correctness guarantees. Our approa... | [
{
"affiliations": [],
"name": "Sangdon Park"
},
{
"affiliations": [],
"name": "Shuo Li"
},
{
"affiliations": [],
"name": "Insup Lee"
},
{
"affiliations": [],
"name": "Osbert Bastani"
}
] | [
{
"authors": [
"A.K. Akametalu",
"J.F. Fisac",
"J.H. Gillula",
"S. Kaynama",
"M.N. Zeilinger",
"C.J. Tomlin"
],
"title": "Reachability-based safe learning with gaussian processes",
"venue": "In 53rd IEEE Conference on Decision and Control,",
"year": 2014
},
... | [
{
"heading": "1 INTRODUCTION",
"text": "Due to the recent success of machine learning, there has been increasing interest in using predictive models such as deep neural networks (DNNs) in safety-critical settings, such as robotics (e.g., obstacle detection (Ren et al., 2015) and forecasting (Kitani et al., ... | 2,021 | PAC CONFIDENCE PREDICTIONS FOR DEEP NEURAL NETWORK CLASSIFIERS |
SP:4c82d9d12ec6a9f171c4281739776da18bcc2906 | [
"of contribution: The authors propose an interesting approach to address the sample-efficiency issue in Neural Architecture Search (NAS). Compared to other existing predictor based methods, the approach distinguishes itself by progressive shrinking the search space. The paper correctly identifies the sampling is an... | Neural Architecture Search (NAS) finds the best network architecture by exploring the architecture-to-performance manifold. It often trains and evaluates a large amount of architectures, causing tremendous computation cost. Recent predictorbased NAS approaches attempt to solve this problem with two key steps: sampling ... | [] | [
{
"authors": [
"Thomas Chau",
"Łukasz Dudziak",
"Mohamed S Abdelfattah",
"Royson Lee",
"Hyeji Kim",
"Nicholas D Lane"
],
"title": "Brp-nas: Prediction-based nas using gcns",
"venue": "arXiv preprint arXiv:2007.08668,",
"year": 2020
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "Neural Architecture Search (NAS) has become a central topic in recent years with great progress (Liu et al., 2018b; Luo et al., 2018; Wu et al., 2019; Howard et al., 2019; Ning et al., 2020; Wei et al., 2020; Luo et al., 2018; Wen et al., 2019; Chau et al., 2020; Luo ... | 2,020 | WEAK NAS PREDICTOR IS ALL YOU NEED |
SP:720f167592297c58d88272599fb66978f3ae8001 | [
"This paper studies the problem of gradient attack in deep learning models. In particular, this paper tries to form a system of linear equations to find a training data point when the gradient of the deep learning model with respect to that data point is available. The algorithm for finding the data point is calle... | Federated learning frameworks have been regarded as a promising approach to break the dilemma between demands on privacy and the promise of learning from large collections of distributed data. Many such frameworks only ask collaborators to share their local update of a common model, i.e. gradients, instead of exposing ... | [
{
"affiliations": [],
"name": "Junyi Zhu"
}
] | [
{
"authors": [
"Peter L Bartlett",
"Michael I Jordan",
"Jon D McAuliffe"
],
"title": "Convexity, classification, and risk bounds",
"venue": "Journal of the American Statistical Association,",
"year": 2006
},
{
"authors": [
"Emiliano De Cristofaro"
],
"titl... | [
{
"heading": "1 INTRODUCTION",
"text": "Distributed and federated learning have become common strategies for training neural networks without transferring data (Jochems et al., 2016; 2017; Konečný et al., 2016; McMahan et al., 2017). Instead, model updates, often in the form of gradients, are exchanged be... | 2,021 | R-GAP: RECURSIVE GRADIENT ATTACK ON PRIVACY |
SP:6cf84af3e1ae0c84dc251ba41a5acb3dc7f61645 | [
"Considering a continuous time RNN with Lipschitz-continuous nonlinearity, the authors formulate sufficient conditions on the parameter matrices for the network to be globally stable, in the sense of a globally attracting fixed point. They provide a specific parameterization for the hidden-to-hidden weight matrices... | Viewing recurrent neural networks (RNNs) as continuous-time dynamical systems, we propose a recurrent unit that describes the hidden state’s evolution with two parts: a well-understood linear component plus a Lipschitz nonlinearity. This particular functional form facilitates stability analysis of the long-term behavio... | [
{
"affiliations": [],
"name": "Omri Azencot"
},
{
"affiliations": [],
"name": "Alejandro Queiruga"
},
{
"affiliations": [],
"name": "Michael W. Mahoney"
}
] | [
{
"authors": [
"Martin Arjovsky",
"Amar Shah",
"Yoshua Bengio"
],
"title": "Unitary evolution recurrent neural networks",
"venue": "In International Conference on Machine Learning,",
"year": 2016
},
{
"authors": [
"Yoshua Bengio",
"Patrice Simard",
"Pa... | [
{
"heading": "1 INTRODUCTION",
"text": "Many interesting problems exhibit temporal structures that can be modeled with recurrent neural networks (RNNs), including problems in robotics, system identification, natural language processing, and machine learning control. In contrast to feed-forward neural networ... | 2,021 | null |
SP:2ad12575818f72f453eb0c04c953a48be56e80e3 | [
"In continual learning settings, one of the important technique for avoiding catastrophe forgetting is to replay data points from the past. For memory efficiency purposes, representative samples can be generated from a generative model, such as GANs, rather than replaying the original samples which can be large in ... | The two main impediments to continual learning are catastrophic forgetting and memory limitations on the storage of data. To cope with these challenges, we propose a novel, cognitively-inspired approach which trains autoencoders with Neural Style Transfer to encode and store images. During training on a new task, recon... | [
{
"affiliations": [],
"name": "CONTINUAL LEARNING"
},
{
"affiliations": [],
"name": "Ali Ayub"
},
{
"affiliations": [],
"name": "Alan R. Wagner"
}
] | [
{
"authors": [
"Ali Ayub",
"Alan R. Wagner"
],
"title": "Centroid based concept learning for rgb-d indoor scene classification",
"venue": "In British Machine Vision Conference (BMVC),",
"year": 2020
},
{
"authors": [
"Ali Ayub",
"Alan R. Wagner"
],
"title"... | [
{
"heading": "1 INTRODUCTION",
"text": "Humans continue to learn new concepts over their lifetime without the need to relearn most previous concepts. Modern machine learning systems, however, require the complete training data to be available at one time (batch learning) (Girshick, 2015). In this paper, we ... | 2,021 | null |
SP:da8ca392a4eb366f4fdedb09d461ef804615b0b2 | [
"In this paper, the authors propose a latent space regression method for analyzing and manipulating the latent space of pre-trained GAN models. Unlike existing optimization-based methods, an explicit latent code regressor is learned to map the input to the latent space. The authors apply this approach to several ap... | In recent years, Generative Adversarial Networks have become ubiquitous in both research and public perception, but how GANs convert an unstructured latent code to a high quality output is still an open question. In this work, we investigate regression into the latent space as a probe to understand the compositional pr... | [
{
"affiliations": [],
"name": "Lucy Chai"
},
{
"affiliations": [],
"name": "Jonas Wulff"
},
{
"affiliations": [],
"name": "Phillip Isola"
}
] | [
{
"authors": [
"Rameen Abdal",
"Yipeng Qin",
"Peter Wonka"
],
"title": "Image2stylegan: How to embed images into the stylegan latent space",
"venue": null,
"year": 2019
},
{
"authors": [
"Rameen Abdal",
"Yipeng Qin",
"Peter Wonka"
],
"title": "... | [
{
"heading": "1 INTRODUCTION",
"text": "Natural scenes are comprised of disparate parts and objects that humans can easily segment and interchange (Biederman, 1987). Recently, unconditional generative adversarial networks (Karras et al., 2017; 2019b;a; Radford et al., 2015) have become capable of mimicking ... | 2,021 | USING LATENT SPACE REGRESSION TO ANALYZE AND LEVERAGE COMPOSITIONALITY IN GANS |
SP:c0e827c33dbc9378404fe2a0949198cb74f13688 | [
"The authors propose a new way to aggregate the embeddings of elements in a set (or sequence) by comparing it with respect to (trainable) reference set(s) via Optimal Transport (OT). The motivation to build such a pooling operation is derived from self-attention and the authors suggest an OT spin to that (e.g., the... | We address the problem of learning on sets of features, motivated by the need of performing pooling operations in long biological sequences of varying sizes, with long-range dependencies, and possibly few labeled data. To address this challenging task, we introduce a parametrized representation of fixed size, which emb... | [
{
"affiliations": [],
"name": "Grégoire Mialon"
},
{
"affiliations": [],
"name": "Dexiong Chen"
},
{
"affiliations": [],
"name": "Alexandre d’Aspremont"
},
{
"affiliations": [],
"name": "Julien Mairal"
}
] | [] | [
{
"heading": "1 INTRODUCTION",
"text": "Many scientific fields such as bioinformatics or natural language processing (NLP) require processing sets of features with positional information (biological sequences, or sentences represented by a set of local features). These objects are delicate to manipulate due... | 2,021 | A TRAINABLE OPTIMAL TRANSPORT EMBEDDING |
SP:a85b6d598513c8e03a013fd20da6b19a1108f71e | [
"This paper extends and explains how to apply the \"free energy principle\" and active inference to RL and imitation learning. They implement a neural network approximation of losses derived this way and test on some control tasks. Importantly the tasks focus on here are imitation + control tasks. That is, there is... | Imitation Learning (IL) and Reinforcement Learning (RL) from high dimensional sensory inputs are often introduced as separate problems, but a more realistic problem setting is how to merge the techniques so that the agent can reduce exploration costs by partially imitating experts at the same time it maximizes its retu... | [] | [
{
"authors": [
"Karl Friston"
],
"title": "The free-energy principle: a unified brain theory",
"venue": "Nature reviews neuroscience,",
"year": 2010
},
{
"authors": [
"Karl Friston",
"James Kilner",
"Lee Harrison"
],
"title": "A free energy principle for t... | [
{
"heading": "1 INTRODUCTION",
"text": "Imitation Learning (IL) is a framework to learn a policy to mimic expert trajectories. As the expert specifies model behaviors, there is no need to do exploration or to design complex reward functions. Reinforcement Learning (RL) does not have these features, so RL ag... | 2,020 | null |
SP:69855e0bec141e9d15eec5cc37022f313e6600b2 | [
"By the first look, this work itself does not introduce any new architecture or novel algorithm. It takes what is considered as the popular choices in generating classifier saliency masks, and conducts quite extensive sets of experiments to dissect the components by their importance. The writing is pretty clear in ... | Saliency maps that identify the most informative regions of an image for a classifier are valuable for model interpretability. A common approach to creating saliency maps involves generating input masks that mask out portions of an image to maximally deteriorate classification performance, or mask in an image to preser... | [
{
"affiliations": [],
"name": "SIMPLIFYING MASKING-BASED"
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] | [
{
"authors": [
"Julius Adebayo",
"Justin Gilmer",
"Michael Muelly",
"Ian Goodfellow",
"Moritz Hardt",
"Been Kim"
],
"title": "Sanity checks for saliency maps",
"venue": "In NeurIPS,",
"year": 2018
},
{
"authors": [
"Chirag Agarwal",
"Anh Ng... | [
{
"heading": "1 INTRODUCTION",
"text": "The success of CNNs (Krizhevsky et al., 2012; Szegedy et al., 2015; He et al., 2016; Tan & Le, 2019) has prompted interest in improving understanding of how these models make their predictions. Particularly in applications such as medical diagnosis, having models expl... | 2,020 | null |
SP:e4e5b4e2bee43c920ed719dc331a370129845268 | [
"The authors propose a model to improve the output distribution of neural nets in image classification problems. Their model is a post hoc procedure and is based on the tree structure of WordNet. The model revises the classifier output based on the distance of the labels in the tree. Intuitively, their solution is ... | There has been increasing interest in building deep hierarchy-aware classifiers that aim to quantify and reduce the severity of mistakes, and not just reduce the number of errors. The idea is to exploit the label hierarchy (e.g., the WordNet ontology) and consider graph distances as a proxy for mistake severity. Surpri... | [
{
"affiliations": [],
"name": "DEEP NETWORKS"
},
{
"affiliations": [],
"name": "Shyamgopal Karthik"
},
{
"affiliations": [],
"name": "Ameya Prabhu"
},
{
"affiliations": [],
"name": "Puneet K. Dokania"
}
] | [
{
"authors": [
"Naoki Abe",
"Bianca Zadrozny",
"John Langford"
],
"title": "An iterative method for multi-class cost-sensitive learning",
"venue": "In KDD,",
"year": 2004
},
{
"authors": [
"Zeynep Akata",
"Scott Reed",
"Daniel Walter",
"Honglak L... | [
{
"heading": "1 INTRODUCTION",
"text": "The conventional performance measure of accuracy for image classification treats all classes other than ground truth as equally wrong. However, some mistakes may have a much higher impact than others in real-world applications. An intuitive example being an autonomous... | 2,021 | null |
SP:7cc59c8f556d03597f7ab391ef14d1a96191a4db | [
"The design of a useful generalization of neural networks on quantum computers has been challenging because the gradient signal will decay exponentially with respect to the depth of the quantum circuit (saturating to exponentially small in system size after the depth is linear in system size). This work provides a ... | Quantum Neural Networks (QNNs) have been recently proposed as generalizations of classical neural networks to achieve the quantum speed-up. Despite the potential to outperform classical models, serious bottlenecks exist for training QNNs; namely, QNNs with random structures have poor trainability due to the vanishing g... | [
{
"affiliations": [],
"name": "TOWARD TRAINABILITY"
}
] | [
{
"authors": [
"Frank Arute",
"Kunal Arya",
"Ryan Babbush",
"Dave Bacon",
"Joseph C Bardin",
"Rami Barends",
"Rupak Biswas",
"Sergio Boixo",
"Fernando GSL Brandao",
"David A Buell"
],
"title": "Quantum supremacy using a programmable superconduc... | [
{
"heading": "1 INTRODUCTION",
"text": "Neural Networks (Hecht-Nielsen, 1992) using gradient-based optimizations have dramatically advanced researches in discriminative models, generative models, and reinforcement learning. To efficiently utilize the parameters and practically improve the trainability, neur... | 2,020 | null |
SP:8a8aa5f245c2fb82beddb19c82dddb8d67f66f8a | [
"In this paper, the authors introduce a class of games called Hidden Convex-Concave where a (stricly) convex-concave potential is composed with smooth maps. On this class of problems, they show that the continuous gradient dynamics converge to (a neighbordhood of) the minimax solutions of the problem. This is an ex... | Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex nonconcave zero-sum games, that we call hidden zero-sum games. In this class, pl... | [
{
"affiliations": [],
"name": "Lampros Flokas"
}
] | [
{
"authors": [
"Jacob Abernethy",
"Kevin A Lai",
"Kfir Y Levy",
"Jun-Kun Wang"
],
"title": "Faster rates for convex-concave games",
"venue": "In COLT,",
"year": 2018
},
{
"authors": [
"Jacob Abernethy",
"Kevin A Lai",
"Andre Wibisono"
],
... | [
{
"heading": "1 Introduction",
"text": "Traditionally, our understanding of convex-concave games revolves around von Neumann’s celebrated minimax theorem, which implies the existence of saddle point solutions with a uniquely defined value. These solutions are called von Nemann solutions and guarantee to eac... | 2,022 | Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent |
SP:5e9b5c3ee27cf90eb73e2672a1bbf18a1b12e791 | [
"This paper shows a correspondence between deep neural networks (DNN) trained with noisy gradients and NNGP. It provides a general analytical form for the finite width correction (FWC) for NNSP expanding around NNGP. Finally, it argues that this FWC can be used to explain why finite width CNNs can improve the perfo... | A recent line of works studied wide deep neural networks (DNNs) by approximating them as Gaussian Processes (GPs). A DNN trained with gradient flow was shown to map to a GP governed by the Neural Tangent Kernel (NTK), whereas earlier works showed that a DNN with an i.i.d. prior over its weights maps to the socalled Neu... | [
{
"affiliations": [],
"name": "NOISY GRADIENTS"
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] | [
{
"authors": [
"Sanjeev Arora",
"Simon S. Du",
"Wei Hu",
"Zhiyuan Li",
"Ruslan Salakhutdinov",
"Ruosong Wang"
],
"title": "On Exact Computation with an Infinitely Wide Neural Net",
"venue": "arXiv e-prints, art",
"year": 2019
},
{
"authors": [
"R... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep neural networks (DNNs) have been rapidly advancing the state-of-the-art in machine learning, yet a complete analytic theory remains elusive. Recently, several exact results were obtained in the highly over-parameterized regime (N →∞ where N denotes the width or n... | 2,020 | null |
SP:95899f38fd0f1789510e67178b587c08a14203f5 | [
"This paper proposes adding regularization terms to encourage diversity of the layer outputs in order to improve the generalization performance. The proposed idea is an extension of Cogswell's work with different regularization terms. In addition, the authors performed detailed generalization analysis based on the ... | During the last decade, neural networks have been intensively used to tackle various problems and they have often led to state-of-the-art results. These networks are composed of multiple jointly optimized layers arranged in a hierarchical structure. At each layer, the aim is to learn to extract hidden patterns needed t... | [] | [
{
"authors": [
"Madhu S Advani",
"Andrew M Saxe",
"Haim Sompolinsky"
],
"title": "High-dimensional dynamics of generalization error in neural networks",
"venue": "Neural Networks,",
"year": 2020
},
{
"authors": [
"Sanjeev Arora",
"Rong Ge",
"Behnam Ney... | [
{
"heading": "1 INTRODUCTION",
"text": "Neural networks are a powerful class of non-linear function approximators that have been successfully used to tackle a wide range of problems. They have enabled breakthroughs in many tasks, such as image classification (Krizhevsky et al., 2012), speech recognition (Hi... | 2,020 | ON NEURAL NETWORK GENERALIZATION VIA PROMOTING WITHIN-LAYER ACTIVATION DIVERSITY |
SP:4fd499ebe9fddb6a3f57663d76bb7bf3b5f29ef7 | [
"The proposed NDP has two main advantages: 1- it has the capability to adapt the incoming data points in time-series (unlike NODE) without retraining, 2- it can provide a measure of uncertainty for the underlying dynamics of the time-series. NDP partitions the global latent context $z$ to a latent position $l$ and ... | Neural Ordinary Differential Equations (NODEs) use a neural network to model the instantaneous rate of change in the state of a system. However, despite their apparent suitability for dynamics-governed time-series, NODEs present a few disadvantages. First, they are unable to adapt to incoming data-points, a fundamental... | [
{
"affiliations": [],
"name": "Alexander Norcliffe"
},
{
"affiliations": [],
"name": "Cristian Bodnar"
},
{
"affiliations": [],
"name": "Ben Day"
},
{
"affiliations": [],
"name": "Jacob Moss"
},
{
"affiliations": [],
"name": "Pietro Liò"
}
] | [
{
"authors": [
"Francesco Paolo Casale",
"Adrian V Dalca",
"Luca Saglietti",
"Jennifer Listgarten",
"Nicolo Fusi"
],
"title": "Gaussian Process Prior Variational Autoencoders",
"venue": "arXiv e-prints, art",
"year": 2018
},
{
"authors": [
"Ricky TQ Ch... | [
{
"heading": "1 INTRODUCTION",
"text": "Many time-series that arise in the natural world, such as the state of a harmonic oscillator, the populations in an ecological network or the spread of a disease, are the product of some underlying dynamics. Sometimes, as in the case of a video of a swinging pendulum,... | 2,021 | NEURAL ODE PROCESSES |
SP:1c2c08605956eb4660a8f8a33ce13e80276582ed | [
"This paper proposes a data-driven approach to choose an informative surrogate sub-dataset, termed \"a \\epsilon-approximation\", from the original data set. A meta-learning algorithm called Kernel Inducing Points (KIP ) is proposed to obtain such sub-datasets for (Linear) Kernel Ridge Regression (KRR), with the p... | One of the most fundamental aspects of any machine learning algorithm is the training data used by the algorithm. We introduce the novel concept of approximation of datasets, obtaining datasets which are much smaller than or are significant corruptions of the original training data while maintaining similar model perfo... | [
{
"affiliations": [],
"name": "Timothy Nguyen"
},
{
"affiliations": [],
"name": "Zhourong Chen"
},
{
"affiliations": [],
"name": "Jaehoon Lee"
}
] | [
{
"authors": [
"Martin Abadi",
"Andy Chu",
"Ian Goodfellow",
"H. Brendan McMahan",
"Ilya Mironov",
"Kunal Talwar",
"Li Zhang"
],
"title": "Deep learning with differential privacy",
"venue": "Proceedings of the 2016 ACM SIGSAC Conference on Computer and Commu... | [
{
"heading": null,
"text": "One of the most fundamental aspects of any machine learning algorithm is the training data used by the algorithm. We introduce the novel concept of - approximation of datasets, obtaining datasets which are much smaller than or are significant corruptions of the original training ... | 2,021 | null |
SP:c06539b9986064977dec933dcce4b81d42f47cc2 | [
"This paper focuses on the problem of multi-agent cooperation in social dilemmas, in which mutual defection is individually rational but collectively suboptimal. The authors use the bias toward status-quo in human psychology to motivate a new training method, called SQLoss: 1) for repeated matrix games, each agent ... | Individual rationality, which involves maximizing expected individual return, does not always lead to optimal individual or group outcomes in multi-agent problems. For instance, in social dilemma situations, Reinforcement Learning (RL) agents trained to maximize individual rewards converge to mutual defection that is i... | [] | [
{
"authors": [
"Dilip Abreu",
"David Pearce",
"Ennio Stacchetti"
],
"title": "Toward a theory of discounted repeated games with imperfect monitoring",
"venue": "URL http://www.jstor.org/stable/2938299",
"year": 1990
},
{
"authors": [
"Robert Axelrod"
],
"t... | [
{
"heading": "1 INTRODUCTION",
"text": "In sequential social dilemmas, individually rational behavior leads to outcomes that are sub-optimal for each individual in the group (Hardin, 1968; Ostrom, 1990; Ostrom et al., 1999; Dietz et al., 2003). Current state-of-the-art Multi-Agent Deep Reinforcement Learnin... | 2,020 | null |
SP:72f379cefb57913386cbd76978943bdc8d0545a7 | [
"The work uses diffusion probabilistic models for conditional speech synthesis tasks, specifically to convert mel-spectrogram to the raw audio waveform. Results from the proposed approach match the state-of-the-art WaveRNN model. The paper is very well-written and it is quite easy to follow. The study of the total ... | This paper introduces WaveGrad, a conditional model for waveform generation which estimates gradients of the data density. The model is built on prior work on score matching and diffusion probabilistic models. It starts from a Gaussian white noise signal and iteratively refines the signal via a gradient-based sampler c... | [
{
"affiliations": [],
"name": "Nanxin Chen"
},
{
"affiliations": [],
"name": "Yu Zhang"
},
{
"affiliations": [],
"name": "Heiga Zen"
},
{
"affiliations": [],
"name": "Ron J. Weiss"
},
{
"affiliations": [],
"name": "Mohammad Norouzi"
},
{
"affiliations"... | [
{
"authors": [
"Yang Ai",
"Zhen-Hua Ling"
],
"title": "Knowledge-and-Data-Driven Amplitude Spectrum Prediction for Hierarchical Neural Vocoders",
"venue": "arXiv preprint arXiv:2004.07832,",
"year": 2020
},
{
"authors": [
"Eric Battenberg",
"RJ Skerry-Ryan",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep generative models have revolutionized speech synthesis (Oord et al., 2016; Sotelo et al., 2017; Wang et al., 2017; Biadsy et al., 2019; Jia et al., 2019; Vasquez & Lewis, 2019). Autoregressive models, in particular, have been popular for raw audio generation than... | 2,021 | WAVEGRAD: ESTIMATING GRADIENTS FOR WAVEFORM GENERATION |
SP:11cd869cd8c6dc657c136545fd2029f0c49843ba | [
"The paper presents a benchmark / dataset, HW-NAS-Bench, for evaluating various neural architecture search algorithms. The benchmark is based on extensive measurements on real hardware. An important goal with the proposal is to support neural architecture searches for non-hardware experts. Further, the paper provid... | HardWare-aware Neural Architecture Search (HW-NAS) has recently gained tremendous attention by automating the design of deep neural networks deployed in more resource-constrained daily life devices. Despite its promising performance, developing optimal HW-NAS solutions can be prohibitively challenging as it requires cr... | [
{
"affiliations": [],
"name": "SEARCH BENCHMARK"
},
{
"affiliations": [],
"name": "Chaojian Li"
},
{
"affiliations": [],
"name": "Zhongzhi Yu"
},
{
"affiliations": [],
"name": "Yonggan Fu"
},
{
"affiliations": [],
"name": "Yongan Zhang"
},
{
"affiliati... | [
{
"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": "The recent performance breakthroughs of deep neural networks (DNNs) have attracted an explosion of research in designing efficient DNNs, aiming to bring powerful yet power-hungry DNNs into more resource-constrained daily life devices for enabling various DNN-powered i... | 2,021 | null |
SP:f65217b47950d0dbf8e77622489d8883211a012d | [
"This paper proposes a novel graph neural network-based architecture. Building upon the theoretical success of graph scattering transforms, the authors propose to learn some aspects of it providing them with more flexibility to adapt to data (recall that graph scattering transforms are built on pre-designed graph w... | Many popular graph neural network (GNN) architectures, which are often considered as the current state of the art, rely on encoding graph structure via smoothness or similarity between neighbors. While this approach performs well on a surprising number of standard benchmarks, the efficacy of such models does not transl... | [] | [
{
"authors": [
"Uri Alon",
"Eran Yahav"
],
"title": "On the bottleneck of graph neural networks and its practical implications",
"venue": "arXiv preprint arXiv:2006.05205,",
"year": 2020
},
{
"authors": [
"Pablo Barceló",
"Egor V Kostylev",
"Mikael Monet",
... | [
{
"heading": "1 INTRODUCTION",
"text": "Geometric deep learning has recently emerged as an increasingly prominent branch of machine learning in general, and deep learning in particular (Bronstein et al., 2017). It is based on the observation that many of the impressive achievements of neural networks come i... | 2,020 | null |
SP:c90a894d965bf8e529df296b9d5c76864aa5f4f9 | [
"This paper describes a neural vocoder based on a diffusion probabilistic model. The model utilizes a fixed-length markov chain to convert between a latent uncorrelated Gaussian vector and a full-length observation. The conversion from observation to latent is fixed and amounts to adding noise at each step. The con... | In this work, we propose DiffWave, a versatile diffusion probabilistic model for conditional and unconditional waveform generation. The model is non-autoregressive, and converts the white noise signal into structured waveform through a Markov chain with a constant number of steps at synthesis. It is efficiently trained... | [
{
"affiliations": [],
"name": "Zhifeng Kong"
},
{
"affiliations": [],
"name": "Wei Ping"
},
{
"affiliations": [],
"name": "Jiaji Huang"
},
{
"affiliations": [],
"name": "Kexin Zhao"
}
] | [
{
"authors": [
"Yang Ai",
"Zhen-Hua Ling"
],
"title": "A neural vocoder with hierarchical generation of amplitude and phase spectra for statistical parametric speech synthesis",
"venue": "IEEE/ACM Transactions on Audio, Speech, and Language Processing,",
"year": 2020
},
{
"au... | [
{
"heading": "1 INTRODUCTION",
"text": "Deep generative models have produced high-fidelity raw audio in speech synthesis and music generation. In previous work, likelihood-based models, including autoregressive models (van den Oord et al., 2016; Kalchbrenner et al., 2018; Mehri et al., 2017) and flow-based ... | 2,021 | DIFFWAVE: A VERSATILE DIFFUSION MODEL FOR AUDIO SYNTHESIS |
SP:efbb0e2e944f1d810a6f0b6bc71e636af9ae9c13 | [
"The authors present a seq2seq model with a sparse transformer encoder and an LSTM decoder. They utilize a learning curriculum wherein the autoregressive decoder is initially trained using teacher forcing and is gradually fed its past predictions as training progresses. The authors introduce a new dataset for long ... | Dancing to music is one of human’s innate abilities since ancient times. In machine learning research, however, synthesizing dance movements from music is a challenging problem. Recently, researchers synthesize human motion sequences through autoregressive models like recurrent neural network (RNN). Such an approach of... | [
{
"affiliations": [],
"name": "CURRICULUM LEARNING"
},
{
"affiliations": [],
"name": "Ruozi Huang"
},
{
"affiliations": [],
"name": "Huang Hu"
},
{
"affiliations": [],
"name": "Wei Wu"
},
{
"affiliations": [],
"name": "Kei Sawada"
},
{
"affiliations": ... | [
{
"authors": [
"Samy Bengio",
"Oriol Vinyals",
"Navdeep Jaitly",
"Noam Shazeer"
],
"title": "Scheduled sampling for sequence prediction with recurrent neural networks",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2015
},
{
"authors": ... | [
{
"heading": "1 INTRODUCTION",
"text": "Arguably, dancing to music is one of human’s innate abilities, as we can spontaneously sway along with the tempo of music we hear. The research in neuropsychology indicates that our brain is hardwired to make us move and synchronize with music regardless of our intent... | 2,021 | DANCE REVOLUTION: LONG-TERM DANCE GENERA- |
SP:18e9f58ab4fc8532cbd298730cff5b7f8ec31a5f | [
"This paper presents the \"Block Skim Transformer\" for extractive question answering tasks. The key idea in this model is using a classifier, on the self-attention distributions of a particular layer, to classify whether a large spans of non-contiguous text (blocks) contain the answer. If a block is rejected by th... | Transformer-based encoder models have achieved promising results on natural language processing (NLP) tasks including question answering (QA). Different from sequence classification or language modeling tasks, hidden states at all positions are used for the final classification in QA. However, we do not always need all... | [
{
"affiliations": [],
"name": "SKIM TRANSFORMER"
}
] | [
{
"authors": [
"Iz Beltagy",
"Matthew E Peters",
"Arman Cohan"
],
"title": "Longformer: The long-document transformer",
"venue": "arXiv preprint arXiv:2004.05150,",
"year": 2020
},
{
"authors": [
"Victor Campos",
"Brendan Jou",
"Xavier Giró-i-Nieto",
... | [
{
"heading": null,
"text": "Transformer-based encoder models have achieved promising results on natural language processing (NLP) tasks including question answering (QA). Different from sequence classification or language modeling tasks, hidden states at all positions are used for the final classification i... | 2,020 | null |
SP:977fc8d3bb7266d1beaecc609a91970783347ed3 | [
"The authors discuss how a classifier’s performance over the initial class sample can be used to extrapolate its expected accuracy on a larger, unobserved set of classes by mean of the dual of the ROC function, swapping the roles of classes and samples. Grounded on such function, the authors develop a novel ANN app... | Multiclass classifiers are often designed and evaluated only on a sample from the classes on which they will eventually be applied. Hence, their final accuracy remains unknown. In this work we study how a classifier’s performance over the initial class sample can be used to extrapolate its expected accuracy on a larger... | [
{
"affiliations": [],
"name": "Yuli Slavutsky"
},
{
"affiliations": [],
"name": "Yuval Benjamini"
}
] | [
{
"authors": [
"Felix Abramovich",
"Marianna Pensky"
],
"title": "Classification with many classes: challenges and pluses",
"venue": "Journal of Multivariate Analysis,",
"year": 2019
},
{
"authors": [
"Brandon Amos",
"Bartosz Ludwiczuk",
"Mahadev Satyanaraya... | [
{
"heading": "1 INTRODUCTION",
"text": "Advances in machine learning and representation learning led to automatic systems that can identify an individual class from very large candidate sets. Examples are abundant in visual object recognition (Russakovsky et al., 2015; Simonyan & Zisserman, 2014), face iden... | 2,021 | PREDICTING CLASSIFICATION ACCURACY WHEN ADDING NEW UNOBSERVED CLASSES |
SP:eb5f64c7d1e303394f4650a14806e60dba1afdd3 | [
"The paper presented an adaptive inference model for efficient action recognition in videos. The core of the model is the dynamic gating of feature channels that controls the fusion between two frame features, whereby the gating is conditioned on the input video and helps to reduce the computational cost at runtime... | Temporal modelling is the key for efficient video action recognition. While understanding temporal information can improve recognition accuracy for dynamic actions, removing temporal redundancy and reusing past features can significantly save computation leading to efficient action recognition. In this paper, we introd... | [
{
"affiliations": [],
"name": "Yue Meng"
},
{
"affiliations": [],
"name": "Rameswar Panda"
},
{
"affiliations": [],
"name": "Chung-Ching Lin"
},
{
"affiliations": [],
"name": "Prasanna Sattigeri"
},
{
"affiliations": [],
"name": "Leonid Karlinsky"
},
{
... | [
{
"authors": [
"Sadjad Asghari-Esfeden",
"Mario Sznaier",
"Octavia Camps"
],
"title": "Dynamic motion representation for human action recognition",
"venue": "In The IEEE Winter Conference on Applications of Computer Vision,",
"year": 2020
},
{
"authors": [
"Emmanu... | [
{
"heading": "1 INTRODUCTION",
"text": "Over the last few years, video action recognition has made rapid progress with the introduction of a number of large-scale video datasets (Carreira & Zisserman, 2017; Monfort et al., 2018; Goyal et al., 2017). Despite impressive results on commonly used benchmark data... | 2,021 | ADAFUSE: ADAPTIVE TEMPORAL FUSION NETWORK FOR EFFICIENT ACTION RECOGNITION |
SP:8b0cee077c1bcdf9a546698dc041654ca6a222ed | [
"This paper is basically unreadable. The sentence structure / grammar is strange, and if that was the only issue it could be overlooked. The paper also does not describe or explain the motivation and interpretation of anything, but instead just lists equations. For example, eta is the parameter that projects a sphe... | We present geometric Bayesian active learning by disagreements (GBALD), a framework that performs BALD on its geometric interpretation interacting with a deep learning model. There are two main components in GBALD: initial acquisitions based on core-set construction and model uncertainty estimation with those initial a... | [] | [
{
"authors": [
"Jordan T Ash",
"Chicheng Zhang",
"Akshay Krishnamurthy",
"John Langford",
"Alekh Agarwal"
],
"title": "Deep batch active learning by diverse, uncertain gradient lower bounds",
"venue": "In International Conference on Learning Representations,",
"year... | [
{
"heading": "1 INTRODUCTION",
"text": "Lack of training labels restricts the performance of deep neural networks (DNNs), though prices of GPU resources were falling fast. Recently, leveraging the abundance of unlabeled data has become a potential solution to relieve this bottleneck whereby expert knowledge... | 2,020 | null |
SP:09bbd1a342033a65e751a8878c23e3fa6facc636 | [
"The authors propose a convolution as a message passing of node features over edges where messages are aggregated weighted by a \"direction\" edge field. Furthermore, the authors propose to use the gradients of Laplace eigenfunctions as direction fields. Presumably, the aggregation is done with different direction ... | In order to overcome the expressive limitations of graph neural networks (GNNs), we propose the first method that exploits vector flows over graphs to develop globally consistent directional and asymmetric aggregation functions. We show that our directional graph networks (DGNs) generalize convolutional neural networks... | [] | [
{
"authors": [
"Uri Alon",
"Eran Yahav"
],
"title": "On the bottleneck of graph neural networks and its practical implications",
"venue": null,
"year": 2020
},
{
"authors": [
"Xavier Bresson",
"Thomas Laurent"
],
"title": "Residual gated graph convnets",
... | [
{
"heading": "1 INTRODUCTION",
"text": "One of the most important distinctions between convolutional neural networks (CNNs) and graph neural networks (GNNs) is that CNNs allow for any convolutional kernel, while most GNN methods are limited to symmetric kernels (also called isotropic kernels in the literatu... | 2,020 | null |
SP:540d8c615b5193239aa43717de8cacc749ccc4c6 | [
"The authors describe a method for representing a continuous signal by a pulse code, in a manner inspired by auditory processing in the brain. The resulting framework is somewhat like matching pursuit except that filters are run a single time in a causal manner to find the spike times (which would be faster than MP... | In many animal sensory pathways, the transformation from external stimuli to spike trains is essentially deterministic. In this context, a new mathematical framework for coding and reconstruction, based on a biologically plausible model of the spiking neuron, is presented. The framework considers encoding of a signal t... | [] | [
{
"authors": [
"Horace B Barlow"
],
"title": "Possible principles underlying the transformations of sensory messages",
"venue": "Sensory Communication,",
"year": 1961
},
{
"authors": [
"Stephen Boyd",
"Leon Chua"
],
"title": "Fading memory and the problem of app... | [
{
"heading": null,
"text": "In many animal sensory pathways, the transformation from external stimuli to spike trains is essentially deterministic. In this context, a new mathematical framework for coding and reconstruction, based on a biologically plausible model of the spiking neuron, is presented. The fr... | 2,020 | null |
SP:725d036c0863e59f6bb0b0bb22cc0ad3a0988126 | [
"Review: This paper studies how to improve contrastive divergence (CD) training of energy-based models (EBMs) by revisiting the gradient term neglected in the traditional CD learning. This paper also introduces some useful techniques, such as data augmentation, multi-scale energy design, and reservoir sampling to i... | We propose several different techniques to improve contrastive divergence training of energy-based models (EBMs). We first show that a gradient term neglected in the popular contrastive divergence formulation is both tractable to estimate and is important to avoid training instabilities in previous models. We further h... | [] | [
{
"authors": [
"Sergey Bartunov",
"Jack W Rae",
"Simon Osindero",
"Timothy P Lillicrap"
],
"title": "Meta-learning deep energy-based memory models",
"venue": "arXiv preprint arXiv:1910.02720,",
"year": 2019
},
{
"authors": [
"Jan Beirlant",
"E. Dudewic... | [
{
"heading": "1 INTRODUCTION",
"text": "Energy-Based models (EBMs) have received an influx of interest recently and have been applied to realistic image generation (Han et al., 2019; Du & Mordatch, 2019), 3D shapes synthesis (Xie et al., 2018b) , out of distribution and adversarial robustness (Lee et al., 2... | 2,020 | null |
SP:6d6e083899bc17a2733aa16efd259ad4ed2076d6 | [
"This paper falls into a class of continual learning methods which accommodate for new tasks by expanding the network architecture, while freezing existing weights. This freezing trivially resolves forgetting. The (hard) problem of determining how to expand the network is tackled with reinforcement learning, largel... | Continual learning with neural networks is an important learning framework in AI that aims to learn a sequence of tasks well. However, it is often confronted with three challenges: (1) overcome the catastrophic forgetting problem, (2) adapt the current network to new tasks, and meanwhile (3) control its model complexit... | [] | [
{
"authors": [
"Rahaf Aljundi",
"Francesca Babiloni",
"Mohamed Elhoseiny",
"Marcus Rohrbach",
"Tinne Tuytelaars"
],
"title": "Memory aware synapses: Learning what (not) to forget",
"venue": "In Proceedings of the European Conference on Computer Vision,",
"year": 201... | [
{
"heading": "1 INTRODUCTION",
"text": "Continual learning, or lifelong learning, refers to the ability of continually learning new tasks and also performing well on learned tasks. It has attracted enormous attention in AI as it mimics a human learning process - constantly acquiring and accumulating knowled... | 2,020 | null |
SP:047761908963bea6350f5d65a253c09f1a626093 | [
"The authors contribute an approach to automatically distinguish between good and bad student assignment submissions by modeling the assignment submissions as MDPs. The authors hypothesize that satisfactory assignments modeled as MDPs will be more alike than they are to unsatisfactory assignments. Therefore this ca... | Contemporary coding education often present students with the task of developing programs that have user interaction and complex dynamic systems, such as mouse based games. While pedagogically compelling, grading such student programs requires dynamic user inputs, therefore they are difficult to grade by unit tests. In... | [] | [
{
"authors": [
"Karl Cobbe",
"Christopher Hesse",
"Jacob Hilton",
"John Schulman"
],
"title": "Leveraging procedural generation to benchmark reinforcement learning",
"venue": "arXiv preprint arXiv:1912.01588,",
"year": 2019
},
{
"authors": [
"Karl Cobbe",
... | [
{
"heading": "1 INTRODUCTION",
"text": "The rise of online coding education platforms accelerates the trend to democratize high quality computer science education for millions of students each year. Corbett (2001) suggests that providing feedback to students can have an enormous impact on efficiently and ef... | 2,020 | null |
SP:2eed06887f51560197590d617b1a37ec6d22e943 | [
"This paper considers the problem of data-free post-training quantization of classfication networks. It proposes three extensions of an existing framework ZeroQ (Cai et al., 2020): (1). in order to generate distilled data for network sensitivity analysis, the \"Retro Synthesis\" method is proposed to turn a random ... | Existing quantization aware training methods attempt to compensate for the quantization loss by leveraging on training data, like most of the post-training quantization methods, and are also time consuming. Both these methods are not effective for privacy constraint applications as they are tightly coupled with trainin... | [] | [
{
"authors": [
"Ron Banner",
"Yury Nahshan",
"Daniel Soudry"
],
"title": "Post training 4-bit quantization of convolutional networks for rapid-deployment",
"venue": "In Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"Chaim Bask... | [
{
"heading": "1 INTRODUCTION",
"text": "Quantization is a widely used and necessary approach to convert heavy Deep Neural Network (DNN) models in Floating Point (FP32) format to a light-weight lower precision format, compatible with edge device inference. The introduction of lower precision computing hardwa... | 2,020 | null |
SP:259b64e62b640ccba4bc82c50e59db7662677e6b | [
"The authors propose a bootstrap framework for understanding generalization in deep learning. In particular, instead of the usual decomposition of test error as training error plus the generalization gap, the bootstrap framework decomposes the empirical test error as online error plus the bootstrap error (the gap ... | We propose a new framework for reasoning about generalization in deep learning. The core idea is to couple the Real World, where optimizers take stochastic gradient steps on the empirical loss, to an Ideal World, where optimizers take steps on the population loss. This leads to an alternate decomposition of test error ... | [
{
"affiliations": [],
"name": "OFFLINE GENERALIZERS"
},
{
"affiliations": [],
"name": "Preetum Nakkiran"
},
{
"affiliations": [],
"name": "Behnam Neyshabur"
},
{
"affiliations": [],
"name": "Hanie Sedghi"
}
] | [
{
"authors": [
"Zeyuan Allen-Zhu",
"Yuanzhi Li",
"Yingyu Liang"
],
"title": "Learning and generalization in overparameterized neural networks, going beyond two layers",
"venue": "In Advances in neural information processing systems,",
"year": 2019
},
{
"authors": [
... | [
{
"heading": "1 INTRODUCTION",
"text": "The goal of a generalization theory in supervised learning is to understand when and why trained models have small test error. The classical framework of generalization decomposes the test error of a model ft as:\nTestError(ft) = TrainError(ft) + [TestError(ft)− Train... | 2,021 | null |
SP:1b984693f1a64c86306aff37d58f9ff188bcf67e | [
"This paper presents a general Self-supervised Time Series representation learning framework. It explores the inter-sample relation reasoning and intra-temporal relation reasoning of time series to capture the underlying structure pattern of the unlabeled time series data. The proposed method achieves new state-of... | Self-supervised learning achieves superior performance in many domains by extracting useful representations from the unlabeled data. However, most of traditional self-supervised methods mainly focus on exploring the inter-sample structure while less efforts have been concentrated on the underlying intra-temporal struct... | [] | [
{
"authors": [
"Anthony Bagnall",
"Jason Lines",
"Jon Hills",
"Aaron Bostrom"
],
"title": "Time-series classification with cote: the collective of transformation-based ensembles",
"venue": "IEEE Transactions on Knowledge and Data Engineering,",
"year": 2015
},
{
"... | [
{
"heading": "1 INTRODUCTION",
"text": "Time series data is ubiquitous and there has been significant progress for time series analysis (Das, 1994) in machine learning, signal processing, and other related areas, with many real-world applications such as healthcare (Stevner et al., 2019), industrial diagnos... | 2,020 | null |
SP:9513f146a764d9e67b7d054692d0a923622ff007 | [
"This paper proposes to use orthogonal weight constraints for autoencoders. The authors demonstrate that under orthogonal weights (hence invertible), more features could be extracted. The theory is conducted under linear cases while the authors claim it can be applied to more complicated scenarios such as higher di... | The pressing need for pretraining algorithms has been diminished by numerous advances in terms of regularization, architectures, and optimizers. Despite this trend, we re-visit the classic idea of unsupervised autoencoder pretraining and propose a modified variant that relies on a full reverse pass trained in conjuncti... | [
{
"affiliations": [],
"name": "REVIVING AUTOENCODER PRETRAINING"
}
] | [
{
"authors": [
"REFERENCES Michele Alberti",
"Mathias Seuret",
"Rolf Ingold",
"Marcus Liwicki"
],
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"venue": "arXiv preprint arXiv:1703.04332,",
"year": 2017
},
{
"authors": [
"Lynton Ardizzone",
"Jakob K... | [
{
"heading": "1 INTRODUCTION",
"text": "While approaches such as greedy layer-wise autoencoder pretraining (Bengio et al., 2007; Vincent et al., 2010; Erhan et al., 2010) arguably paved the way for many fundamental concepts of today’s methodologies in deep learning, the pressing need for pretraining neural ... | 2,020 | null |
SP:70fc08b1b6161c770b5019272c2eaa0d2e3c39ee | [
"This paper raises and studies concerns about the generalization of 3D human motion prediction approaches across unseen motion categories. The authors address this problem by augmenting existing architectures with a VAE framework. More precisely, an encoder network that is responsible for summarizing the seed seque... | The task of predicting human motion is complicated by the natural heterogeneity and compositionality of actions, necessitating robustness to distributional shifts as far as out-of-distribution (OoD). Here we formulate a new OoD benchmark based on the Human3.6M and CMU motion capture datasets, and introduce a hybrid fra... | [] | [
{
"authors": [
"Emre Aksan",
"Manuel Kaufmann",
"Otmar Hilliges"
],
"title": "Structured prediction helps 3d human motion modelling",
"venue": "In Proceedings of the IEEE International Conference on Computer Vision,",
"year": 2019
},
{
"authors": [
"Alexandre Alah... | [
{
"heading": "1 INTRODUCTION",
"text": "Human motion is naturally intelligible as a time-varying graph of connected joints constrained by locomotor anatomy and physiology. Its prediction allows the anticipation of actions with applications across healthcare (Geertsema et al., 2018; Kakar et al., 2005), phys... | 2,020 | null |
SP:8f1c7fabe235bdf095007948007509102dd0c126 | [
"The authors address the problem of discrete keypoint matching. For an input pair of images, the task is to match the unannotated (but given as part of the input) keypoints. The main contribution is identifying the bottleneck of the current SOTA algorithm: a fixed connectivity construction given by Delauney triangu... | Graph matching (GM) has been traditionally modeled as a deterministic optimization problem characterized by an affinity matrix under pre-defined graph topology. Though there have been several attempts on learning more effective node-level affinity/representation for matching, they still heavily rely on the initial grap... | [] | [
{
"authors": [
"Yoshua Bengio",
"Nicholas Léonard",
"Aaron Courville"
],
"title": "Estimating or propagating gradients through stochastic neurons for conditional computation",
"venue": "arXiv preprint arXiv:1308.3432,",
"year": 2013
},
{
"authors": [
"Christopher ... | [
{
"heading": "1 INTRODUCTION",
"text": "Being a long standing NP-hard problem (Loiola et al., 2007), graph matching (GM) has received persistent attention from the machine learning and optimization communities for many years. Concretely, for two graphs with n nodes for each, graph matching seeks to solve1:\... | 2,020 | null |
SP:879ce870f09e422aced7d008abc42fe5a8db29bc | [
"The paper proposes a method for stabilizing the training of GAN as well as overcoming the problem of mode collapse by optimizing several auxiliary models. The first step is to learn a latent space using an autoencoder. Then, this latent space is \"intervened\" by a predefined set of $K$ transformations to generate... | In this paper we propose a novel approach for stabilizing the training process of Generative Adversarial Networks as well as alleviating the mode collapse problem. The main idea is to incorporate a regularization term that we call intervention into the objective. We refer to the resulting generative model as Interventi... | [] | [
{
"authors": [
"M Arjovsky",
"L Bottou"
],
"title": "Towards principled methods for training generative adversarial networks. arxiv 2017",
"venue": "arXiv preprint arXiv:1701.04862",
"year": 2017
},
{
"authors": [
"Martin Arjovsky",
"Soumith Chintala",
"Léon... | [
{
"heading": "1 INTRODUCTION",
"text": "As one of the most important advances in generative models in recent years, Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have been attracting great attention in the machine learning community. GANs aim to train a generator network that transforms s... | 2,020 | null |
SP:a9c70bdca13ee3800c633589a6ee028701e5bf51 | [
"This work proposed a dimensionality reduction algorithm called Uniform Manifold Approximation with Two-phase Optimization (UMATO), which is an improved version of UMAP (Ref. [3] see below). UMATO has a two-phase optimization approach: global optimization to obtain the overall skeleton of data & local optimization ... | We present a dimensionality reduction algorithm called Uniform Manifold Approximation with Two-phase Optimization (UMATO) which produces less biased global structures in the embedding results and is robust over diverse initialization methods than previous methods such as t-SNE and UMAP. We divide the optimization into ... | [] | [
{
"authors": [
"Josh Barnes",
"Piet Hut"
],
"title": "A hierarchical o (n log n) force-calculation",
"venue": null,
"year": 1986
},
{
"authors": [
"Mikhail Belkin",
"Partha Niyogi"
],
"title": "Laplacian eigenmaps and spectral techniques for embedding and ... | [
{
"heading": "1 INTRODUCTION",
"text": "We present a novel dimensionality reduction method, Uniform Manifold Approximation with Twophase Optimization (UMATO) to obtain less biased and robust embedding over diverse initialization methods. One effective way of understanding high-dimensional data in various do... | 2,020 | null |
SP:fd70696898c5c725ad789565265274a37a6c2ca0 | [
"This paper presents a reduction approach to tackle the optimization problem of constrained RL. They propose a Frank-Wolfe type algorithm for the task, which avoids many shortcomings of previous methods, such as the memory complexity. They prove that their algorithm can find an $\\epsilon$-approximate solution with... | Many applications of reinforcement learning (RL) optimize a long-term reward subject to risk, safety, budget, diversity or other constraints. Though constrained RL problem has been studied to incorporate various constraints, existing methods either tie to specific families of RL algorithms or require storing infinitely... | [] | [
{
"authors": [
"Jacob Abernethy",
"Peter L Bartlett",
"Elad Hazan"
],
"title": "Blackwell approachability and no-regret learning are equivalent",
"venue": "In Proceedings of the 24th Annual Conference on Learning Theory, pp",
"year": 2011
},
{
"authors": [
"Jacob ... | [
{
"heading": "1 INTRODUCTION",
"text": "Contemporary approaches in reinforcement learning (RL) largely focus on optimizing the behavior of an agent against a single reward function. RL algorithms like value function methods (Zou et al., 2019; Zheng et al., 2018) or policy optimization methods (Chen et al., ... | 2,020 | null |
SP:df5fec4899d97f7d5df259a013f467e038895669 | [
"The paper proposes a post-hoc uncertainty tuning pipeline for Bayesian neural networks. After getting the point estimate, it adds extra dimensions to the weight matrices and hidden layers, which has no effect on the network output, with the hope that it would influence the variance of the original network weights ... | Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly “train” the uncertainty in a decoupled way to... | [] | [
{
"authors": [
"Felix Dangel",
"Frederik Kunstner",
"Philipp Hennig"
],
"title": "BackPACK: Packing more into Backprop",
"venue": "In ICLR,",
"year": 2020
},
{
"authors": [
"Yarin Gal",
"Zoubin Ghahramani"
],
"title": "Dropout as a Bayesian Approxima... | [
{
"heading": null,
"text": "Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly “trai... | 2,020 | null |
SP:2a2368b5bc6b59f66af75ea37f4cbc19c8fcf50f | [
"In this paper, the authors studied the possibility of sparsity exploration in Recurrent Neural Networks (RNNs) training. The main contributions include two parts: (1) Selfish-RNN training algorithm in Section 3.1 (2) SNT-ASGD optimizer in Section 3.2. The key idea of the Selfish-RNN training algorithm is a non-uni... | Sparse neural networks have been widely applied to reduce the necessary resource requirements to train and deploy over-parameterized deep neural networks. For inference acceleration, methods that induce sparsity from a pre-trained dense network (dense-to-sparse) work effectively. Recently, dynamic sparse training (DST)... | [] | [
{
"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": [
"Rui... | [
{
"heading": "1 INTRODUCTION",
"text": "Recurrent neural networks (RNNs) (Elman, 1990), with a variant of long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997), have been highly successful in various fields, including language modeling (Mikolov et al., 2010), machine translation (Kalchbrenner & Blu... | 2,020 | null |
SP:60d704b4a1555e24c09963617c879a15d8f3c805 | [
"This paper proposes a spatial-temporal graph neural network, which is designed to adaptively capture the complex spatial-temporal dependency. Further, the authors design a spatial-temporal attention module, which aims to capture multi-scale correlations. For multi-step prediction instead of one-step prediction, th... | Spatial-temporal data forecasting is of great importance for industries such as telecom network operation and transportation management. However, spatialtemporal data are inherent with complex spatial-temporal correlations and behaves heterogeneities among the spatial and temporal aspects, which makes the forecasting r... | [] | [
{
"authors": [
"Lei Bai",
"Lina Yao",
"Salil Kanhere",
"Xianzhi Wang",
"Quan Sheng"
],
"title": "Stg2seq: Spatialtemporal graph to sequence model for multi-step passenger demand forecasting",
"venue": "arXiv preprint arXiv:1905.10069,",
"year": 2019
},
{
"au... | [
{
"heading": "1 INTRODUCTION",
"text": "Spatial-temporal data forecasting has attracted attention from researchers due to its wide range of applications and the same specific characteristics of spatial-temporal data. Typical applications include mobile traffic forecast (He et al., 2019), traffic road condit... | 2,020 | null |
SP:a99af0f9e848f4f9068ad407612745a85a262644 | [
"This paper extends NTK to RNN to explain behavior of RNNs in overparametrized case. It’s a good extension study and interesting to see RNN with infinite-width limit converges to a kernel. The paper proves the same RNTK formula when the weights are shared and not shared. The proposed sensitivity for computationally... | The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network... | [
{
"affiliations": [],
"name": "Sina Alemohammad"
},
{
"affiliations": [],
"name": "Zichao Wang"
},
{
"affiliations": [],
"name": "Randall Balestriero"
},
{
"affiliations": [],
"name": "Richard G. Baraniuk"
}
] | [
{
"authors": [
"Zeyuan Allen-Zhu",
"Yuanzhi Li",
"Yingyu Liang"
],
"title": "Learning and generalization in overparameterized neural networks, going beyond two layers",
"venue": "Advances in Neural Information Processing Systems,",
"year": 2019
},
{
"authors": [
"... | [
{
"heading": "1 INTRODUCTION",
"text": "The overparameterization of modern deep neural networks (DNNs) has resulted in not only remarkably good generalization performance on unseen data (Novak et al., 2018; Neyshabur et al., 2019; Belkin et al., 2019) but also guarantees that gradient descent learning can f... | 2,021 | THE RECURRENT NEURAL TANGENT KERNEL |
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