Daily Papers

Exploiting the power of pre-trained models, prompt-based approaches stand out compared to other continual learning solutions in effectively preventing catastrophic forgetting, even with very few learnable parameters and without the need for a memory buffer. While existing prompt-based continual learning methods excel in leveraging prompts for state-of-the-art performance, they often lack a theoretical explanation for the effectiveness of prompting. This paper conducts a theoretical analysis to unravel how prompts bestow such advantages in continual learning, thus offering a new perspective on prompt design. We first show that the attention block of pre-trained models like Vision Transformers inherently encodes a special mixture of experts architecture, characterized by linear experts and quadratic gating score functions. This realization drives us to provide a novel view on prefix tuning, reframing it as the addition of new task-specific experts, thereby inspiring the design of a novel gating mechanism termed Non-linear Residual Gates (NoRGa). Through the incorporation of non-linear activation and residual connection, NoRGa enhances continual learning performance while preserving parameter efficiency. The effectiveness of NoRGa is substantiated both theoretically and empirically across diverse benchmarks and pretraining paradigms. Our code is publicly available at https://github.com/Minhchuyentoancbn/MoE_PromptCL

Gating mechanisms have been widely utilized, from early models like LSTMs and Highway Networks to recent state space models, linear attention, and also softmax attention. Yet, existing literature rarely examines the specific effects of gating. In this work, we conduct comprehensive experiments to systematically investigate gating-augmented softmax attention variants. Specifically, we perform a comprehensive comparison over 30 variants of 15B Mixture-of-Experts (MoE) models and 1.7B dense models trained on a 3.5 trillion token dataset. Our central finding is that a simple modification-applying a head-specific sigmoid gate after the Scaled Dot-Product Attention (SDPA)-consistently improves performance. This modification also enhances training stability, tolerates larger learning rates, and improves scaling properties. By comparing various gating positions and computational variants, we attribute this effectiveness to two key factors: (1) introducing non-linearity upon the low-rank mapping in the softmax attention, and (2) applying query-dependent sparse gating scores to modulate the SDPA output. Notably, we find this sparse gating mechanism mitigates 'attention sink' and enhances long-context extrapolation performance, and we also release related https://github.com/qiuzh20/gated_attention{codes} and https://huggingface.co/QwQZh/gated_attention{models} to facilitate future research.

We consider the problem of learning a target function corresponding to a single hidden layer neural network, with a quadratic activation function after the first layer, and random weights. We consider the asymptotic limit where the input dimension and the network width are proportionally large. Recent work [Cui & al '23] established that linear regression provides Bayes-optimal test error to learn such a function when the number of available samples is only linear in the dimension. That work stressed the open challenge of theoretically analyzing the optimal test error in the more interesting regime where the number of samples is quadratic in the dimension. In this paper, we solve this challenge for quadratic activations and derive a closed-form expression for the Bayes-optimal test error. We also provide an algorithm, that we call GAMP-RIE, which combines approximate message passing with rotationally invariant matrix denoising, and that asymptotically achieves the optimal performance. Technically, our result is enabled by establishing a link with recent works on optimal denoising of extensive-rank matrices and on the ellipsoid fitting problem. We further show empirically that, in the absence of noise, randomly-initialized gradient descent seems to sample the space of weights, leading to zero training loss, and averaging over initialization leads to a test error equal to the Bayes-optimal one.

Diffusion models have emerged as a powerful tool rivaling GANs in generating high-quality samples with improved fidelity, flexibility, and robustness. A key component of these models is to learn the score function through score matching. Despite empirical success on various tasks, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. As a first step toward answering this question, this paper establishes a mathematical framework for analyzing score estimation using neural networks trained by gradient descent. Our analysis covers both the optimization and the generalization aspects of the learning procedure. In particular, we propose a parametric form to formulate the denoising score-matching problem as a regression with noisy labels. Compared to the standard supervised learning setup, the score-matching problem introduces distinct challenges, including unbounded input, vector-valued output, and an additional time variable, preventing existing techniques from being applied directly. In this paper, we show that with proper designs, the evolution of neural networks during training can be accurately modeled by a series of kernel regression tasks. Furthermore, by applying an early-stopping rule for gradient descent and leveraging recent developments in neural tangent kernels, we establish the first generalization error (sample complexity) bounds for learning the score function with neural networks, despite the presence of noise in the observations. Our analysis is grounded in a novel parametric form of the neural network and an innovative connection between score matching and regression analysis, facilitating the application of advanced statistical and optimization techniques.

Activation functions are core components of all deep learning architectures. Currently, the most popular activation functions are smooth ReLU variants like GELU and SiLU. These are self-gated activation functions where the range of the gating function is between zero and one. In this paper, we explore the viability of using arctan as a gating mechanism. A self-gated activation function that uses arctan as its gating function has a monotonically increasing first derivative. To make this activation function competitive, it is necessary to introduce a trainable parameter for every MLP block to expand the range of the gating function beyond zero and one. We find that this technique also improves existing self-gated activation functions. We conduct an empirical evaluation of Expanded ArcTan Linear Unit (xATLU), Expanded GELU (xGELU), and Expanded SiLU (xSiLU) and show that they outperform existing activation functions within a transformer architecture. Additionally, expanded gating ranges show promising results in improving first-order Gated Linear Units (GLU).

By learning the gradient of smoothed data distributions, diffusion models can iteratively generate samples from complex distributions. The learned score function enables their generalization capabilities, but how the learned score relates to the score of the underlying data manifold remains largely unclear. Here, we aim to elucidate this relationship by comparing learned neural scores to the scores of two kinds of analytically tractable distributions: Gaussians and Gaussian mixtures. The simplicity of the Gaussian model makes it theoretically attractive, and we show that it admits a closed-form solution and predicts many qualitative aspects of sample generation dynamics. We claim that the learned neural score is dominated by its linear (Gaussian) approximation for moderate to high noise scales, and supply both theoretical and empirical arguments to support this claim. Moreover, the Gaussian approximation empirically works for a larger range of noise scales than naive theory suggests it should, and is preferentially learned early in training. At smaller noise scales, we observe that learned scores are better described by a coarse-grained (Gaussian mixture) approximation of training data than by the score of the training distribution, a finding consistent with generalization. Our findings enable us to precisely predict the initial phase of trained models' sampling trajectories through their Gaussian approximations. We show that this allows the skipping of the first 15-30% of sampling steps while maintaining high sample quality (with a near state-of-the-art FID score of 1.93 on CIFAR-10 unconditional generation). This forms the foundation of a novel hybrid sampling method, termed analytical teleportation, which can seamlessly integrate with and accelerate existing samplers, including DPM-Solver-v3 and UniPC. Our findings suggest ways to improve the design and training of diffusion models.

The adversarial vulnerability of Deep Neural Networks (DNNs) has been well-known and widely concerned, often under the context of learning top-1 attacks (e.g., fooling a DNN to classify a cat image as dog). This paper shows that the concern is much more serious by learning significantly more aggressive ordered top-K clear-box~ This is often referred to as white/black-box attacks in the literature. We choose to adopt neutral terminology, clear/opaque-box attacks in this paper, and omit the prefix clear-box for simplicity. targeted attacks proposed in Adversarial Distillation. We propose a novel and rigorous quadratic programming (QP) method of learning ordered top-K attacks with low computing cost, dubbed as QuadAttacK. Our QuadAttacK directly solves the QP to satisfy the attack constraint in the feature embedding space (i.e., the input space to the final linear classifier), which thus exploits the semantics of the feature embedding space (i.e., the principle of class coherence). With the optimized feature embedding vector perturbation, it then computes the adversarial perturbation in the data space via the vanilla one-step back-propagation. In experiments, the proposed QuadAttacK is tested in the ImageNet-1k classification using ResNet-50, DenseNet-121, and Vision Transformers (ViT-B and DEiT-S). It successfully pushes the boundary of successful ordered top-K attacks from K=10 up to K=20 at a cheap budget (1times 60) and further improves attack success rates for K=5 for all tested models, while retaining the performance for K=1.

Mixture-of-experts (MoE) model incorporates the power of multiple submodels via gating functions to achieve greater performance in numerous regression and classification applications. From a theoretical perspective, while there have been previous attempts to comprehend the behavior of that model under the regression settings through the convergence analysis of maximum likelihood estimation in the Gaussian MoE model, such analysis under the setting of a classification problem has remained missing in the literature. We close this gap by establishing the convergence rates of density estimation and parameter estimation in the softmax gating multinomial logistic MoE model. Notably, when part of the expert parameters vanish, these rates are shown to be slower than polynomial rates owing to an inherent interaction between the softmax gating and expert functions via partial differential equations. To address this issue, we propose using a novel class of modified softmax gating functions which transform the input value before delivering them to the gating functions. As a result, the previous interaction disappears and the parameter estimation rates are significantly improved.

Top-K sparse softmax gating mixture of experts has been widely used for scaling up massive deep-learning architectures without increasing the computational cost. Despite its popularity in real-world applications, the theoretical understanding of that gating function has remained an open problem. The main challenge comes from the structure of the top-K sparse softmax gating function, which partitions the input space into multiple regions with distinct behaviors. By focusing on a Gaussian mixture of experts, we establish theoretical results on the effects of the top-K sparse softmax gating function on both density and parameter estimations. Our results hinge upon defining novel loss functions among parameters to capture different behaviors of the input regions. When the true number of experts k_{ast} is known, we demonstrate that the convergence rates of density and parameter estimations are both parametric on the sample size. However, when k_{ast} becomes unknown and the true model is over-specified by a Gaussian mixture of k experts where k > k_{ast}, our findings suggest that the number of experts selected from the top-K sparse softmax gating function must exceed the total cardinality of a certain number of Voronoi cells associated with the true parameters to guarantee the convergence of the density estimation. Moreover, while the density estimation rate remains parametric under this setting, the parameter estimation rates become substantially slow due to an intrinsic interaction between the softmax gating and expert functions.

Curriculum Learning is the presentation of samples to the machine learning model in a meaningful order instead of a random order. The main challenge of Curriculum Learning is determining how to rank these samples. The ranking of the samples is expressed by the scoring function. In this study, scoring functions were compared using data set features, using the model to be trained, and using another model and their ensemble versions. Experiments were performed for 4 images and 4 text datasets. No significant differences were found between scoring functions for text datasets, but significant improvements were obtained in scoring functions created using transfer learning compared to classical model training and other scoring functions for image datasets. It shows that different new scoring functions are waiting to be found for text classification tasks.

We consider a contextual combinatorial bandit problem where in each round a learning agent selects a subset of arms and receives feedback on the selected arms according to their scores. The score of an arm is an unknown function of the arm's feature. Approximating this unknown score function with deep neural networks, we propose algorithms: Combinatorial Neural UCB (CN-UCB) and Combinatorial Neural Thompson Sampling (CN-TS). We prove that CN-UCB achieves mathcal{O}(d T) or mathcal{O}(tilde{d T K}) regret, where d is the effective dimension of a neural tangent kernel matrix, K is the size of a subset of arms, and T is the time horizon. For CN-TS, we adapt an optimistic sampling technique to ensure the optimism of the sampled combinatorial action, achieving a worst-case (frequentist) regret of mathcal{O}(d TK). To the best of our knowledge, these are the first combinatorial neural bandit algorithms with regret performance guarantees. In particular, CN-TS is the first Thompson sampling algorithm with the worst-case regret guarantees for the general contextual combinatorial bandit problem. The numerical experiments demonstrate the superior performances of our proposed algorithms.

Understanding the parameter estimation of softmax gating Gaussian mixture of experts has remained a long-standing open problem in the literature. It is mainly due to three fundamental theoretical challenges associated with the softmax gating function: (i) the identifiability only up to the translation of parameters; (ii) the intrinsic interaction via partial differential equations between the softmax gating and the expert functions in the Gaussian density; (iii) the complex dependence between the numerator and denominator of the conditional density of softmax gating Gaussian mixture of experts. We resolve these challenges by proposing novel Voronoi loss functions among parameters and establishing the convergence rates of maximum likelihood estimator (MLE) for solving parameter estimation in these models. When the true number of experts is unknown and over-specified, our findings show a connection between the convergence rate of the MLE and a solvability problem of a system of polynomial equations.

Models of neural responses to stimuli with complex spatiotemporal correlation structure often assume that neurons are only selective for a small number of linear projections of a potentially high-dimensional input. Here we explore recent modeling approaches where the neural response depends on the quadratic form of the input rather than on its linear projection, that is, the neuron is sensitive to the local covariance structure of the signal preceding the spike. To infer this quadratic dependence in the presence of arbitrary (e.g. naturalistic) stimulus distribution, we review several inference methods, focussing in particular on two information-theory-based approaches (maximization of stimulus energy or of noise entropy) and a likelihood-based approach (Bayesian spike-triggered covariance, extensions of generalized linear models). We analyze the formal connection between the likelihood-based and information-based approaches to show how they lead to consistent inference. We demonstrate the practical feasibility of these procedures by using model neurons responding to a flickering variance stimulus.

We consider the problem of subset selection where one is given multiple rankings of items and the goal is to select the highest ``quality'' subset. Score functions from the multiwinner voting literature have been used to aggregate rankings into quality scores for subsets. We study this setting of subset selection problems when, in addition, rankings may contain systemic or unconscious biases toward a group of items. For a general model of input rankings and biases, we show that requiring the selected subset to satisfy group fairness constraints can improve the quality of the selection with respect to unbiased rankings. Importantly, we show that for fairness constraints to be effective, different multiwinner score functions may require a drastically different number of rankings: While for some functions, fairness constraints need an exponential number of rankings to recover a close-to-optimal solution, for others, this dependency is only polynomial. This result relies on a novel notion of ``smoothness'' of submodular functions in this setting that quantifies how well a function can ``correctly'' assess the quality of items in the presence of bias. The results in this paper can be used to guide the choice of multiwinner score functions for the subset selection setting considered here; we additionally provide a tool to empirically enable this.

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form h = g circ p where p : R^d rightarrow R is a degree k polynomial and g: R rightarrow R is a degree q polynomial. This function class generalizes the single-index model, which corresponds to k=1, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree k polynomials p, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target h up to vanishing test error in mathcal{O}(d^k) samples and polynomial time. This is a strict improvement over kernel methods, which require widetilde Theta(d^{kq}) samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of p being a quadratic. When p is indeed a quadratic, we achieve the information-theoretically optimal sample complexity mathcal{O}(d^2), which is an improvement over prior work~nichani2023provable requiring a sample size of widetildeTheta(d^4). Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature p with mathcal{O}(d^k) samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

Dense-to-sparse gating mixture of experts (MoE) has recently become an effective alternative to a well-known sparse MoE. Rather than fixing the number of activated experts as in the latter model, which could limit the investigation of potential experts, the former model utilizes the temperature to control the softmax weight distribution and the sparsity of the MoE during training in order to stabilize the expert specialization. Nevertheless, while there are previous attempts to theoretically comprehend the sparse MoE, a comprehensive analysis of the dense-to-sparse gating MoE has remained elusive. Therefore, we aim to explore the impacts of the dense-to-sparse gate on the maximum likelihood estimation under the Gaussian MoE in this paper. We demonstrate that due to interactions between the temperature and other model parameters via some partial differential equations, the convergence rates of parameter estimations are slower than any polynomial rates, and could be as slow as O(1/log(n)), where n denotes the sample size. To address this issue, we propose using a novel activation dense-to-sparse gate, which routes the output of a linear layer to an activation function before delivering them to the softmax function. By imposing linearly independence conditions on the activation function and its derivatives, we show that the parameter estimation rates are significantly improved to polynomial rates.

Risk scores are simple classification models that let users make quick risk predictions by adding and subtracting a few small numbers. These models are widely used in medicine and criminal justice, but are difficult to learn from data because they need to be calibrated, sparse, use small integer coefficients, and obey application-specific operational constraints. In this paper, we present a new machine learning approach to learn risk scores. We formulate the risk score problem as a mixed integer nonlinear program, and present a cutting plane algorithm for non-convex settings to efficiently recover its optimal solution. We improve our algorithm with specialized techniques to generate feasible solutions, narrow the optimality gap, and reduce data-related computation. Our approach can fit risk scores in a way that scales linearly in the number of samples, provides a certificate of optimality, and obeys real-world constraints without parameter tuning or post-processing. We benchmark the performance benefits of this approach through an extensive set of numerical experiments, comparing to risk scores built using heuristic approaches. We also discuss its practical benefits through a real-world application where we build a customized risk score for ICU seizure prediction in collaboration with the Massachusetts General Hospital.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

Existing Score-Based Models (SBMs) can be categorized into constrained SBMs (CSBMs) or unconstrained SBMs (USBMs) according to their parameterization approaches. CSBMs model probability density functions as Boltzmann distributions, and assign their predictions as the negative gradients of some scalar-valued energy functions. On the other hand, USBMs employ flexible architectures capable of directly estimating scores without the need to explicitly model energy functions. In this paper, we demonstrate that the architectural constraints of CSBMs may limit their modeling ability. In addition, we show that USBMs' inability to preserve the property of conservativeness may lead to degraded performance in practice. To address the above issues, we propose Quasi-Conservative Score-Based Models (QCSBMs) for keeping the advantages of both CSBMs and USBMs. Our theoretical derivations demonstrate that the training objective of QCSBMs can be efficiently integrated into the training processes by leveraging the Hutchinson's trace estimator. In addition, our experimental results on the CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets validate the effectiveness of QCSBMs. Finally, we justify the advantage of QCSBMs using an example of a one-layered autoencoder.

This paper studies attack detection for discrete-time linear systems with stochastic process noise that produce both a vulnerable (i.e., attackable) linear measurement and a secured (i.e., unattackable) quadratic measurement. The motivating application of this model is a dynamic-game setting where the quadratic measurement is interpreted as a system-level utility or reward, and control inputs into the linear system are interpreted as control policies that, once applied, are known to all game participants and which steer the system towards a game-theoretic equilibrium (e.g., Nash equilibrium). To detect attacks on the linear channel, we develop a novel quadratic-utility-aware observer that leverages the secured quadratic output and enforces measurement consistency via a projection step. We establish three properties for this observer: feasibility of the true state, prox-regularity of the quadratic-constraint set, and a monotone error-reduction guarantee in the noise-free case. To detect adversarial manipulation, we compare linear and quadratic observer trajectories using a wild bootstrap maximum mean discrepancy (MMD) test that provides valid inference under temporal dependence. We validate our framework using numerical experiments of a pursuit-evasion game, where the quadratic observer preserves estimation accuracy under linear-sensor attacks, while the statistical test detects distributional divergence between the observers' trajectories.

Inspired by the diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are important issues not well addressed. Theoretically, the superior expressivity of a quadratic network over either a conventional network or a conventional network via quadratic activation is not fully elucidated, which makes the use of quadratic networks not well grounded. Practically, although a quadratic network can be trained via generic backpropagation, it can be subject to a higher risk of collapse than the conventional counterpart. To address these issues, we first apply the spline theory and a measure from algebraic geometry to give two theorems that demonstrate better model expressivity of a quadratic network than the conventional counterpart with or without quadratic activation. Then, we propose an effective training strategy referred to as ReLinear to stabilize the training process of a quadratic network, thereby unleashing the full potential in its associated machine learning tasks. Comprehensive experiments on popular datasets are performed to support our findings and confirm the performance of quadratic deep learning. We have shared our code in https://github.com/FengleiFan/ReLinear.

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

Recent Graphical User Interface (GUI) agents replicate the R1-Zero paradigm, coupling online Reinforcement Learning (RL) with explicit chain-of-thought reasoning prior to object grounding and thereby achieving substantial performance gains. In this paper, we first conduct extensive analysis experiments of three key components of that training pipeline: input design, output evaluation, and policy update-each revealing distinct challenges arising from blindly applying general-purpose RL without adapting to GUI grounding tasks. Input design: Current templates encourage the model to generate chain-of-thought reasoning, but longer chains unexpectedly lead to worse grounding performance. Output evaluation: Reward functions based on hit signals or box area allow models to exploit box size, leading to reward hacking and poor localization quality. Policy update: Online RL tends to overfit easy examples due to biases in length and sample difficulty, leading to under-optimization on harder cases. To address these issues, we propose three targeted solutions. First, we adopt a Fast Thinking Template that encourages direct answer generation, reducing excessive reasoning during training. Second, we incorporate a box size constraint into the reward function to mitigate reward hacking. Third, we revise the RL objective by adjusting length normalization and adding a difficulty-aware scaling factor, enabling better optimization on hard samples. Our GUI-G1-3B, trained on 17K public samples with Qwen2.5-VL-3B-Instruct, achieves 90.3% accuracy on ScreenSpot and 37.1% on ScreenSpot-Pro. This surpasses all prior models of similar size and even outperforms the larger UI-TARS-7B, establishing a new state-of-the-art in GUI agent grounding. The project repository is available at https://github.com/Yuqi-Zhou/GUI-G1.

In distributed deep learning with data parallelism, synchronizing gradients at each training step can cause a huge communication overhead, especially when many nodes work together to train large models. Local gradient methods, such as Local SGD, address this issue by allowing workers to compute locally for H steps without synchronizing with others, hence reducing communication frequency. While H has been viewed as a hyperparameter to trade optimization efficiency for communication cost, recent research indicates that setting a proper H value can lead to generalization improvement. Yet, selecting a proper H is elusive. This work proposes a theory-grounded method for determining H, named the Quadratic Synchronization Rule (QSR), which recommends dynamically setting H in proportion to 1{eta^2} as the learning rate eta decays over time. Extensive ImageNet experiments on ResNet and ViT show that local gradient methods with QSR consistently improve the test accuracy over other synchronization strategies. Compared with the standard data parallel training, QSR enables Local AdamW on ViT-B to cut the training time on 16 or 64 GPUs down from 26.7 to 20.2 hours or from 8.6 to 5.5 hours and, at the same time, achieves 1.16% or 0.84% higher top-1 validation accuracy.

Value-based methods constitute a fundamental methodology in planning and deep reinforcement learning (RL). In this paper, we propose to exploit the underlying structures of the state-action value function, i.e., Q function, for both planning and deep RL. In particular, if the underlying system dynamics lead to some global structures of the Q function, one should be capable of inferring the function better by leveraging such structures. Specifically, we investigate the low-rank structure, which widely exists for big data matrices. We verify empirically the existence of low-rank Q functions in the context of control and deep RL tasks. As our key contribution, by leveraging Matrix Estimation (ME) techniques, we propose a general framework to exploit the underlying low-rank structure in Q functions. This leads to a more efficient planning procedure for classical control, and additionally, a simple scheme that can be applied to any value-based RL techniques to consistently achieve better performance on "low-rank" tasks. Extensive experiments on control tasks and Atari games confirm the efficacy of our approach. Code is available at https://github.com/YyzHarry/SV-RL.

We introduce for continual learning Autodiff Quadratic Consolidation (AQC), which approximates the previous loss function with a quadratic function, and Neural Consolidation (NC), which approximates the previous loss function with a neural network. Although they are not scalable to large neural networks, they can be used with a fixed pre-trained feature extractor. We empirically study these methods in class-incremental learning, for which regularization-based methods produce unsatisfactory results, unless combined with replay. We find that for small datasets, quadratic approximation of the previous loss function leads to poor results, even with full Hessian computation, and NC could significantly improve the predictive performance, while for large datasets, when used with a fixed pre-trained feature extractor, AQC provides superior predictive performance. We also find that using tanh-output features can improve the predictive performance of AQC. In particular, in class-incremental Split MNIST, when a Convolutional Neural Network (CNN) with tanh-output features is pre-trained on EMNIST Letters and used as a fixed pre-trained feature extractor, AQC can achieve predictive performance comparable to joint training.

There is a long history, as well as a recent explosion of interest, in statistical and generative modeling approaches based on score functions -- derivatives of the log-likelihood of a distribution. In seminal works, Hyv\"arinen proposed vanilla score matching as a way to learn distributions from data by computing an estimate of the score function of the underlying ground truth, and established connections between this method and established techniques like Contrastive Divergence and Pseudolikelihood estimation. It is by now well-known that vanilla score matching has significant difficulties learning multimodal distributions. Although there are various ways to overcome this difficulty, the following question has remained unanswered -- is there a natural way to sample multimodal distributions using just the vanilla score? Inspired by a long line of related experimental works, we prove that the Langevin diffusion with early stopping, initialized at the empirical distribution, and run on a score function estimated from data successfully generates natural multimodal distributions (mixtures of log-concave distributions).

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" [Lewkowycz et al. 2020] that arises when training such models with large learning rates. We then empirically show that the behaviour of neural quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models can be an effective tool for analysis of neural networks.

We deal with the combinatorial problem of learning directed acyclic graph (DAG) structure from observational data adhering to a linear structural equation model (SEM). Leveraging advances in differentiable, nonconvex characterizations of acyclicity, recent efforts have advocated a continuous constrained optimization paradigm to efficiently explore the space of DAGs. Most existing methods employ lasso-type score functions to guide this search, which (i) require expensive penalty parameter retuning when the unknown SEM noise variances change across problem instances; and (ii) implicitly rely on limiting homoscedasticity assumptions. In this work, we propose a new convex score function for sparsity-aware learning of linear DAGs, which incorporates concomitant estimation of scale and thus effectively decouples the sparsity parameter from the exogenous noise levels. Regularization via a smooth, nonconvex acyclicity penalty term yields CoLiDE (Concomitant Linear DAG Estimation), a regression-based criterion amenable to efficient gradient computation and closed-form estimation of noise variances in heteroscedastic scenarios. Our algorithm outperforms state-of-the-art methods without incurring added complexity, especially when the DAGs are larger and the noise level profile is heterogeneous. We also find CoLiDE exhibits enhanced stability manifested via reduced standard deviations in several domain-specific metrics, underscoring the robustness of our novel linear DAG estimator.

We propose the Bayes-UCBVI algorithm for reinforcement learning in tabular, stage-dependent, episodic Markov decision process: a natural extension of the Bayes-UCB algorithm by Kaufmann et al. (2012) for multi-armed bandits. Our method uses the quantile of a Q-value function posterior as upper confidence bound on the optimal Q-value function. For Bayes-UCBVI, we prove a regret bound of order O(H^3SAT) where H is the length of one episode, S is the number of states, A the number of actions, T the number of episodes, that matches the lower-bound of Ω(H^3SAT) up to poly-log terms in H,S,A,T for a large enough T. To the best of our knowledge, this is the first algorithm that obtains an optimal dependence on the horizon H (and S) without the need for an involved Bernstein-like bonus or noise. Crucial to our analysis is a new fine-grained anti-concentration bound for a weighted Dirichlet sum that can be of independent interest. We then explain how Bayes-UCBVI can be easily extended beyond the tabular setting, exhibiting a strong link between our algorithm and Bayesian bootstrap (Rubin, 1981).

In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the d-dimensional Boolean hypercube labeled by the quadratic ``XOR'' function y = -x_ix_j, it is possible to train to a population error o(1) with d :polylog(d) samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of O(d) for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a signal-finding phase where the network is small and many of the neurons evolve independently to find features, and a signal-heavy phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

Integral reinforcement learning (IntRL) demands the precise computation of the utility function's integral at its policy evaluation (PEV) stage. This is achieved through quadrature rules, which are weighted sums of utility functions evaluated from state samples obtained in discrete time. Our research reveals a critical yet underexplored phenomenon: the choice of the computational method -- in this case, the quadrature rule -- can significantly impact control performance. This impact is traced back to the fact that computational errors introduced in the PEV stage can affect the policy iteration's convergence behavior, which in turn affects the learned controller. To elucidate how computation impacts control, we draw a parallel between IntRL's policy iteration and Newton's method applied to the Hamilton-Jacobi-Bellman equation. In this light, computational error in PEV manifests as an extra error term in each iteration of Newton's method, with its upper bound proportional to the computational error. Further, we demonstrate that when the utility function resides in a reproducing kernel Hilbert space (RKHS), the optimal quadrature is achievable by employing Bayesian quadrature with the RKHS-inducing kernel function. We prove that the local convergence rates for IntRL using the trapezoidal rule and Bayesian quadrature with a Mat\'ern kernel to be O(N^{-2}) and O(N^{-b}), where N is the number of evenly-spaced samples and b is the Mat\'ern kernel's smoothness parameter. These theoretical findings are finally validated by two canonical control tasks.

We consider a family of convex quadratic programs in which the coefficients of the linear objective term and the righthand side of the constraints are affine functions of a parameter. It is well known that the solution of such a parametrized quadratic program is a piecewise affine function of the parameter. The number of (polyhedral) regions in the solution map can grow exponentially in problem size, but when the number of regions is moderate, a so-called explicit solver is practical. Such a solver computes the coefficients of the affine functions and the linear inequalities defining the polyhedral regions offline; to solve a problem instance online it simply evaluates this explicit solution map. Potential advantages of an explicit solver over a more general purpose iterative solver can include transparency, interpretability, reliability, and speed. In this paper we describe how code generation can be used to automatically generate an explicit solver from a high level description of a parametrized quadratic program. Our method has been implemented in the open-source software CVXPYgen, which is part of CVXPY, a domain specific language for general convex optimization.

We consider the question of learning Q-function in a sample efficient manner for reinforcement learning with continuous state and action spaces under a generative model. If Q-function is Lipschitz continuous, then the minimal sample complexity for estimating ε-optimal Q-function is known to scale as Ω(1{ε^{d_1+d_2 +2}}) per classical non-parametric learning theory, where d_1 and d_2 denote the dimensions of the state and action spaces respectively. The Q-function, when viewed as a kernel, induces a Hilbert-Schmidt operator and hence possesses square-summable spectrum. This motivates us to consider a parametric class of Q-functions parameterized by its "rank" r, which contains all Lipschitz Q-functions as r to infty. As our key contribution, we develop a simple, iterative learning algorithm that finds ε-optimal Q-function with sample complexity of O(1{ε^{max(d_1, d_2)+2}}) when the optimal Q-function has low rank r and the discounting factor γ is below a certain threshold. Thus, this provides an exponential improvement in sample complexity. To enable our result, we develop a novel Matrix Estimation algorithm that faithfully estimates an unknown low-rank matrix in the ell_infty sense even in the presence of arbitrary bounded noise, which might be of interest in its own right. Empirical results on several stochastic control tasks confirm the efficacy of our "low-rank" algorithms.

GossipSub is a new peer-to-peer communication protocol designed to counter attacks from misbehaving peers by controlling what information is sent and to whom, via a score function computed by each peer that captures positive and negative behaviors of its neighbors. The score function depends on several parameters (weights, caps, thresholds) that can be configured by applications using GossipSub. The specification for GossipSub is written in English and its resilience to attacks from misbehaving peers is supported empirically by emulation testing using an implementation in Golang. In this work we take a foundational approach to understanding the resilience of GossipSub to attacks from misbehaving peers. We build the first formal model of GossipSub, using the ACL2s theorem prover. Our model is officially endorsed by the GossipSub developers. It can simulate GossipSub networks of arbitrary size and topology, with arbitrarily configured peers, and can be used to prove and disprove theorems about the protocol. We formalize fundamental security properties stating that the score function is fair, penalizes bad behavior, and rewards good behavior. We prove that the score function is always fair, but can be configured in ways that either penalize good behavior or ignore bad behavior. Using our model, we run GossipSub with the specific configurations for two popular real-world applications: the FileCoin and Eth2.0 blockchains. We show that all properties hold for FileCoin. However, given any Eth2.0 network (of any topology and size) with any number of potentially misbehaving peers, we can synthesize attacks where these peers are able to continuously misbehave by never forwarding topic messages, while maintaining positive scores so that they are never pruned from the network by GossipSub.

In this paper we introduce the SCoRe (Submodular Combinatorial Representation Learning) framework, a novel approach in representation learning that addresses inter-class bias and intra-class variance. SCoRe provides a new combinatorial viewpoint to representation learning, by introducing a family of loss functions based on set-based submodular information measures. We develop two novel combinatorial formulations for loss functions, using the Total Information and Total Correlation, that naturally minimize intra-class variance and inter-class bias. Several commonly used metric/contrastive learning loss functions like supervised contrastive loss, orthogonal projection loss, and N-pairs loss, are all instances of SCoRe, thereby underlining the versatility and applicability of SCoRe in a broad spectrum of learning scenarios. Novel objectives in SCoRe naturally model class-imbalance with up to 7.6\% improvement in classification on CIFAR-10-LT, CIFAR-100-LT, MedMNIST, 2.1% on ImageNet-LT, and 19.4% in object detection on IDD and LVIS (v1.0), demonstrating its effectiveness over existing approaches.

The success of Deepseek-R1 has drawn the LLM community's attention to reinforcement learning (RL) methods like GRPO. However, such rule-based 0/1 outcome reward methods lack the capability to regulate the intermediate reasoning processes during chain-of-thought (CoT) generation, leading to severe overthinking phenomena. In response, recent studies have designed reward functions to reinforce models' behaviors in producing shorter yet correct completions. Nevertheless, we observe that these length-penalty reward functions exacerbate RL training instability: as the completion length decreases, model accuracy abruptly collapses, often occurring early in training. To address this issue, we propose a simple yet effective solution GRPO-lambda, an efficient and stabilized variant of GRPO, which dynamically adjusts the reward strategy by monitoring the correctness ratio among completions within each query-sampled group. A low correctness ratio indicates the need to avoid length penalty that compromises CoT quality, triggering a switch to length-agnostic 0/1 rewards that prioritize reasoning capability. A high ratio maintains length penalties to boost efficiency. Experimental results show that our approach avoids training instability caused by length penalty while maintaining the optimal accuracy-efficiency trade-off. On the GSM8K, GPQA, MATH-500, AMC 2023, and AIME 2024 benchmarks, it improves average accuracy by 1.48% while reducing CoT sequence length by 47.3%.

Residual connections are central to modern deep neural networks, enabling stable optimization and efficient information flow across depth. In this work, we propose SCORE (Skip-Connection ODE Recurrent Embedding), a discrete recurrent alternative to classical layer stacking. Instead of composing multiple independent layers, SCORE iteratively applies a single shared neural block using an ODE (Ordinary Differential Equation)-inspired contractive update: ht+1 = (1 - dt) * ht + dt * F(ht) This formulation can be interpreted as a depth-by-iteration refinement process, where the step size dt explicitly controls stability and update magnitude. Unlike continuous Neural ODE approaches, SCORE uses a fixed number of discrete iterations and standard backpropagation without requiring ODE solvers or adjoint methods. We evaluate SCORE across graph neural networks (ESOL molecular solubility), multilayer perceptrons, and Transformer-based language models (nanoGPT). Across architectures, SCORE generally improves convergence speed and often accelerates training. SCORE is reducing parameter count through shared weights. In practice, simple Euler integration provides the best trade-off between computational cost and performance, while higher-order integrators yield marginal gains at increased compute. These results suggest that controlled recurrent depth with contractive residual updates offers a lightweight and effective alternative to classical stacking in deep neural networks.

We present a unified representation of the most popular neural network activation functions. Adopting Mittag-Leffler functions of fractional calculus, we propose a flexible and compact functional form that is able to interpolate between various activation functions and mitigate common problems in training neural networks such as vanishing and exploding gradients. The presented gated representation extends the scope of fixed-shape activation functions to their adaptive counterparts whose shape can be learnt from the training data. The derivatives of the proposed functional form can also be expressed in terms of Mittag-Leffler functions making it a suitable candidate for gradient-based backpropagation algorithms. By training multiple neural networks of different complexities on various datasets with different sizes, we demonstrate that adopting a unified gated representation of activation functions offers a promising and affordable alternative to individual built-in implementations of activation functions in conventional machine learning frameworks.

While score-based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference-time behavior in a principled manner, e.g. for composing multiple pretrained models. Existing classifier-free guidance methods use a simple heuristic to mix conditional and unconditional scores to approximately sample from conditional distributions. However, such methods do not approximate the intermediate distributions, necessitating additional 'corrector' steps. In this work, we provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models. We derive a weighted simulation scheme which we call Feynman-Kac Correctors (FKCs) based on the celebrated Feynman-Kac formula by carefully accounting for terms in the appropriate partial differential equations (PDEs). To simulate these PDEs, we propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality. We empirically demonstrate the utility of our methods by proposing amortized sampling via inference-time temperature annealing, improving multi-objective molecule generation using pretrained models, and improving classifier-free guidance for text-to-image generation. Our code is available at https://github.com/martaskrt/fkc-diffusion.

Bayesian optimisation is a popular approach for optimising expensive black-box functions. The next location to be evaluated is selected via maximising an acquisition function that balances exploitation and exploration. Gaussian processes, the surrogate models of choice in Bayesian optimisation, are often used with a constant prior mean function equal to the arithmetic mean of the observed function values. We show that the rate of convergence can depend sensitively on the choice of mean function. We empirically investigate 8 mean functions (constant functions equal to the arithmetic mean, minimum, median and maximum of the observed function evaluations, linear, quadratic polynomials, random forests and RBF networks), using 10 synthetic test problems and two real-world problems, and using the Expected Improvement and Upper Confidence Bound acquisition functions. We find that for design dimensions ge5 using a constant mean function equal to the worst observed quality value is consistently the best choice on the synthetic problems considered. We argue that this worst-observed-quality function promotes exploitation leading to more rapid convergence. However, for the real-world tasks the more complex mean functions capable of modelling the fitness landscape may be effective, although there is no clearly optimum choice.

This paper re-examines a continuous optimization framework dubbed NOTEARS for learning Bayesian networks. We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials. Next, focusing on a one-parameter-per-edge setting, it is shown that the Karush-Kuhn-Tucker (KKT) optimality conditions for the NOTEARS formulation cannot be satisfied except in a trivial case, which explains a behavior of the associated algorithm. We then derive the KKT conditions for an equivalent reformulation, show that they are indeed necessary, and relate them to explicit constraints that certain edges be absent from the graph. If the score function is convex, these KKT conditions are also sufficient for local minimality despite the non-convexity of the constraint. Informed by the KKT conditions, a local search post-processing algorithm is proposed and shown to substantially and universally improve the structural Hamming distance of all tested algorithms, typically by a factor of 2 or more. Some combinations with local search are both more accurate and more efficient than the original NOTEARS.

We introduce a Cascaded Diffusion Model (Cas-DM) that improves a Denoising Diffusion Probabilistic Model (DDPM) by effectively incorporating additional metric functions in training. Metric functions such as the LPIPS loss have been proven highly effective in consistency models derived from the score matching. However, for the diffusion counterparts, the methodology and efficacy of adding extra metric functions remain unclear. One major challenge is the mismatch between the noise predicted by a DDPM at each step and the desired clean image that the metric function works well on. To address this problem, we propose Cas-DM, a network architecture that cascades two network modules to effectively apply metric functions to the diffusion model training. The first module, similar to a standard DDPM, learns to predict the added noise and is unaffected by the metric function. The second cascaded module learns to predict the clean image, thereby facilitating the metric function computation. Experiment results show that the proposed diffusion model backbone enables the effective use of the LPIPS loss, leading to state-of-the-art image quality (FID, sFID, IS) on various established benchmarks.

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: MultALMC, which uses multiscale annealed Langevin dynamics, and MultCDiff, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student's t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

Constructing prediction sets with coverage guarantees for unobserved outcomes is a core problem in modern statistics. Methods for predictive inference have been developed for a wide range of settings, but usually only consider test data points one at a time. Here we study the problem of distribution-free predictive inference for a batch of multiple test points, aiming to construct prediction sets for functions -- such as the mean or median -- of any number of unobserved test datapoints. This setting includes constructing simultaneous prediction sets with a high probability of coverage, and selecting datapoints satisfying a specified condition while controlling the number of false claims. For the general task of predictive inference on a function of a batch of test points, we introduce a methodology called batch predictive inference (batch PI), and provide a distribution-free coverage guarantee under exchangeability of the calibration and test data. Batch PI requires the quantiles of a rank ordering function defined on certain subsets of ranks. While computing these quantiles is NP-hard in general, we show that it can be done efficiently in many cases of interest, most notably for batch score functions with a compositional structure -- which includes examples of interest such as the mean -- via a dynamic programming algorithm that we develop. Batch PI has advantages over naive approaches (such as partitioning the calibration data or directly extending conformal prediction) in many settings, as it can deliver informative prediction sets even using small calibration sample sizes. We illustrate that our procedures provide informative inference across the use cases mentioned above, through experiments on both simulated data and a drug-target interaction dataset.

Control barrier functions (CBFs) enable guaranteed safe multi-agent navigation in the continuous domain. The resulting navigation performance, however, is highly sensitive to the underlying hyperparameters. Traditional approaches consider fixed CBFs (where parameters are tuned apriori), and hence, typically do not perform well in cluttered and highly dynamic environments: conservative parameter values can lead to inefficient agent trajectories, or even failure to reach goal positions, whereas aggressive parameter values can lead to infeasible controls. To overcome these issues, in this paper, we propose online CBFs, whereby hyperparameters are tuned in real-time, as a function of what agents perceive in their immediate neighborhood. Since the explicit relationship between CBFs and navigation performance is hard to model, we leverage reinforcement learning to learn CBF-tuning policies in a model-free manner. Because we parameterize the policies with graph neural networks (GNNs), we are able to synthesize decentralized agent controllers that adjust parameter values locally, varying the degree of conservative and aggressive behaviors across agents. Simulations as well as real-world experiments show that (i) online CBFs are capable of solving navigation scenarios that are infeasible for fixed CBFs, and (ii), that they improve navigation performance by adapting to other agents and changes in the environment.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Reinforcement Learning with Verifiable Rewards (RLVR) strengthens LLM reasoning, but training often oscillates between {entropy collapse} and {entropy explosion}. We trace both hazards to the mean baseline used in value-free RL (e.g., GRPO and DAPO), which improperly penalizes negative-advantage samples under reward outliers. We propose {Quantile Advantage Estimation} (QAE), replacing the mean with a group-wise K-quantile baseline. QAE induces a response-level, two-regime gate: on hard queries (p <= 1 - K) it reinforces rare successes, while on easy queries (p > 1 - K) it targets remaining failures. Under first-order softmax updates, we prove {two-sided entropy safety}, giving lower and upper bounds on one-step entropy change that curb explosion and prevent collapse. Empirically, this minimal modification stabilizes entropy, sparsifies credit assignment (with tuned K, roughly 80% of responses receive zero advantage), and yields sustained pass@1 gains on Qwen3-8B/14B-Base across AIME 2024/2025 and AMC 2023. These results identify {baseline design} -- rather than token-level heuristics -- as the primary mechanism for scaling RLVR.

Block coordinate descent (BCD) methods are widely used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can significantly improve the progress made by each BCD iteration. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with sparse dependencies between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active-set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

We consider the problem of conformal prediction under covariate shift. Given labeled data from a source domain and unlabeled data from a covariate shifted target domain, we seek to construct prediction sets with valid marginal coverage in the target domain. Most existing methods require estimating the unknown likelihood ratio function, which can be prohibitive for high-dimensional data such as images. To address this challenge, we introduce the likelihood ratio regularized quantile regression (LR-QR) algorithm, which combines the pinball loss with a novel choice of regularization in order to construct a threshold function without directly estimating the unknown likelihood ratio. We show that the LR-QR method has coverage at the desired level in the target domain, up to a small error term that we can control. Our proofs draw on a novel analysis of coverage via stability bounds from learning theory. Our experiments demonstrate that the LR-QR algorithm outperforms existing methods on high-dimensional prediction tasks, including a regression task for the Communities and Crime dataset, an image classification task from the WILDS repository, and an LLM question-answering task on the MMLU benchmark.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

We study the representation complexity of model-based and model-free reinforcement learning (RL) in the context of circuit complexity. We prove theoretically that there exists a broad class of MDPs such that their underlying transition and reward functions can be represented by constant depth circuits with polynomial size, while the optimal Q-function suffers an exponential circuit complexity in constant-depth circuits. By drawing attention to the approximation errors and building connections to complexity theory, our theory provides unique insights into why model-based algorithms usually enjoy better sample complexity than model-free algorithms from a novel representation complexity perspective: in some cases, the ground-truth rule (model) of the environment is simple to represent, while other quantities, such as Q-function, appear complex. We empirically corroborate our theory by comparing the approximation error of the transition kernel, reward function, and optimal Q-function in various Mujoco environments, which demonstrates that the approximation errors of the transition kernel and reward function are consistently lower than those of the optimal Q-function. To the best of our knowledge, this work is the first to study the circuit complexity of RL, which also provides a rigorous framework for future research.

This paper investigates the use of prior computation to estimate the value function to improve sample efficiency in on-policy policy gradient methods in reinforcement learning. Our approach is to estimate the value function from prior computations, such as from the Q-network learned in DQN or the value function trained for different but related environments. In particular, we learn a new value function for the target task while combining it with a value estimate from the prior computation. Finally, the resulting value function is used as a baseline in the policy gradient method. This use of a baseline has the theoretical property of reducing variance in gradient computation and thus improving sample efficiency. The experiments show the successful use of prior value estimates in various settings and improved sample efficiency in several tasks.

The assessment of safety performance plays a pivotal role in the development and deployment of connected and automated vehicles (CAVs). A common approach involves designing testing scenarios based on prior knowledge of CAVs (e.g., surrogate models), conducting tests in these scenarios, and subsequently evaluating CAVs' safety performances. However, substantial differences between CAVs and the prior knowledge can significantly diminish the evaluation efficiency. In response to this issue, existing studies predominantly concentrate on the adaptive design of testing scenarios during the CAV testing process. Yet, these methods have limitations in their applicability to high-dimensional scenarios. To overcome this challenge, we develop an adaptive testing environment that bolsters evaluation robustness by incorporating multiple surrogate models and optimizing the combination coefficients of these surrogate models to enhance evaluation efficiency. We formulate the optimization problem as a regression task utilizing quadratic programming. To efficiently obtain the regression target via reinforcement learning, we propose the dense reinforcement learning method and devise a new adaptive policy with high sample efficiency. Essentially, our approach centers on learning the values of critical scenes displaying substantial surrogate-to-real gaps. The effectiveness of our method is validated in high-dimensional overtaking scenarios, demonstrating that our approach achieves notable evaluation efficiency.

Score-based diffusion modeling is a generative machine learning algorithm that can be used to sample from complex distributions. They achieve this by learning a score function, i.e., the gradient of the log-probability density of the data, and reversing a noising process using the same. Once trained, score-based diffusion models not only generate new samples but also enable zero-shot conditioning of the generated samples on observed data. This promises a novel paradigm for data and model fusion, wherein the implicitly learned distributions of pretrained score-based diffusion models can be updated given the availability of online data in a Bayesian formulation. In this article, we apply such a concept to the super-resolution of a high-dimensional dynamical system, given the real-time availability of low-resolution and experimentally observed sparse sensor measurements from multimodal data. Additional analysis on how score-based sampling can be used for uncertainty estimates is also provided. Our experiments are performed for a super-resolution task that generates the ERA5 atmospheric dataset given sparse observations from a coarse-grained representation of the same and/or from unstructured experimental observations of the IGRA radiosonde dataset. We demonstrate accurate recovery of the high dimensional state given multiple sources of low-fidelity measurements. We also discover that the generative model can balance the influence of multiple dataset modalities during spatiotemporal reconstructions.

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.

Gradient descent has proven to be a powerful and effective technique for optimization in numerous machine learning applications. Recent advances in computational neuroscience have shown that learning in standard gradient descent optimization formulation is not consistent with learning in biological systems. This has opened up interesting avenues for building biologically inspired learning techniques. One such approach is inspired by Dale's law, which states that inhibitory and excitatory synapses do not swap roles during the course of learning. The resulting exponential gradient descent optimization scheme leads to log-normally distributed synaptic weights. Interestingly, the density that satisfies the Fokker-Planck equation corresponding to the stochastic differential equation (SDE) with geometric Brownian motion (GBM) is the log-normal density. Leveraging this connection, we start with the SDE governing geometric Brownian motion, and show that discretizing the corresponding reverse-time SDE yields a multiplicative update rule, which surprisingly, coincides with the sampling equivalent of the exponential gradient descent update founded on Dale's law. Furthermore, we propose a new formalism for multiplicative denoising score-matching, subsuming the loss function proposed by Hyvaerinen for non-negative data. Indeed, log-normally distributed data is positive and the proposed score-matching formalism turns out to be a natural fit. This allows for training of score-based models for image data and results in a novel multiplicative update scheme for sample generation starting from a log-normal density. Experimental results on MNIST, Fashion MNIST, and Kuzushiji datasets demonstrate generative capability of the new scheme. To the best of our knowledge, this is the first instance of a biologically inspired generative model employing multiplicative updates, founded on geometric Brownian motion.

Automated model selection is often proposed to users to choose which machine learning model (or method) to apply to a given regression task. In this paper, we show that combining different regression models can yield better results than selecting a single ('best') regression model, and outline an efficient method that obtains optimally weighted convex linear combination from a heterogeneous set of regression models. More specifically, in this paper, a heuristic weight optimization, used in a preceding conference paper, is replaced by an exact optimization algorithm using convex quadratic programming. We prove convexity of the quadratic programming formulation for the straightforward formulation and for a formulation with weighted data points. The novel weight optimization is not only (more) exact but also more efficient. The methods we develop in this paper are implemented and made available via github-open source. They can be executed on commonly available hardware and offer a transparent and easy to interpret interface. The results indicate that the approach outperforms model selection methods on a range of data sets, including data sets with mixed variable type from drug discovery applications.

Reversing a diffusion process by learning its score forms the heart of diffusion-based generative modeling and for estimating properties of scientific systems. The diffusion processes that are tractable center on linear processes with a Gaussian stationary distribution. This limits the kinds of models that can be built to those that target a Gaussian prior or more generally limits the kinds of problems that can be generically solved to those that have conditionally linear score functions. In this work, we introduce a family of tractable denoising score matching objectives, called local-DSM, built using local increments of the diffusion process. We show how local-DSM melded with Taylor expansions enables automated training and score estimation with nonlinear diffusion processes. To demonstrate these ideas, we use automated-DSM to train generative models using non-Gaussian priors on challenging low dimensional distributions and the CIFAR10 image dataset. Additionally, we use the automated-DSM to learn the scores for nonlinear processes studied in statistical physics.

Bayesian optimization (BO) is a powerful framework to optimize black-box expensive-to-evaluate functions via sequential interactions. In several important problems (e.g. drug discovery, circuit design, neural architecture search, etc.), though, such functions are defined over large combinatorial and unstructured spaces. This makes existing BO algorithms not feasible due to the intractable maximization of the acquisition function over these domains. To address this issue, we propose GameOpt, a novel game-theoretical approach to combinatorial BO. GameOpt establishes a cooperative game between the different optimization variables, and selects points that are game equilibria of an upper confidence bound acquisition function. These are stable configurations from which no variable has an incentive to deviate- analog to local optima in continuous domains. Crucially, this allows us to efficiently break down the complexity of the combinatorial domain into individual decision sets, making GameOpt scalable to large combinatorial spaces. We demonstrate the application of GameOpt to the challenging protein design problem and validate its performance on four real-world protein datasets. Each protein can take up to 20^{X} possible configurations, where X is the length of a protein, making standard BO methods infeasible. Instead, our approach iteratively selects informative protein configurations and very quickly discovers highly active protein variants compared to other baselines.

Score-based diffusion models learn to reverse a stochastic differential equation that maps data to noise. However, for complex tasks, numerical error can compound and result in highly unnatural samples. Previous work mitigates this drift with thresholding, which projects to the natural data domain (such as pixel space for images) after each diffusion step, but this leads to a mismatch between the training and generative processes. To incorporate data constraints in a principled manner, we present Reflected Diffusion Models, which instead reverse a reflected stochastic differential equation evolving on the support of the data. Our approach learns the perturbed score function through a generalized score matching loss and extends key components of standard diffusion models including diffusion guidance, likelihood-based training, and ODE sampling. We also bridge the theoretical gap with thresholding: such schemes are just discretizations of reflected SDEs. On standard image benchmarks, our method is competitive with or surpasses the state of the art without architectural modifications and, for classifier-free guidance, our approach enables fast exact sampling with ODEs and produces more faithful samples under high guidance weight.

With the growing prominence of the Mixture of Experts (MoE) architecture in developing large-scale foundation models, we investigate the Hierarchical Mixture of Experts (HMoE), a specialized variant of MoE that excels in handling complex inputs and improving performance on targeted tasks. Our analysis highlights the advantages of using the Laplace gating function over the traditional Softmax gating within the HMoE frameworks. We theoretically demonstrate that applying the Laplace gating function at both levels of the HMoE model helps eliminate undesirable parameter interactions caused by the Softmax gating and, therefore, accelerates the expert convergence as well as enhances the expert specialization. Empirical validation across diverse scenarios supports these theoretical claims. This includes large-scale multimodal tasks, image classification, and latent domain discovery and prediction tasks, where our modified HMoE models show great performance improvements compared to the conventional HMoE models.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

In this paper, we have studied the performance and role of local optimizers in quantum variational circuits. We studied the performance of the two most popular optimizers and compared their results with some popular classical machine learning algorithms. The classical algorithms we used in our study are support vector machine (SVM), gradient boosting (GB), and random forest (RF). These were compared with a variational quantum classifier (VQC) using two sets of local optimizers viz AQGD and COBYLA. For experimenting with VQC, IBM Quantum Experience and IBM Qiskit was used while for classical machine learning models, sci-kit learn was used. The results show that machine learning on noisy immediate scale quantum machines can produce comparable results as on classical machines. For our experiments, we have used a popular restaurant sentiment analysis dataset. The extracted features from this dataset and then after applying PCA reduced the feature set into 5 features. Quantum ML models were trained using 100 epochs and 150 epochs on using EfficientSU2 variational circuit. Overall, four Quantum ML models were trained and three Classical ML models were trained. The performance of the trained models was evaluated using standard evaluation measures viz, Accuracy, Precision, Recall, F-Score. In all the cases AQGD optimizer-based model with 100 Epochs performed better than all other models. It produced an accuracy of 77% and an F-Score of 0.785 which were highest across all the trained models.

We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the known lower bounds for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the recently proposed min-max regression procedure, thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all p-th power error functions for p in (2, infty). Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

Ensuring safety in autonomous systems requires controllers that satisfy hard, state-wise constraints without relying on online interaction. While existing Safe Offline RL methods typically enforce soft expected-cost constraints, they do not guarantee forward invariance. Conversely, Control Barrier Functions (CBFs) provide rigorous safety guarantees but usually depend on expert-designed barrier functions or full knowledge of the system dynamics. We introduce Value-Guided Offline Control Barrier Functions (V-OCBF), a framework that learns a neural CBF entirely from offline demonstrations. Unlike prior approaches, V-OCBF does not assume access to the dynamics model; instead, it derives a recursive finite-difference barrier update, enabling model-free learning of a barrier that propagates safety information over time. Moreover, V-OCBF incorporates an expectile-based objective that avoids querying the barrier on out-of-distribution actions and restricts updates to the dataset-supported action set. The learned barrier is then used with a Quadratic Program (QP) formulation to synthesize real-time safe control. Across multiple case studies, V-OCBF yields substantially fewer safety violations than baseline methods while maintaining strong task performance, highlighting its scalability for offline synthesis of safety-critical controllers without online interaction or hand-engineered barriers.

We study the effect of mini-batching on the loss landscape of deep neural networks using spiked, field-dependent random matrix theory. We demonstrate that the magnitude of the extremal values of the batch Hessian are larger than those of the empirical Hessian. We also derive similar results for the Generalised Gauss-Newton matrix approximation of the Hessian. As a consequence of our theorems we derive an analytical expressions for the maximal learning rates as a function of batch size, informing practical training regimens for both stochastic gradient descent (linear scaling) and adaptive algorithms, such as Adam (square root scaling), for smooth, non-convex deep neural networks. Whilst the linear scaling for stochastic gradient descent has been derived under more restrictive conditions, which we generalise, the square root scaling rule for adaptive optimisers is, to our knowledge, completely novel. %For stochastic second-order methods and adaptive methods, we derive that the minimal damping coefficient is proportional to the ratio of the learning rate to batch size. We validate our claims on the VGG/WideResNet architectures on the CIFAR-100 and ImageNet datasets. Based on our investigations of the sub-sampled Hessian we develop a stochastic Lanczos quadrature based on the fly learning rate and momentum learner, which avoids the need for expensive multiple evaluations for these key hyper-parameters and shows good preliminary results on the Pre-Residual Architecure for CIFAR-100.

Most value function learning algorithms in reinforcement learning are based on the mean squared (projected) Bellman error. However, squared errors are known to be sensitive to outliers, both skewing the solution of the objective and resulting in high-magnitude and high-variance gradients. To control these high-magnitude updates, typical strategies in RL involve clipping gradients, clipping rewards, rescaling rewards, or clipping errors. While these strategies appear to be related to robust losses -- like the Huber loss -- they are built on semi-gradient update rules which do not minimize a known loss. In this work, we build on recent insights reformulating squared Bellman errors as a saddlepoint optimization problem and propose a saddlepoint reformulation for a Huber Bellman error and Absolute Bellman error. We start from a formalization of robust losses, then derive sound gradient-based approaches to minimize these losses in both the online off-policy prediction and control settings. We characterize the solutions of the robust losses, providing insight into the problem settings where the robust losses define notably better solutions than the mean squared Bellman error. Finally, we show that the resulting gradient-based algorithms are more stable, for both prediction and control, with less sensitivity to meta-parameters.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating C^s smooth functions with s >0 and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.

This paper introduces a novel Hamiltonian-inspired neural network approach to credit scoring, designed to address the challenges of class imbalance and out-of-time (OOT) prediction in financial risk management. Drawing from concepts in Hamiltonian mechanics, we develop a symplectic optimizer and a new loss function to capture the complex dynamics of credit risk evolution. Using the Freddie Mac Single-Family Loan-Level Dataset, we evaluate our model's performance against other machine learning approaches. Our method shows superior discriminative power in OOT scenarios, as measured by the Area Under the Curve (AUC), indicating better ranking ability and robustness to class imbalance. The Hamiltonian-inspired approach shows particular strength in maintaining consistent performance between in-sample and OOT test sets, suggesting improved generalization to future, unseen data. These findings suggest that physics-inspired techniques offer a promising direction for developing more robust and reliable credit scoring models, particularly in uncertain economic situations.

Quantile regression is a fundamental problem in statistical learning motivated by a need to quantify uncertainty in predictions, or to model a diverse population without being overly reductive. For instance, epidemiological forecasts, cost estimates, and revenue predictions all benefit from being able to quantify the range of possible values accurately. As such, many models have been developed for this problem over many years of research in statistics, machine learning, and related fields. Rather than proposing yet another (new) algorithm for quantile regression we adopt a meta viewpoint: we investigate methods for aggregating any number of conditional quantile models, in order to improve accuracy and robustness. We consider weighted ensembles where weights may vary over not only individual models, but also over quantile levels, and feature values. All of the models we consider in this paper can be fit using modern deep learning toolkits, and hence are widely accessible (from an implementation point of view) and scalable. To improve the accuracy of the predicted quantiles (or equivalently, prediction intervals), we develop tools for ensuring that quantiles remain monotonically ordered, and apply conformal calibration methods. These can be used without any modification of the original library of base models. We also review some basic theory surrounding quantile aggregation and related scoring rules, and contribute a few new results to this literature (for example, the fact that post sorting or post isotonic regression can only improve the weighted interval score). Finally, we provide an extensive suite of empirical comparisons across 34 data sets from two different benchmark repositories.

Reinforcement learning (RL) has gained increasing interest since the demonstration it was able to reach human performance on video game benchmarks using deep Q-learning (DQN). The current consensus for training neural networks on such complex environments is to rely on gradient-based optimization. Although alternative Bayesian deep learning methods exist, most of them still rely on gradient-based optimization, and they typically do not scale on benchmarks such as the Atari game environment. Moreover none of these approaches allow performing the analytical inference for the weights and biases defining the neural network. In this paper, we present how we can adapt the temporal difference Q-learning framework to make it compatible with the tractable approximate Gaussian inference (TAGI), which allows learning the parameters of a neural network using a closed-form analytical method. Throughout the experiments with on- and off-policy reinforcement learning approaches, we demonstrate that TAGI can reach a performance comparable to backpropagation-trained networks while using fewer hyperparameters, and without relying on gradient-based optimization.

This paper considers the problem of combinatorial multi-armed bandits with semi-bandit feedback and a cardinality constraint on the super-arm size. Existing algorithms for solving this problem typically involve two key sub-routines: (1) a parameter estimation routine that sequentially estimates a set of base-arm parameters, and (2) a super-arm selection policy for selecting a subset of base arms deemed optimal based on these parameters. State-of-the-art algorithms assume access to an exact oracle for super-arm selection with unbounded computational power. At each instance, this oracle evaluates a list of score functions, the number of which grows as low as linearly and as high as exponentially with the number of arms. This can be prohibitive in the regime of a large number of arms. This paper introduces a novel realistic alternative to the perfect oracle. This algorithm uses a combination of group-testing for selecting the super arms and quantized Thompson sampling for parameter estimation. Under a general separability assumption on the reward function, the proposed algorithm reduces the complexity of the super-arm-selection oracle to be logarithmic in the number of base arms while achieving the same regret order as the state-of-the-art algorithms that use exact oracles. This translates to at least an exponential reduction in complexity compared to the oracle-based approaches.

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over K episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order mathcal O(K^{2/3}) (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to mathcal O(sqrt K) in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves mathcal O(K^{8/9}) regret and greatly improves over the best existing bound mathcal O(K^{14/15}). This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

Scoring systems are linear classification models that only require users to add, subtract and multiply a few small numbers in order to make a prediction. These models are in widespread use by the medical community, but are difficult to learn from data because they need to be accurate and sparse, have coprime integer coefficients, and satisfy multiple operational constraints. We present a new method for creating data-driven scoring systems called a Supersparse Linear Integer Model (SLIM). SLIM scoring systems are built by solving an integer program that directly encodes measures of accuracy (the 0-1 loss) and sparsity (the ell_0-seminorm) while restricting coefficients to coprime integers. SLIM can seamlessly incorporate a wide range of operational constraints related to accuracy and sparsity, and can produce highly tailored models without parameter tuning. We provide bounds on the testing and training accuracy of SLIM scoring systems, and present a new data reduction technique that can improve scalability by eliminating a portion of the training data beforehand. Our paper includes results from a collaboration with the Massachusetts General Hospital Sleep Laboratory, where SLIM was used to create a highly tailored scoring system for sleep apnea screening

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

Diffusion and flow-matching models achieve remarkable generative performance but at the cost of many sampling steps, this slows inference and limits applicability to time-critical tasks. The ReFlow procedure can accelerate sampling by straightening generation trajectories. However, ReFlow is an iterative procedure, typically requiring training on simulated data, and results in reduced sample quality. To mitigate sample deterioration, we examine the design space of ReFlow and highlight potential pitfalls in prior heuristic practices. We then propose seven improvements for training dynamics, learning and inference, which are verified with thorough ablation studies on CIFAR10 32 times 32, AFHQv2 64 times 64, and FFHQ 64 times 64. Combining all our techniques, we achieve state-of-the-art FID scores (without / with guidance, resp.) for fast generation via neural ODEs: 2.23 / 1.98 on CIFAR10, 2.30 / 1.91 on AFHQv2, 2.84 / 2.67 on FFHQ, and 3.49 / 1.74 on ImageNet-64, all with merely 9 neural function evaluations.

The rapid progress of multi-modal large language models (MLLMs) has boosted the task of image quality assessment (IQA). However, a key challenge arises from the inherent mismatch between the discrete token outputs of MLLMs and the continuous nature of quality scores required by IQA tasks. This discrepancy significantly hinders the performance of MLLM-based IQA methods. Previous approaches that convert discrete token predictions into continuous scores often suffer from conversion errors. Moreover, the semantic confusion introduced by level tokens (e.g., ``good'') further constrains the performance of MLLMs on IQA tasks and degrades their original capabilities for related tasks. To tackle these problems, we provide a theoretical analysis of the errors inherent in previous approaches and, motivated by this analysis, propose a simple yet effective framework, Q-Scorer. This framework incorporates a lightweight regression module and IQA-specific score tokens into the MLLM pipeline. Extensive experiments demonstrate that Q-Scorer achieves state-of-the-art performance across multiple IQA benchmarks, generalizes well to mixed datasets, and further improves when combined with other methods.

Large pretrained models, coupled with fine-tuning, are slowly becoming established as the dominant architecture in machine learning. Even though these models offer impressive performance, their practical application is often limited by the prohibitive amount of resources required for every inference. Early-exiting dynamic neural networks (EDNN) circumvent this issue by allowing a model to make some of its predictions from intermediate layers (i.e., early-exit). Training an EDNN architecture is challenging as it consists of two intertwined components: the gating mechanism (GM) that controls early-exiting decisions and the intermediate inference modules (IMs) that perform inference from intermediate representations. As a result, most existing approaches rely on thresholding confidence metrics for the gating mechanism and strive to improve the underlying backbone network and the inference modules. Although successful, this approach has two fundamental shortcomings: 1) the GMs and the IMs are decoupled during training, leading to a train-test mismatch; and 2) the thresholding gating mechanism introduces a positive bias into the predictive probabilities, making it difficult to readily extract uncertainty information. We propose a novel architecture that connects these two modules. This leads to significant performance improvements on classification datasets and enables better uncertainty characterization capabilities.

State-of-the-art neural network training methods depend on the gradient of the network function. Therefore, they cannot be applied to networks whose activation functions do not have useful derivatives, such as binary and discrete-time spiking neural networks. To overcome this problem, the activation function's derivative is commonly substituted with a surrogate derivative, giving rise to surrogate gradient learning (SGL). This method works well in practice but lacks theoretical foundation. The neural tangent kernel (NTK) has proven successful in the analysis of gradient descent. Here, we provide a generalization of the NTK, which we call the surrogate gradient NTK, that enables the analysis of SGL. First, we study a naive extension of the NTK to activation functions with jumps, demonstrating that gradient descent for such activation functions is also ill-posed in the infinite-width limit. To address this problem, we generalize the NTK to gradient descent with surrogate derivatives, i.e., SGL. We carefully define this generalization and expand the existing key theorems on the NTK with mathematical rigor. Further, we illustrate our findings with numerical experiments. Finally, we numerically compare SGL in networks with sign activation function and finite width to kernel regression with the surrogate gradient NTK; the results confirm that the surrogate gradient NTK provides a good characterization of SGL.

Motivated by the problem of fast processing of attention matrices, we study fast algorithms for computing matrix-vector products for asymmetric Gaussian Kernel matrices Kin R^{ntimes n}. K's columns are indexed by a set of n keys k_1,k_2ldots, k_nin R^d, rows by a set of n queries q_1,q_2,ldots,q_nin R^d , and its i,j entry is K_{ij} = e^{-|q_i-k_j|_2^2/2sigma^2} for some bandwidth parameter sigma>0. Given a vector xin R^n and error parameter epsilon>0, our task is to output a yin R^n such that |Kx-y|_2leq epsilon |x|_2 in time subquadratic in n and linear in d. Our algorithms rely on the following modelling assumption about the matrices K: the sum of the entries of K scales linearly in n, as opposed to worst case quadratic growth. We validate this assumption experimentally, for Gaussian kernel matrices encountered in various settings such as fast attention computation in LLMs. We obtain the first subquadratic-time algorithm that works under this assumption, for unrestricted vectors.

We show how to obtain improved active learning methods in the agnostic (adversarial noise) setting by combining marginal leverage score sampling with non-independent sampling strategies that promote spatial coverage. In particular, we propose an easily implemented method based on the pivotal sampling algorithm, which we test on problems motivated by learning-based methods for parametric PDEs and uncertainty quantification. In comparison to independent sampling, our method reduces the number of samples needed to reach a given target accuracy by up to 50%. We support our findings with two theoretical results. First, we show that any non-independent leverage score sampling method that obeys a weak one-sided ell_{infty} independence condition (which includes pivotal sampling) can actively learn d dimensional linear functions with O(dlog d) samples, matching independent sampling. This result extends recent work on matrix Chernoff bounds under ell_{infty} independence, and may be of interest for analyzing other sampling strategies beyond pivotal sampling. Second, we show that, for the important case of polynomial regression, our pivotal method obtains an improved bound of O(d) samples.

Stochastically Extended Adversarial (SEA) model is introduced by Sachs et al. [2022] as an interpolation between stochastic and adversarial online convex optimization. Under the smoothness condition, they demonstrate that the expected regret of optimistic follow-the-regularized-leader (FTRL) depends on the cumulative stochastic variance sigma_{1:T}^2 and the cumulative adversarial variation Sigma_{1:T}^2 for convex functions. They also provide a slightly weaker bound based on the maximal stochastic variance sigma_{max}^2 and the maximal adversarial variation Sigma_{max}^2 for strongly convex functions. Inspired by their work, we investigate the theoretical guarantees of optimistic online mirror descent (OMD) for the SEA model. For convex and smooth functions, we obtain the same O(sigma_{1:T^2}+Sigma_{1:T^2}) regret bound, without the convexity requirement of individual functions. For strongly convex and smooth functions, we establish an O(min{log (sigma_{1:T}^2+Sigma_{1:T}^2), (sigma_{max}^2 + Sigma_{max}^2) log T}) bound, better than their O((sigma_{max}^2 + Sigma_{max}^2) log T) bound. For exp-concave and smooth functions, we achieve a new O(dlog(sigma_{1:T}^2+Sigma_{1:T}^2)) bound. Owing to the OMD framework, we can further extend our result to obtain dynamic regret guarantees, which are more favorable in non-stationary online scenarios. The attained results allow us to recover excess risk bounds of the stochastic setting and regret bounds of the adversarial setting, and derive new guarantees for many intermediate scenarios.

Recent works have shown a reduction from contextual bandits to online regression under a realizability assumption [Foster and Rakhlin, 2020, Foster and Krishnamurthy, 2021]. In this work, we investigate the use of neural networks for such online regression and associated Neural Contextual Bandits (NeuCBs). Using existing results for wide networks, one can readily show a {O}(T) regret for online regression with square loss, which via the reduction implies a {O}(K T^{3/4}) regret for NeuCBs. Departing from this standard approach, we first show a O(log T) regret for online regression with almost convex losses that satisfy QG (Quadratic Growth) condition, a generalization of the PL (Polyak-\L ojasiewicz) condition, and that have a unique minima. Although not directly applicable to wide networks since they do not have unique minima, we show that adding a suitable small random perturbation to the network predictions surprisingly makes the loss satisfy QG with unique minima. Based on such a perturbed prediction, we show a {O}(log T) regret for online regression with both squared loss and KL loss, and subsequently convert these respectively to mathcal{O}(KT) and mathcal{O}(KL^* + K) regret for NeuCB, where L^* is the loss of the best policy. Separately, we also show that existing regret bounds for NeuCBs are Omega(T) or assume i.i.d. contexts, unlike this work. Finally, our experimental results on various datasets demonstrate that our algorithms, especially the one based on KL loss, persistently outperform existing algorithms.

Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper is devoted to the study of the problem of model selection among a collection of GLoME or BLoME models characterized by the number of mixture components, the complexity of Gaussian mean experts, and the hidden block-diagonal structures of the covariance matrices, in a penalized maximum likelihood estimation framework. In particular, we establish non-asymptotic risk bounds that take the form of weak oracle inequalities, provided that lower bounds for the penalties hold. The good empirical behavior of our models is then demonstrated on synthetic and real datasets.

We propose a Monte Carlo sampler from the reverse diffusion process. Unlike the practice of diffusion models, where the intermediary updates -- the score functions -- are learned with a neural network, we transform the score matching problem into a mean estimation one. By estimating the means of the regularized posterior distributions, we derive a novel Monte Carlo sampling algorithm called reverse diffusion Monte Carlo (rdMC), which is distinct from the Markov chain Monte Carlo (MCMC) methods. We determine the sample size from the error tolerance and the properties of the posterior distribution to yield an algorithm that can approximately sample the target distribution with any desired accuracy. Additionally, we demonstrate and prove under suitable conditions that sampling with rdMC can be significantly faster than that with MCMC. For multi-modal target distributions such as those in Gaussian mixture models, rdMC greatly improves over the Langevin-style MCMC sampling methods both theoretically and in practice. The proposed rdMC method offers a new perspective and solution beyond classical MCMC algorithms for the challenging complex distributions.

Probing studies what information is encoded in a frozen LLM's layer representations by training a lightweight predictor on top of them. Beyond analysis, probes are often used operationally in probe-then-steer pipelines: a learned concept vector is extracted from a probe and injected via additive activation steering by adding it to a layer representation during the forward pass. The effectiveness of this pipeline hinges on estimating concept vectors that are accurate, directionally stable under ablation, and inexpensive to obtain. Motivated by these desiderata, we propose RAPTOR (Ridge-Adaptive Logistic Probe), a simple L2-regularized logistic probe whose validation-tuned ridge strength yields concept vectors from normalized weights. Across extensive experiments on instruction-tuned LLMs and human-written concept datasets, RAPTOR matches or exceeds strong baselines in accuracy while achieving competitive directional stability and substantially lower training cost; these quantitative results are supported by qualitative downstream steering demonstrations. Finally, using the Convex Gaussian Min-max Theorem (CGMT), we provide a mechanistic characterization of ridge logistic regression in an idealized Gaussian teacher-student model in the high-dimensional few-shot regime, explaining how penalty strength mediates probe accuracy and concept-vector stability and yielding structural predictions that qualitatively align with trends observed on real LLM embeddings.

In this work, a method of random parameters generation for randomized learning of a single-hidden-layer feedforward neural network is proposed. The method firstly, randomly selects the slope angles of the hidden neurons activation functions from an interval adjusted to the target function, then randomly rotates the activation functions, and finally distributes them across the input space. For complex target functions the proposed method gives better results than the approach commonly used in practice, where the random parameters are selected from the fixed interval. This is because it introduces the steepest fragments of the activation functions into the input hypercube, avoiding their saturation fragments.

Recent studies of gradient descent with large step sizes have shown that there is often a regime with an initial increase in the largest eigenvalue of the loss Hessian (progressive sharpening), followed by a stabilization of the eigenvalue near the maximum value which allows convergence (edge of stability). These phenomena are intrinsically non-linear and do not happen for models in the constant Neural Tangent Kernel (NTK) regime, for which the predictive function is approximately linear in the parameters. As such, we consider the next simplest class of predictive models, namely those that are quadratic in the parameters, which we call second-order regression models. For quadratic objectives in two dimensions, we prove that this second-order regression model exhibits progressive sharpening of the NTK eigenvalue towards a value that differs slightly from the edge of stability, which we explicitly compute. In higher dimensions, the model generically shows similar behavior, even without the specific structure of a neural network, suggesting that progressive sharpening and edge-of-stability behavior aren't unique features of neural networks, and could be a more general property of discrete learning algorithms in high-dimensional non-linear models.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

Diffusion models have emerged as the principal paradigm for generative modeling across various domains. During training, they learn the score function, which in turn is used to generate samples at inference. They raise a basic yet unsolved question: which score do they actually learn? In principle, a diffusion model that matches the empirical score in the entire data space would simply reproduce the training data, failing to generate novel samples. Recent work addresses this question by arguing that diffusion models underfit the empirical score due to training-time inductive biases. In this work, we refine this perspective, introducing the notion of selective underfitting: instead of underfitting the score everywhere, better diffusion models more accurately approximate the score in certain regions of input space, while underfitting it in others. We characterize these regions and design empirical interventions to validate our perspective. Our results establish that selective underfitting is essential for understanding diffusion models, yielding new, testable insights into their generalization and generative performance.

In this work, we study policy poisoning through state manipulation, also known as sensor spoofing, and focus specifically on the case of an agent forming a control policy through batch learning in a linear-quadratic (LQ) system. In this scenario, an attacker aims to trick the learner into implementing a targeted malicious policy by manipulating the batch data before the agent begins its learning process. An attack model is crafted to carry out the poisoning strategically, with the goal of modifying the batch data as little as possible to avoid detection by the learner. We establish an optimization framework to guide the design of such policy poisoning attacks. The presence of bi-linear constraints in the optimization problem requires the design of a computationally efficient algorithm to obtain a solution. Therefore, we develop an iterative scheme based on the Alternating Direction Method of Multipliers (ADMM) which is able to return solutions that are approximately optimal. Several case studies are used to demonstrate the effectiveness of the algorithm in carrying out the sensor-based attack on the batch-learning agent in LQ control systems.

We revisit the structure learning problem for dynamic Bayesian networks and propose a method that simultaneously estimates contemporaneous (intra-slice) and time-lagged (inter-slice) relationships between variables in a time-series. Our approach is score-based, and revolves around minimizing a penalized loss subject to an acyclicity constraint. To solve this problem, we leverage a recent algebraic result characterizing the acyclicity constraint as a smooth equality constraint. The resulting algorithm, which we call DYNOTEARS, outperforms other methods on simulated data, especially in high-dimensions as the number of variables increases. We also apply this algorithm on real datasets from two different domains, finance and molecular biology, and analyze the resulting output. Compared to state-of-the-art methods for learning dynamic Bayesian networks, our method is both scalable and accurate on real data. The simple formulation and competitive performance of our method make it suitable for a variety of problems where one seeks to learn connections between variables across time.

Architecture optimization, which is a technique for finding an efficient neural network that meets certain requirements, generally reduces to a set of multiple-choice selection problems among alternative sub-structures or parameters. The discrete nature of the selection problem, however, makes this optimization difficult. To tackle this problem we introduce a novel concept of a trainable gate function. The trainable gate function, which confers a differentiable property to discretevalued variables, allows us to directly optimize loss functions that include non-differentiable discrete values such as 0-1 selection. The proposed trainable gate can be applied to pruning. Pruning can be carried out simply by appending the proposed trainable gate functions to each intermediate output tensor followed by fine-tuning the overall model, using any gradient-based training methods. So the proposed method can jointly optimize the selection of the pruned channels while fine-tuning the weights of the pruned model at the same time. Our experimental results demonstrate that the proposed method efficiently optimizes arbitrary neural networks in various tasks such as image classification, style transfer, optical flow estimation, and neural machine translation.

Quantized neural networks employ reduced precision representations for both weights and activations. This quantization process significantly reduces the memory requirements and computational complexity of the network. Binary Neural Networks (BNNs) are the extreme quantization case, representing values with just one bit. Since the sign function is typically used to map real values to binary values, smooth approximations are introduced to mimic the gradients during error backpropagation. Thus, the mismatch between the forward and backward models corrupts the direction of the gradient, causing training inconsistency problems and performance degradation. In contrast to current BNN approaches, we propose to employ a binary periodic (BiPer) function during binarization. Specifically, we use a square wave for the forward pass to obtain the binary values and employ the trigonometric sine function with the same period of the square wave as a differentiable surrogate during the backward pass. We demonstrate that this approach can control the quantization error by using the frequency of the periodic function and improves network performance. Extensive experiments validate the effectiveness of BiPer in benchmark datasets and network architectures, with improvements of up to 1% and 0.69% with respect to state-of-the-art methods in the classification task over CIFAR-10 and ImageNet, respectively. Our code is publicly available at https://github.com/edmav4/BiPer.

The Reinforcement Learning (RL) building blocks, i.e. Q-functions and policy networks, usually take elements from the cartesian product of two domains as input. In particular, the input of the Q-function is both the state and the action, and in multi-task problems (Meta-RL) the policy can take a state and a context. Standard architectures tend to ignore these variables' underlying interpretations and simply concatenate their features into a single vector. In this work, we argue that this choice may lead to poor gradient estimation in actor-critic algorithms and high variance learning steps in Meta-RL algorithms. To consider the interaction between the input variables, we suggest using a Hypernetwork architecture where a primary network determines the weights of a conditional dynamic network. We show that this approach improves the gradient approximation and reduces the learning step variance, which both accelerates learning and improves the final performance. We demonstrate a consistent improvement across different locomotion tasks and different algorithms both in RL (TD3 and SAC) and in Meta-RL (MAML and PEARL).

We present SURE-Score: an approach for learning score-based generative models using training samples corrupted by additive Gaussian noise. When a large training set of clean samples is available, solving inverse problems via score-based (diffusion) generative models trained on the underlying fully-sampled data distribution has recently been shown to outperform end-to-end supervised deep learning. In practice, such a large collection of training data may be prohibitively expensive to acquire in the first place. In this work, we present an approach for approximately learning a score-based generative model of the clean distribution, from noisy training data. We formulate and justify a novel loss function that leverages Stein's unbiased risk estimate to jointly denoise the data and learn the score function via denoising score matching, while using only the noisy samples. We demonstrate the generality of SURE-Score by learning priors and applying posterior sampling to ill-posed inverse problems in two practical applications from different domains: compressive wireless multiple-input multiple-output channel estimation and accelerated 2D multi-coil magnetic resonance imaging reconstruction, where we demonstrate competitive reconstruction performance when learning at signal-to-noise ratio values of 0 and 10 dB, respectively.

Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the important conditions for the success of neural networks is the choice of an appropriate activation function introducing non-linearity into the model. Many types of these functions have been proposed in the literature in the past, but there is no single comprehensive source containing their exhaustive overview. The absence of this overview, even in our experience, leads to redundancy and the unintentional rediscovery of already existing activation functions. To bridge this gap, our paper presents an extensive survey involving 400 activation functions, which is several times larger in scale than previous surveys. Our comprehensive compilation also references these surveys; however, its main goal is to provide the most comprehensive overview and systematization of previously published activation functions with links to their original sources. The secondary aim is to update the current understanding of this family of functions.

We examine a simple stochastic strategy for adapting well-known single-point acquisition functions to allow batch active learning. Unlike acquiring the top-K points from the pool set, score- or rank-based sampling takes into account that acquisition scores change as new data are acquired. This simple strategy for adapting standard single-sample acquisition strategies can even perform just as well as compute-intensive state-of-the-art batch acquisition functions, like BatchBALD or BADGE, while using orders of magnitude less compute. In addition to providing a practical option for machine learning practitioners, the surprising success of the proposed method in a wide range of experimental settings raises a difficult question for the field: when are these expensive batch acquisition methods pulling their weight?

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) infinite-width BNNs are particularly promising, especially in high dimensions.

Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding epsilon-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to p-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is Omegabig(epsilon^{-1+p{p}}big) regarding the first setting, and Omega(epsilon^{-4}) regarding the second setting (or Omega(epsilon^{-3}) if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding epsilon-stationary points of nonconvex functions with p-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.

Multivariate probabilistic time series forecasts are commonly evaluated via proper scoring rules, i.e., functions that are minimal in expectation for the ground-truth distribution. However, this property is not sufficient to guarantee good discrimination in the non-asymptotic regime. In this paper, we provide the first systematic finite-sample study of proper scoring rules for time-series forecasting evaluation. Through a power analysis, we identify the "region of reliability" of a scoring rule, i.e., the set of practical conditions where it can be relied on to identify forecasting errors. We carry out our analysis on a comprehensive synthetic benchmark, specifically designed to test several key discrepancies between ground-truth and forecast distributions, and we gauge the generalizability of our findings to real-world tasks with an application to an electricity production problem. Our results reveal critical shortcomings in the evaluation of multivariate probabilistic forecasts as commonly performed in the literature.

Reinforcement Learning with Verifiable Rewards (RLVR) reliably improves the reasoning performance of large language models, yet it appears to modify only a small fraction of parameters. We revisit this paradox and show that sparsity is a surface artifact of a model-conditioned optimization bias: for a fixed pretrained model, updates consistently localize to preferred parameter regions, highly consistent across runs and largely invariant to datasets and RL recipes. We mechanistically explain these dynamics with a Three-Gate Theory: Gate I (KL Anchor) imposes a KL-constrained update; Gate II (Model Geometry) steers the step off principal directions into low-curvature, spectrum-preserving subspaces; and Gate III (Precision) hides micro-updates in non-preferred regions, making the off-principal bias appear as sparsity. We then validate this theory and, for the first time, provide a parameter-level characterization of RLVR's learning dynamics: RLVR learns off principal directions in weight space, achieving gains via minimal spectral drift, reduced principal-subspace rotation, and off-principal update alignment. In contrast, SFT targets principal weights, distorts the spectrum, and even lags RLVR. Together, these results provide the first parameter-space account of RLVR's training dynamics, revealing clear regularities in how parameters evolve. Crucially, we show that RL operates in a distinct optimization regime from SFT, so directly adapting SFT-era parameter-efficient fine-tuning (PEFT) methods can be flawed, as evidenced by our case studies on advanced sparse fine-tuning and LoRA variants. We hope this work charts a path toward a white-box understanding of RLVR and the design of geometry-aware, RLVR-native learning algorithms, rather than repurposed SFT-era heuristics.

Recent advances in one-step generative models typically follow a two-stage process: first training a teacher diffusion model and then distilling it into a one-step student model. This distillation process traditionally relies on both the teacher model's score function to compute the distillation loss and its weights for student initialization. In this paper, we explore whether one-step generative models can be trained directly without this distillation process. First, we show that the teacher's score function is not essential and propose a family of distillation methods that achieve competitive results without relying on score estimation. Next, we demonstrate that initialization from teacher weights is indispensable in successful training. Surprisingly, we find that this benefit is not due to improved ``input-output" mapping but rather the learned feature representations, which dominate distillation quality. Our findings provide a better understanding of the role of initialization in one-step model training and its impact on distillation quality.

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this assumption implies that the target of learning is a conservative vector field, which is not enforced by the neural network architectures used in practice. We present numerical evidence that trained diffusion networks violate both integral and differential constraints required of true score functions, demonstrating that the learned vector fields are not conservative. Despite this, the models perform remarkably well as generative mechanisms. To explain this apparent paradox, we advocate a new theoretical perspective: diffusion training is better understood as flow matching to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate for adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

Graphical User Interface (GUI) grounding maps natural language instructions to precise interface locations for autonomous interaction. Current reinforcement learning approaches use binary rewards that treat elements as hit-or-miss targets, creating sparse signals that ignore the continuous nature of spatial interactions. Motivated by human clicking behavior that naturally forms Gaussian distributions centered on target elements, we introduce GUI Gaussian Grounding Rewards (GUI-G^2), a principled reward framework that models GUI elements as continuous Gaussian distributions across the interface plane. GUI-G^2 incorporates two synergistic mechanisms: Gaussian point rewards model precise localization through exponentially decaying distributions centered on element centroids, while coverage rewards assess spatial alignment by measuring the overlap between predicted Gaussian distributions and target regions. To handle diverse element scales, we develop an adaptive variance mechanism that calibrates reward distributions based on element dimensions. This framework transforms GUI grounding from sparse binary classification to dense continuous optimization, where Gaussian distributions generate rich gradient signals that guide models toward optimal interaction positions. Extensive experiments across ScreenSpot, ScreenSpot-v2, and ScreenSpot-Pro benchmarks demonstrate that GUI-G^2, substantially outperforms state-of-the-art method UI-TARS-72B, with the most significant improvement of 24.7% on ScreenSpot-Pro. Our analysis reveals that continuous modeling provides superior robustness to interface variations and enhanced generalization to unseen layouts, establishing a new paradigm for spatial reasoning in GUI interaction tasks.

Many existing conditional score-based data generation methods utilize Bayes' theorem to decompose the gradients of a log posterior density into a mixture of scores. These methods facilitate the training procedure of conditional score models, as a mixture of scores can be separately estimated using a score model and a classifier. However, our analysis indicates that the training objectives for the classifier in these methods may lead to a serious score mismatch issue, which corresponds to the situation that the estimated scores deviate from the true ones. Such an issue causes the samples to be misled by the deviated scores during the diffusion process, resulting in a degraded sampling quality. To resolve it, we formulate a novel training objective, called Denoising Likelihood Score Matching (DLSM) loss, for the classifier to match the gradients of the true log likelihood density. Our experimental evidence shows that the proposed method outperforms the previous methods on both Cifar-10 and Cifar-100 benchmarks noticeably in terms of several key evaluation metrics. We thus conclude that, by adopting DLSM, the conditional scores can be accurately modeled, and the effect of the score mismatch issue is alleviated.

The Quantum-Inspired Stacked Integrated Concept Graph Model (QISICGM) is an innovative machine learning framework that harnesses quantum-inspired techniques to predict diabetes risk with exceptional accuracy and efficiency. Utilizing the PIMA Indians Diabetes dataset augmented with 2,000 synthetic samples to mitigate class imbalance (total: 2,768 samples, 1,949 positives), QISICGM integrates a self-improving concept graph with a stacked ensemble comprising Random Forests (RF), Extra Trees (ET), transformers, convolutional neural networks (CNNs), and feed-forward neural networks (FFNNs). This approach achieves an out-of-fold (OOF) F1 score of 0.8933 and an AUC of 0.8699, outperforming traditional methods. Quantum inspired elements, such as phase feature mapping and neighborhood sequence modeling, enrich feature representations, enabling CPU-efficient inference at 8.5 rows per second. This paper presents a detailed architecture, theoretical foundations, code insights, and performance evaluations, including visualizations from the outputs subfolder. The open-source implementation (v1.0.0) is available at https://github.com/keninayoung/QISICGM, positioning QISICGM as a potential benchmark for AI-assisted clinical triage in diabetes and beyond. Ultimately, this work emphasizes trustworthy AI through calibration, interpretability, and open-source reproducibility.

When trying to solve a computational problem, we are often faced with a choice between algorithms that are guaranteed to return the right answer but differ in their runtime distributions (e.g., SAT solvers, sorting algorithms). This paper aims to lay theoretical foundations for such choices by formalizing preferences over runtime distributions. It might seem that we should simply prefer the algorithm that minimizes expected runtime. However, such preferences would be driven by exactly how slow our algorithm is on bad inputs, whereas in practice we are typically willing to cut off occasional, sufficiently long runs before they finish. We propose a principled alternative, taking a utility-theoretic approach to characterize the scoring functions that describe preferences over algorithms. These functions depend on the way our value for solving our problem decreases with time and on the distribution from which captimes are drawn. We describe examples of realistic utility functions and show how to leverage a maximum-entropy approach for modeling underspecified captime distributions. Finally, we show how to efficiently estimate an algorithm's expected utility from runtime samples.

This study develops and empirically validates a Mixture of Experts (MoE) framework for stock price prediction across heterogeneous volatility regimes using real market data. The proposed model combines a Recurrent Neural Network (RNN) optimized for high-volatility stocks with a linear regression model tailored to stable equities. A volatility-aware gating mechanism dynamically weights the contributions of each expert based on asset classification. Using a dataset of 30 publicly traded U.S. stocks spanning diverse sectors, the MoE approach consistently outperforms both standalone models. Specifically, it achieves up to 33% improvement in MSE for volatile assets and 28% for stable assets relative to their respective baselines. Stratified evaluation across volatility classes demonstrates the model's ability to adapt complexity to underlying market dynamics. These results confirm that no single model suffices across market regimes and highlight the advantage of adaptive architectures in financial prediction. Future work should explore real-time gate learning, dynamic volatility segmentation, and applications to portfolio optimization.

The objective of continual learning (CL) is to learn tasks sequentially without retraining on earlier tasks. However, when subjected to CL, traditional neural networks exhibit catastrophic forgetting and limited generalization. To overcome these problems, we introduce a novel method called 'Gate and Obstruct Network' (GateON). GateON combines learnable gating of activity and online estimation of parameter relevance to safeguard crucial knowledge from being overwritten. Our method generates partially overlapping pathways between tasks which permits forward and backward transfer during sequential learning. GateON addresses the issue of network saturation after parameter fixation by a re-activation mechanism of fixed neurons, enabling large-scale continual learning. GateON is implemented on a wide range of networks (fully-connected, CNN, Transformers), has low computational complexity, effectively learns up to 100 MNIST learning tasks, and achieves top-tier results for pre-trained BERT in CL-based NLP tasks.

This paper presents a deep machine learning architecture, the "polyharmonic cascade" -- a sequence of packages of polyharmonic splines, where each layer is rigorously derived from the theory of random functions and the principles of indifference. This makes it possible to approximate nonlinear functions of arbitrary complexity while preserving global smoothness and a probabilistic interpretation. For the polyharmonic cascade, a training method alternative to gradient descent is proposed: instead of directly optimizing the coefficients, one solves a single global linear system on each batch with respect to the function values at fixed "constellations" of nodes. This yields synchronized updates of all layers, preserves the probabilistic interpretation of individual layers and theoretical consistency with the original model, and scales well: all computations reduce to 2D matrix operations efficiently executed on a GPU. Fast learning without overfitting on MNIST is demonstrated.

The popular Q-learning algorithm is known to overestimate action values under certain conditions. It was not previously known whether, in practice, such overestimations are common, whether they harm performance, and whether they can generally be prevented. In this paper, we answer all these questions affirmatively. In particular, we first show that the recent DQN algorithm, which combines Q-learning with a deep neural network, suffers from substantial overestimations in some games in the Atari 2600 domain. We then show that the idea behind the Double Q-learning algorithm, which was introduced in a tabular setting, can be generalized to work with large-scale function approximation. We propose a specific adaptation to the DQN algorithm and show that the resulting algorithm not only reduces the observed overestimations, as hypothesized, but that this also leads to much better performance on several games.

Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.

Deep learning models are susceptible to adversarial samples in white and black-box environments. Although previous studies have shown high attack success rates, coupling DNN models with interpretation models could offer a sense of security when a human expert is involved, who can identify whether a given sample is benign or malicious. However, in white-box environments, interpretable deep learning systems (IDLSes) have been shown to be vulnerable to malicious manipulations. In black-box settings, as access to the components of IDLSes is limited, it becomes more challenging for the adversary to fool the system. In this work, we propose a Query-efficient Score-based black-box attack against IDLSes, QuScore, which requires no knowledge of the target model and its coupled interpretation model. QuScore is based on transfer-based and score-based methods by employing an effective microbial genetic algorithm. Our method is designed to reduce the number of queries necessary to carry out successful attacks, resulting in a more efficient process. By continuously refining the adversarial samples created based on feedback scores from the IDLS, our approach effectively navigates the search space to identify perturbations that can fool the system. We evaluate the attack's effectiveness on four CNN models (Inception, ResNet, VGG, DenseNet) and two interpretation models (CAM, Grad), using both ImageNet and CIFAR datasets. Our results show that the proposed approach is query-efficient with a high attack success rate that can reach between 95% and 100% and transferability with an average success rate of 69% in the ImageNet and CIFAR datasets. Our attack method generates adversarial examples with attribution maps that resemble benign samples. We have also demonstrated that our attack is resilient against various preprocessing defense techniques and can easily be transferred to different DNN models.

Neural algorithmic reasoning is an emerging research direction that endows neural networks with the ability to mimic algorithmic executions step-by-step. A common paradigm in existing designs involves the use of historical embeddings in predicting the results of future execution steps. Our observation in this work is that such historical dependence intrinsically contradicts the Markov nature of algorithmic reasoning tasks. Based on this motivation, we present our ForgetNet, which does not use historical embeddings and thus is consistent with the Markov nature of the tasks. To address challenges in training ForgetNet at early stages, we further introduce G-ForgetNet, which uses a gating mechanism to allow for the selective integration of historical embeddings. Such an enhanced capability provides valuable computational pathways during the model's early training phase. Our extensive experiments, based on the CLRS-30 algorithmic reasoning benchmark, demonstrate that both ForgetNet and G-ForgetNet achieve better generalization capability than existing methods. Furthermore, we investigate the behavior of the gating mechanism, highlighting its degree of alignment with our intuitions and its effectiveness for robust performance.