Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical or biological processes and epidemiology. A typical challenge in practical problems modeled by SRNs is that only a few state variables can be dynamically observed. Given the measurement trajectories, one can estimate the conditional probability distribution of unobserved (hidden) state variables by solving a stochastic filtering problem. In this setting, the conditional distribution evolves over time according to an extensive or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. The current numerical filtering techniques, such as the Filtered Finite State Projection (DAmbrosio et al., 2022), are hindered by the curse of dimensionality, significantly affecting their computational performance. To address these limitations, we propose to use a dimensionality reduction technique based on the Markovian projection (MP), initially introduced for forward problems (Ben Hammouda et al., 2024). In this work, we explore how to adapt the existing MP approach to the filtering problem and introduce a novel version of the MP, the Filtered MP, that guarantees the consistency of the resulting estimator. The novel method combines a particle filter with reduced variance and solving the filtering equations in a low-dimensional space, exploiting the advantages of both approaches. The analysis and empirical results highlight the superior computational efficiency of projection methods compared to the existing filtered finite state projection in the large dimensional setting.

Singular learning models with non-positive Fisher information matrices include neural networks, reduced-rank regression, Boltzmann machines, normal mixture models, and others. These models have been widely used in the development of learning machines. However, theoretical analysis is still in its early stages. In this paper, we examine learning coefficients, which indicate the general learning efficiency of deep linear learning models and three-layer neural network models with ReLU units. Finally, we extend the results to include the case of the Softmax function.

Permutation tests are a popular choice for distinguishing distributions and testing independence, due to their exact, finite-sample control of false positives and their minimax optimality when paired with U-statistics. However, standard permutation tests are also expensive, requiring a test statistic to be computed hundreds or thousands of times to detect a separation between distributions. In this work, we offer a simple approach to accelerate testing: group your datapoints into bins and permute only those bins. For U and V-statistics, we prove that these cheap permutation tests have two remarkable properties. First, by storing appropriate sufficient statistics, a cheap test can be run in time comparable to evaluating a single test statistic. Second, cheap permutation power closely approximates standard permutation power. As a result, cheap tests inherit the exact false positive control and minimax optimality of standard permutation tests while running in a fraction of the time. We complement these findings with improved power guarantees for standard permutation testing and experiments demonstrating the benefits of cheap permutations over standard maximum mean discrepancy (MMD), Hilbert-Schmidt independence criterion (HSIC), random Fourier feature, Wilcoxon-Mann-Whitney, cross-MMD, and cross-HSIC tests.

Predicting masked from visible parts of an image is a powerful self-supervised approach for visual representation learning. However, the common practice of masking random patches of pixels exhibits certain failure modes, which can prevent learning meaningful high-level features, as required for downstream tasks. We propose an alternative masking strategy that operates on a suitable transformation of the data rather than on the raw pixels. Specifically, we perform principal component analysis and then randomly mask a subset of components, which accounts for a fixed ratio of the data variance. The learning task then amounts to reconstructing the masked components from the visible ones. Compared to local patches of pixels, the principal components of images carry more global information. We thus posit that predicting masked from visible components involves more high-level features, allowing our masking strategy to extract more useful representations. This is corroborated by our empirical findings which demonstrate improved image classification performance for component over pixel masking. Our method thus constitutes a simple and robust data-driven alternative to traditional masked image modeling approaches.

Using continuous-time stochastic differential equation (SDE) proxies to stochastic optimization algorithms has proven fruitful for understanding their generalization abilities. A significant part of these approaches are based on the so-called ``entropy flows'', which greatly simplify the generalization analysis. Unfortunately, such well-structured entropy flows cannot be obtained for most discrete-time algorithms, and the existing SDE approaches remain limited to specific noise and algorithmic structures. We aim to alleviate this issue by introducing a generic framework for analyzing the generalization error of Markov algorithms through `Poissonization', a continuous-time approximation of discrete-time processes with formal approximation guarantees. Through this approach, we first develop a novel entropy flow, which directly leads to PAC-Bayesian generalization bounds. We then draw novel links to modified versions of the celebrated logarithmic Sobolev inequalities (LSI), identify cases where such LSIs are satisfied, and obtain improved bounds. Beyond its generality, our framework allows exploiting specific properties of learning algorithms. In particular, we incorporate the noise structure of different algorithm types - namely, those with additional noise injections (noisy) and those without (non-noisy) - through various technical tools. This illustrates the capacity of our methods to achieve known (yet, Poissonized) and new generalization bounds.

Simulation-based inference (SBI) is a method to perform inference on a variety of complex scientific models with challenging inference (inverse) problems. Bayesian Optimal Experimental Design (BOED) aims to efficiently use experimental resources to make better inferences. Various stochastic gradient-based BOED methods have been proposed as an alternative to Bayesian optimization and other experimental design heuristics to maximize information gain from an experiment. We demonstrate a link via mutual information bounds between SBI and stochastic gradient-based variational inference methods that permits BOED to be used in SBI applications as SBI-BOED. This link allows simultaneous optimization of experimental designs and optimization of amortized inference functions. We evaluate the pitfalls of naive design optimization using this method in a standard SBI task and demonstrate the utility of a well-chosen design distribution in BOED. We compare this approach on SBI-based models in real-world simulators in epidemiology and biology, showing notable improvements in inference.

Despite the impressive progress in statistical Optimal Transport (OT) in recent years, there has been little interest in the study of the \emph{sequential learning} of OT. Surprisingly so, as this problem is both practically motivated and a challenging extension of existing settings such as linear bandits. This article considers (for the first time) the stochastic bandit problem of learning to solve generic Kantorovich and entropic OT problems from repeated interactions when the marginals are known but the cost is unknown. We provide $\tilde{\mathcal O}(\sqrt{T})$ regret algorithms for both problems by extending linear bandits on Hilbert spaces. These results provide a reduction to infinite-dimensional linear bandits. To deal with the dimension, we provide a method to exploit the intrinsic regularity of the cost to learn, yielding corresponding regret bounds which interpolate between $\tilde{\mathcal O}(\sqrt{T})$ and $\tilde{\mathcal O}(T)$.

Previous influential work showed that infinite width limits of neural networks in the lazy training regime are described by kernel machines. Here, we show that neural networks trained in the rich, feature learning infinite-width regime in two different settings are also described by kernel machines, but with data-dependent kernels. For both cases, we provide explicit expressions for the kernel predictors and prescriptions to numerically calculate them. To derive the first predictor, we study the large-width limit of feature-learning Bayesian networks, showing how feature learning leads to task-relevant adaptation of layer kernels and preactivation densities. The saddle point equations governing this limit result in a min-max optimization problem that defines the kernel predictor. To derive the second predictor, we study gradient flow training of randomly initialized networks trained with weight decay in the infinite-width limit using dynamical mean field theory (DMFT). The fixed point equations of the arising DMFT defines the task-adapted internal representations and the kernel predictor. We compare our kernel predictors to kernels derived from lazy regime and demonstrate that our adaptive kernels achieve lower test loss on benchmark datasets.

In recent years, there has been much interest in understanding the generalization behavior of interpolating predictors, which overfit on noisy training data. Whereas standard analyses are concerned with whether a method is consistent or not, recent observations have shown that even inconsistent predictors can generalize well. In this work, we revisit the classic interpolating Nadaraya-Watson (NW) estimator (also known as Shepard's method), and study its generalization capabilities through this modern viewpoint. In particular, by varying a single bandwidth-like hyperparameter, we prove the existence of multiple overfitting behaviors, ranging non-monotonically from catastrophic, through benign, to tempered. Our results highlight how even classical interpolating methods can exhibit intricate generalization behaviors. Numerical experiments complement our theory, demonstrating the same phenomena.

Empirical risk minimization (ERM) is a cornerstone of modern machine learning (ML), supported by advances in optimization theory that ensure efficient solutions with provable algorithmic convergence rates, which measure the speed at which optimization algorithms approach a solution, and statistical learning rates, which characterize how well the solution generalizes to unseen data. Privacy, memory, computational, and communications constraints increasingly necessitate data collection, processing, and storage across network-connected devices. In many applications, these networks operate in decentralized settings where a central server cannot be assumed, requiring decentralized ML algorithms that are both efficient and resilient. Decentralized learning, however, faces significant challenges, including an increased attack surface for adversarial interference during decentralized learning processes. This paper focuses on the man-in-the-middle (MITM) attack, which can cause models to deviate significantly from their intended ERM solutions. To address this challenge, we propose RESIST (Resilient dEcentralized learning using conSensus gradIent deScenT), an optimization algorithm designed to be robust against adversarially compromised communication links. RESIST achieves algorithmic and statistical convergence for strongly convex, Polyak-Lojasiewicz, and nonconvex ERM problems. Experimental results demonstrate the robustness and scalability of RESIST for real-world decentralized learning in adversarial environments.