Stochastic approximation (SA) is a fundamental iterative framework with broad applications in reinforcement learning and optimization. Classical analyses typically rely on martingale difference or Markov noise with bounded second moments, but many practical settings, including finance and communications, frequently encounter heavy-tailed and long-range dependent (LRD) noise. In this work, we study SA for finding the root of a strongly monotone operator under these non-classical noise models. We establish the first finite-time moment bounds in both settings, providing explicit convergence rates that quantify the impact of heavy tails and temporal dependence. Our analysis employs a noise-averaging argument that regularizes the impact of noise without modifying the iteration. Finally, we apply our general framework to stochastic gradient descent (SGD) and gradient play, and corroborate our finite-time analysis through numerical experiments.

A major limitation of clustering approaches is their lack of explainability: methods rarely provide insight into which features drive the grouping of similar observations. To address this limitation, we propose an ensemble-based clustering framework that integrates bagging and feature dropout to generate feature importance scores, in analogy with feature importance mechanisms in supervised random forests. By leveraging multiple bootstrap resampling schemes and aggregating the resulting partitions, the method improves stability and robustness of the cluster definition, particularly in small-sample or noisy settings. Feature importance is assessed through an information-theoretic approach: at each step, the mutual information between each feature and the estimated cluster labels is computed and weighted by a measure of clustering validity to emphasize well-formed partitions, before being aggregated into a final score. The method outputs both a consensus partition and a corresponding measure of feature importance, enabling a unified interpretation of clustering structure and variable relevance. Its effectiveness is demonstrated on multiple simulated and real-world datasets.

It is currently difficult to distill discrete diffusion models. In contrast, continuous diffusion literature has many distillation approaches methods that can reduce sampling steps to a handful. Our method, Discrete Moment Matching Distillation (D-MMD), leverages ideas that have been highly successful in the continuous domain. Whereas previous discrete distillation methods collapse, D-MMD maintains high quality and diversity (given sufficient sampling steps). This is demonstrated on both text and image datasets. Moreover, the newly distilled generators can outperform their teachers.

Abstract: We use the law of total variance to generate multiple expansions for the posterior predictive variance. These expansions are sums of terms involving conditional expectations and conditional variances and provide a quantification of the sources of predictive uncertainty. Since the posterior predictive variance is fixed given the model, it represents a constant quantity that is conserved over these expansions. The terms in the expansions can be assessed in absolute or relative sense to understand the main contributors to the length of prediction intervals. We quantify the term-wise uncertainty across expansions varying in the number of terms and the order of conditionates. In particular, given that a specific term in one expansion is small or zero, we identify the other terms in other expansions that must also be small or zero. We illustrate this approach to predictive model assessment in several well-known models. The Setting and Intuition Everyone uses prediction intervals (PI's) but few examine their structure or more precisely how they should be interpreted in the context of a model with multiple components. Often PI's seem overconfident (too narrow) or useless (too wide). Both frequentist and Bayesian practitioners routinely report PI's.

Efficient and scalable non-parametric or semi-parametric regression analysis and density estimation are of crucial importance to the fields of statistics and machine learning. However, available methods are limited in their ability to handle large-scale data. We address this issue by developing a novel coreset construction for multivariate conditional transformation models (MCTMs) to enhance their scalability and training efficiency. To the best of our knowledge, these are the first coresets for semi-parametric distributional models. Our approach yields substantial data reduction via importance sampling. It ensures with high probability that the log-likelihood remains within multiplicative error bounds of $(1\pm\varepsilon)$ and thereby maintains statistical model accuracy. Compared to conventional full-parametric models, where coresets have been incorporated before, our semi-parametric approach exhibits enhanced adaptability, particularly in scenarios where complex distributions and non-linear relationships are present, but not fully understood. To address numerical problems associated with normalizing logarithmic terms, we follow a geometric approximation based on the convex hull of input data. This ensures feasible, stable, and accurate inference in scenarios involving large amounts of data. Numerical experiments demonstrate substantially improved computational efficiency when handling large and complex datasets, thus laying the foundation for a broad range of applications within the statistics and machine learning communities.

Prior work on partial labels learning (PLL) has shown that learning is possible even when each instance is associated with a bag of labels, rather than a single accurate but costly label. However, the necessary conditions for learning with partial labels remain unclear, and existing PLL methods are effective only in specific scenarios. In this work, we mathematically characterize the settings in which PLL is feasible. In addition, we present PL A-$k$NN, an adaptive nearest-neighbors algorithm for PLL that is effective in general scenarios and enjoys strong performance guarantees. Experimental results corroborate that PL A-$k$NN can outperform state-of-the-art methods in general PLL scenarios.

Over the last decades, two distinct approaches have been instrumental to our understanding of the computational complexity of statistical estimation. The statistical physics literature predicts algorithmic hardness through local stability and monotonicity properties of the Franz--Parisi (FP) potential \cite{franz1995recipes,franz1997phase}, while the mathematically rigorous literature characterizes hardness via the limitations of restricted algorithmic classes, most notably low-degree polynomial estimators \cite{hopkins2017efficient}. For many inference models, these two perspectives yield strikingly consistent predictions, giving rise to a long-standing open problem of establishing a precise mathematical relationship between them. In this work, we show that for estimation problems the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential for a broad family of Gaussian additive models (GAMs) with signal-to-noise ratio $λ$. In particular, subject to a low-degree conjecture for GAMs, our results imply that the polynomial-time limits of these models are directly implied by the monotonicity of the annealed FP potential, in conceptual agreement with predictions from the physics literature dating back to the 1990s.

Finite-context models (FCMs) are widely used for compressing symbolic sequences such as DNA, where predictive performance depends critically on the context length k and smoothing parameter α. In practice, these hyperparameters are typically selected through exhaustive search, which is computationally expensive and scales poorly with model complexity. This paper proposes a statistically grounded two-step sequential approach for efficient hyperparameter selection in FCMs. The key idea is to decompose the joint optimization problem into two independent stages. First, the context length k is estimated using categorical serial dependence measures, including Cramér's ν, Cohen's \k{appa} and partial mutual information (pami). Second, the smoothing parameter α is estimated via maximum likelihood conditional on the selected context length k. Simulation experiments were conducted on synthetic symbolic sequences generated by FCMs across multiple (k, α) configurations, considering a four-letter alphabet and different sample sizes. Results show that the dependence measures are substantially more sensitive to variations in k than in α, supporting the sequential estimation strategy. As expected, the accuracy of the hyperparameter estimation improves with increasing sample size. Furthermore, the proposed method achieves compression performance comparable to exhaustive grid search in terms of average bitrate (bits per symbol), while substantially reducing computational cost. Overall, the results on simulated data show that the proposed sequential approach is a practical and computationally efficient alternative to exhaustive hyperparameter tuning in FCMs.

Federated learning enables collaborative training without sharing raw data, but struggles under client heterogeneity and streaming distribution shifts, where drift and novel data can impair convergence and cause forgetting. We propose a federated associative-memory framework that learns shared archetypes in heterogeneous, continual settings, where client data are independent but not necessarily balanced. Each client encodes its experience as a low-rank Hebbian operator, sent to a central server for aggregation and factorization into global archetypes. This approach preserves privacy, avoids centralized replay buffers, and is robust to small, noisy, or evolving datasets. We cast aggregation as a low-rank-plus-noise spectral inference problem, deriving theoretical thresholds for detectability and retrieval robustness. An entropy-based controller balances stability and plasticity in streaming regimes. Experiments with heterogeneous clients, drift, and novelty show improved global archetype reconstruction and associative retrieval, supporting the spectral view of federated consolidation.

We establish convergence of the training dynamics of residual neural networks (ResNets) to their joint infinite depth L, hidden width M, and embedding dimension D limit. Specifically, we consider ResNets with two-layer perceptron blocks in the maximal local feature update (MLU) regime and prove that, after a bounded number of training steps, the error between the ResNet and its large-scale limit is O(1/L + sqrt(D/(L M)) + 1/sqrt(D)). This error rate is empirically tight when measured in embedding space. For a budget of P = Theta(L M D) parameters, this yields a convergence rate O(P^(-1/6)) for the scalings of (L, M, D) that minimize the bound. Our analysis exploits in an essential way the depth-two structure of residual blocks and applies formally to a broad class of state-of-the-art architectures, including Transformers with bounded key-query dimension. From a technical viewpoint, this work completes the program initiated in the companion paper [Chi25] where it is proved that for a fixed embedding dimension D, the training dynamics converges to a Mean ODE dynamics at rate O(1/L + sqrt(D)/sqrt(L M)). Here, we study the large-D limit of this Mean ODE model and establish convergence at rate O(1/sqrt(D)), yielding the above bound by a triangle inequality. To handle the rich probabilistic structure of the limit dynamics and obtain one of the first rigorous quantitative convergence for a DMFT-type limit, we combine the cavity method with propagation of chaos arguments at a functional level on so-called skeleton maps, which express the weight updates as functions of CLT-type sums from the past.