arXiv subjects

Morgane Austern

Publications and source records attributed to Morgane Austern.

At least 19 recordsLinked to original sources

arXiv2026

Wasserstein-p Bounds via Cumulant-Based Edgeworth Expansion for $α$-Mixing Random Fields

We establish normal-approximation bounds in the Wasserstein-$p$ distance for $α$-mixing random fields, extending results previously available mainly for independent or locally dependent variables. For standardized sums indexed by $T\subset\mathbb{Z}^d$, we obtain rates $O(|T|^{-β})$, where $β\in(0,1/2]$ depends on $p$, the moment order, the dimension, and the mixing decay; the optimal square-root rate holds under sufficiently fast polynomial mixing. For integer $p\geq2$, a refined bound retains the stronger mixing power produced by the expansion, covers the endpoint of finite $(p+2)$-th moments, and identifies exact logarithmic corrections at critical decay. Under finite-range dependence, the endpoint moment condition gives the optimal rate for every real $p\geq1$. The proof combines a cumulant-based Edgeworth expansion with Stein's method. Its main technical tool is a constructive graph calculus that organizes high-order dependence and preserves cancellations before mixing bounds are applied. The resulting Wasserstein estimates also imply two-regime non-uniform normal-approximation bounds with sharper sample-size decay in the far tail. A Lean 4/Mathlib companion machine-checks the principal theorem interfaces and the complete paper-internal genogram argument for the expansions, relative to seven explicitly identified literature and standard inputs.

math.PR↗

arXiv2026

The Zero Pattern of a Design Matrix Drives Multiple Descent in Over-parameterized Regression

Over-parameterized linear regression has been widely studied over the last decade. However, most existing works assume that the covariates are independent and that their covariance matrices are non-degenerate. In this paper, we relax both assumptions and derive deterministic equivalents for the prediction risk in a vanishing-ridge regime. We show that degeneracy of the covariance matrices and dependence can lead to multiple descent, and characterize where the corresponding peaks can occur. Our proofs use a novel graph representation of the variance profile. We show that maximum matchings and the Dulmage--Mendelsohn decomposition of the associated bipartite graph identify the configurations at which the variance becomes singular.

math.ST↗

arXiv2026

Wasserstein-p Bounds in the Central Limit Theorem Under Weak Dependence

The central limit theorem is one of the most fundamental results in probability and has been successfully extended to locally dependent data and strongly-mixing random fields. In this paper, we establish its rate of convergence for transport distances, namely for arbitrary $p\ge1$ we obtain an upper bound for the Wasserstein-$p$ distance for locally dependent random variables and strongly mixing stationary random fields. Our proofs adapt the Stein dependency neighborhood method to the Wasserstein-$p$ distance and as a by-product we establish high-order local expansions of the Stein equation for dependent random variables. Finally, we demonstrate how our results can be used to obtain tail bounds that are asymptotically tight, and decrease polynomially fast, for the empirical average of weakly dependent random variables.

math.PR↗

arXiv2026

Universality of High-Dimensional Logistic Regression and a Novel CGMT under Dependence with Applications to Data Augmentation

Over the last decade, a wave of research has characterized the exact asymptotic risk of many high-dimensional models in the proportional regime. Two foundational results have driven this progress: Gaussian universality, which shows that the asymptotic risk of estimators trained on non-Gaussian and Gaussian data is equivalent, and the convex Gaussian min-max theorem (CGMT), which characterizes the risk under Gaussian settings. However, these results rely on the assumption that the data consists of independent random vectors--an assumption that significantly limits its applicability to many practical setups. In this paper, we address this limitation by generalizing both results to the dependent setting. More precisely, we prove that Gaussian universality still holds for high-dimensional logistic regression under block dependence, $m$-dependence and special cases of mixing, and establish a novel CGMT framework that accommodates for correlation across both the covariates and observations. Using these results, we establish the impact of data augmentation, a widespread practice in deep learning, on the asymptotic risk.

math.ST↗

arXiv2026

Efficient Concentration with Gaussian Approximation

Concentration inequalities for the sample mean, like those due to Bernstein, Hoeffding, and Bentkus, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem (CLT) provides asymptotic confidence intervals with optimal width, but these are invalid for all sample sizes. To resolve this tension, we develop new computable concentration inequalities for bounded variables with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. These bounds enable the construction of efficient confidence intervals with correct coverage for any sample size and efficient empirical Berry-Esseen bounds that require no prior knowledge of the population variance. We derive our inequalities by tightly bounding non-uniform Kolmogorov and Wasserstein distances to a Gaussian using zero-bias couplings and Stein's method of exchangeable pairs and demonstrate practical improvements over the Bernstein, Hoeffding, Bentkus, Berry-Esseen, Feller-Cramér, Romano-Wolf, empirical Bernstein, empirical Bentkus, and coin-betting inequalities.

math.PR↗

arXiv2026

Computable Bounds for Strong Approximations with Applications

The Komlós$\unicode{x2013}$Major$\unicode{x2013}$Tusnády (KMT) inequality for partial sums is one of the most celebrated results in probability theory. Yet its practical application has been hindered by a lack of practical constants. This paper addresses this limitation for bounded i.i.d. random variables. At the cost of an additional logarithmic factor, we propose a computable version of the KMT inequality that depends only on the variables' range and standard deviation. We also derive an empirical version of the inequality that achieves nominal coverage even when the standard deviation is unknown. We then demonstrate the practicality of our bounds through applications to online change point detection and first hitting time probabilities. As a byproduct of our analysis, we obtain a Cramér-type moderate deviation bound for normalized centered partial sums.

math.ST↗

arXiv2026

Inference on Optimal Policy Values and Other Irregular Functionals via Softmax Smoothing

Constructing confidence intervals for the value of an (unknown) optimal treatment policy is a fundamental problem in causal inference. Insight into the optimal policy value can guide the development of reward-maximizing, individualized treatment regimes. However, because the functional that defines the optimal value is non-differentiable, standard semi-parametric approaches for performing inference fail to be directly applicable. Many existing works circumvent non-differentiability by making the unrealistic assumption of zero probability of treatment non-response, i.e. that every unit responds (either positively or negatively) to an assigned treatment. Further, works that don't circumvent this restriction rely on refitting nuisance models a number of times proportional to the sample size. In this paper, we construct and analyze a simple, softmax smoothing-based estimator for the value of an optimal treatment policy. Our estimator applies in both static and dynamic treatment regimes, only requires fitting a constant number of nuisance models, and is statistically efficient when there is zero probability of non-response to treatment. Also, while our estimator does not require making semi-parametric restrictions, it can exploit them when they exist. We further show how our softmax smoothing approach can be used to estimate general parameters that are specified as a maximum of scores involving nuisance components, and look at conditional Balke and Pearl bounds and $L^1$ calibration error as salient examples.

econ.EM↗

arXiv2026

Graph Attention Network for Node Regression on Random Geometric Graphs with Erdős--Rényi contamination

Graph attention networks (GATs) are widely used and often appear robust to noise in node covariates and edges, yet rigorous statistical guarantees demonstrating a provable advantage of GATs over non-attention graph neural networks~(GNNs) are scarce. We partially address this gap for node regression with graph-based errors-in-variables models under simultaneous covariate and edge corruption: responses are generated from latent node-level covariates, but only noise-perturbed versions of the latent covariates are observed; and the sample graph is a random geometric graph created from the node covariates but contaminated by independent Erdős--Rényi edges. We propose and analyze a carefully designed, task-specific GAT that constructs denoised proxy features for regression. We prove that regressing the response variables on the proxies achieves lower error asymptotically in (a) estimating the regression coefficient compared to the ordinary least squares (OLS) estimator on the noisy node covariates, and (b) predicting the response for an unlabelled node compared to a vanilla graph convolutional network~(GCN) -- under mild growth conditions. Our analysis leverages high-dimensional geometric tail bounds and concentration for neighbourhood counts and sample covariances. We verify our theoretical findings through experiments on synthetically generated data. We also perform experiments on real-world graphs and demonstrate the effectiveness of the attention mechanism in several node regression tasks.

stat.ML↗

arXiv2025

Poisson-Process Topic Model for Integrating Knowledge from Pre-trained Language Models

Topic modeling is traditionally applied to word counts without accounting for the context in which words appear. Recent advancements in large language models (LLMs) offer contextualized word embeddings, which capture deeper meaning and relationships between words. We aim to leverage such embeddings to improve topic modeling. We use a pre-trained LLM to convert each document into a sequence of word embeddings. This sequence is then modeled as a Poisson point process, with its intensity measure expressed as a convex combination of $K$ base measures, each corresponding to a topic. To estimate these topics, we propose a flexible algorithm that integrates traditional topic modeling methods, enhanced by net-rounding applied before and kernel smoothing applied after. One advantage of this framework is that it treats the LLM as a black box, requiring no fine-tuning of its parameters. Another advantage is its ability to seamlessly integrate any traditional topic modeling approach as a plug-in module, without the need for modifications Assuming each topic is a $β$-Hölder smooth intensity measure on the embedded space, we establish the rate of convergence of our method. We also provide a minimax lower bound and show that the rate of our method matches with the lower bound when $β\leq 1$. Additionally, we apply our method to several datasets, providing evidence that it offers an advantage over traditional topic modeling approaches.

stat.ML↗

arXiv2025

Gaussian and Non-Gaussian Universality of Data Augmentation

We provide universality results that quantify how data augmentation affects the variance and limiting distribution of estimates through simple surrogates, and analyze several specific models in detail. The results confirm some observations made in machine learning practice, but also lead to unexpected findings: Data augmentation may increase rather than decrease the uncertainty of estimates, such as the empirical prediction risk. It can act as a regularizer, but fails to do so in certain high-dimensional problems, and it may shift the double-descent peak of an empirical risk. Overall, the analysis shows that several properties data augmentation has been attributed with are not either true or false, but rather depend on a combination of factors -- notably the data distribution, the properties of the estimator, and the interplay of sample size, number of augmentations, and dimension. As our main theoretical tool, we develop an adaptation of Lindeberg's technique for block dependence. The resulting universality regime may be Gaussian or non-Gaussian.

cs.LG↗

arXiv2025

Gaussian universality for approximately polynomial functions of high-dimensional data

Gaussian universality results assert that the properties of many estimators remain unchanged when the input data are replaced by Gaussians. Such results have gained popularity in high-dimensional statistics and machine learning, as Gaussianity often substantially simplifies downstream analyses. Yet, an open question remains on when universality may cease to hold. To address this, we establish nearly optimal upper and lower bounds for Gaussian universality approximation, measured in Kolmogorov distance, over the class of approximately polynomial functions of high-dimensional random vectors. The upper bounds adapt the invariance principle of Mossel, O'Donnell and Oleszkiewicz (2010) for high-dimensional vectors and functions beyond multilinear forms. As applications, we obtain a delta method for high-dimensional data with non-Gaussian limits, a necessary and sufficient condition for asymptotic normality, and simple estimators that are asymptotically normal but for which bootstrap fails to be consistent. We also extend recent results on the high-dimensional degeneracy of non-degenerate U-statistics, phase transition of MMD in two-sample tests with imbalanced data, and confidence spheres for high-dimensional averages. Our lower bound is constructive and shows that, for polynomials of even degree $m$, universality holds up to $m=o(\log n)$. As a corollary, the Gaussian polynomial approximation error of $Ω(n^{-1/6m})$ is not improvable for even-degree U-statistics and V-statistics. Our results also explain how universality results for U-statistics and V-statistics differ significantly in their dependence on dimensions.

math.PR↗

arXiv2025

Bounding Hellinger Distance with Stein's Method

This work introduces a new, explicit bound on the Hellinger distance between a continuous random variable and a Gaussian with matching mean and variance. As example applications, we derive a quantitative Hellinger central limit theorem and efficient concentration inequalities for U-statistics.

math.PR↗

arXiv2025

Perfect Recovery for Random Geometric Graph Matching with Shallow Graph Neural Networks

We study the graph matching problem in the presence of vertex feature information using shallow graph neural networks. Specifically, given two graphs that are independent perturbations of a single random geometric graph with sparse binary features, the task is to recover an unknown one-to-one mapping between the vertices of the two graphs. We show under certain conditions on the sparsity and noise level of the feature vectors, a carefully designed two-layer graph neural network can, with high probability, recover the correct mapping between the vertices with the help of the graph structure. Additionally, we prove that our condition on the noise parameter is tight up to logarithmic factors. Finally, we compare the performance of the graph neural network to directly solving an assignment problem using the noisy vertex features and demonstrate that when the noise level is at least constant, this direct matching fails to achieve perfect recovery, whereas the graph neural network can tolerate noise levels growing as fast as a power of the size of the graph. Our theoretical findings are further supported by numerical studies as well as real-world data experiments.

cs.LG↗

arXiv2024

Statistical Guarantees for Link Prediction using Graph Neural Networks

This paper derives statistical guarantees for the performance of Graph Neural Networks (GNNs) in link prediction tasks on graphs generated by a graphon. We propose a linear GNN architecture (LG-GNN) that produces consistent estimators for the underlying edge probabilities. We establish a bound on the mean squared error and give guarantees on the ability of LG-GNN to detect high-probability edges. Our guarantees hold for both sparse and dense graphs. Finally, we demonstrate some of the shortcomings of the classical GCN architecture, as well as verify our results on real and synthetic datasets.

cs.LG↗

arXiv2024

Poisson approximation for stochastic processes summed over amenable groups

We generalize the Poisson limit theorem to binary functions of random objects whose law is invariant under the action of an amenable group. Examples include stationary random fields, exchangeable sequences, and exchangeable graphs. A celebrated result of E. Lindenstrauss shows that normalized sums over certain increasing subsets of such groups approximate expectations. Our results clarify that the corresponding unnormalized sums of binary statistics are asymptotically Poisson, provided suitable mixing conditions hold. They extend further to randomly subsampled sums and also show that strict invariance of the distribution is not needed if the requisite mixing condition defined by the group holds. We illustrate the results with applications to random fields, Cayley graphs, and Poisson processes on groups.

math.PR↗

arXiv2023

Inference on Optimal Dynamic Policies via Softmax Approximation

Estimating optimal dynamic policies from offline data is a fundamental problem in dynamic decision making. In the context of causal inference, the problem is known as estimating the optimal dynamic treatment regime. Even though there exists a plethora of methods for estimation, constructing confidence intervals for the value of the optimal regime and structural parameters associated with it is inherently harder, as it involves non-linear and non-differentiable functionals of unknown quantities that need to be estimated. Prior work resorted to sub-sample approaches that can deteriorate the quality of the estimate. We show that a simple soft-max approximation to the optimal treatment regime, for an appropriately fast growing temperature parameter, can achieve valid inference on the truly optimal regime. We illustrate our result for a two-period optimal dynamic regime, though our approach should directly extend to the finite horizon case. Our work combines techniques from semi-parametric inference and $g$-estimation, together with an appropriate triangular array central limit theorem, as well as a novel analysis of the asymptotic influence and asymptotic bias of softmax approximations.

econ.EM↗

arXiv2023

Wasserstein-p Bounds in the Central Limit Theorem Under Local Dependence

The central limit theorem (CLT) is one of the most fundamental results in probability; and establishing its rate of convergence has been a key question since the 1940s. For independent random variables, a series of recent works established optimal error bounds under the Wasserstein-p distance (with p>=1). In this paper, we extend those results to locally dependent random variables, which include m-dependent random fields and U-statistics. Under conditions on the moments and the dependency neighborhoods, we derive optimal rates in the CLT for the Wasserstein-p distance. Our proofs rely on approximating the empirical average of dependent observations by the empirical average of i.i.d. random variables. To do so, we expand the Stein equation to arbitrary orders by adapting the Stein's dependency neighborhood method. Finally we illustrate the applicability of our results by obtaining efficient tail bounds.

math.PR↗

arXiv2023

Asymptotics of Network Embeddings Learned via Subsampling

Network data are ubiquitous in modern machine learning, with tasks of interest including node classification, node clustering and link prediction. A frequent approach begins by learning an Euclidean embedding of the network, to which algorithms developed for vector-valued data are applied. For large networks, embeddings are learned using stochastic gradient methods where the sub-sampling scheme can be freely chosen. Despite the strong empirical performance of such methods, they are not well understood theoretically. Our work encapsulates representation methods using a subsampling approach, such as node2vec, into a single unifying framework. We prove, under the assumption that the graph is exchangeable, that the distribution of the learned embedding vectors asymptotically decouples. Moreover, we characterize the asymptotic distribution and provided rates of convergence, in terms of the latent parameters, which includes the choice of loss function and the embedding dimension. This provides a theoretical foundation to understand what the embedding vectors represent and how well these methods perform on downstream tasks. Notably, we observe that typically used loss functions may lead to shortcomings, such as a lack of Fisher consistency.

stat.ML↗