Statistics (Machine Learning) Papers

ALT Synthetic data are widely used for training modern AI systems. However, their statistical understanding remains limited. It is well recognized in statistics that repeated self-training on synthetic data can amplify errors inherited from pre-trained models, leading to model collapse. Verification plays a crucial role in selecting high-quality synthetic data to help prevent model collapse and improve model performance. We study verified self-training in a linear regression framework, where a pre-trained estimator generates synthetic data with injected generation noise and a verifier selects samples according to a verification threshold. Post-training on the accepted samples updates either all coordinates (full fine-tuning) or only an active subset (partial fine-tuning). Our analysis reveals three potentially counterintuitive effects. We establish that tighter verification does not always lead to lower prediction risk. We also find that a suitable level of injected generation noise can re

ALT Recent advances in Time Series Foundation Models (TSFMs) increasingly rely on axial attention mechanisms, but despite their empirical success, the theoretical properties of axial attention remain largely unexplored. To bridge this gap, we extend the Single Location Regression framework introduced in Marion et al. [2025] to 2D structured data, in order to study sparse information settings where a target depends on a small number of tokens. We formulate an analytically tractable axial attention-type predictor and rigorously analyze its statistical properties and training dynamics. We prove that under a specific asymptotic regime and with knowledge of the underlying oracle parameters, our predictor achieves Bayes optimality, whereas a class of semi-linear simplifications strictly fails to do so, underlining the role of the inner non-linearity. We prove that for a specific temperature scaling, Projected Gradient Descent converges globally to the optimal oracle parameters. Extensive numeric

ALT We study the problem of fair ranking in recommender systems. We provide a formal probabilistic framework in which ranking policies are distributions over permutations, and the utility of a user is the exposure given to the items they interact with, relative to that of an ideal ranking. We consider utility parity across user groups, as well as its category-aware refinement, conditional utility parity, which requires parity within each category of items. Following a recent view of fairness as distributional stability, we show that both criteria are equivalent to the stability of the utility under changes in the composition of user groups. Optimizing over mixtures of ranking policies leads to a second-order cone program, for which we provide generalization bounds. A numerical study validates our theoretical findings.

ALT Self-supervised contrastive learning (SSCL) learns representations by contrasting augmented views of the same sample, but it can exhibit feature suppression, where dominant or easy-to-learn factors suppress other class-relevant signals, degrading the performance of representations, particularly in high-dimensional data with heterogeneous features. We introduce Fisher Contrastive Learning (FCL), which uses a regularized Fisher trace criterion to address feature suppression. FCL learns embeddings by maximizing the discriminative strength of Fisher discriminant functions, producing representations that preserve class-relevant information implied by the augmentation process. We prove that the representations learned by FCL could alleviate the feature suppression effects. We demonstrate FCL on simulated benchmarks designed to induce feature suppression and on a real-world biomedical application, pretraining on large white blood cell microscopy image dataset and transfer learning for diagnos

ALT Reinforcement learning (RL) is increasingly used to train robust, adaptable systems across healthcare, e-commerce, and robotics. In these settings, legal, ethical, or logistical concerns may preclude certain key variables that drive decisions from being collected during deployment time. This gives rise to a setting known as runtime confounding: the logged data hold full observability, but deployment time lacks knowledge of certain variables, known as runtime confounders. Unlike single timepoint and non-adaptive settings, runtime confounding under reinforcement learning has the potential to propagate over time, as earlier decisions affect future states, observations, and action assignments, leading to biased learning. We address these problems with three contributions: we (1) develop policy learning algorithms under runtime-confounding, (2) develop semi-parametric efficient off-policy estimation procedures, and (3) provide diagnostics for understanding the price of runtime confounding.

ALT Probabilistic meshless methods solve partial differential equations (PDEs) by placing a Gaussian process prior on the solution and conditioning it on the PDE at a finite set of collocation nodes. The posterior provides both an approximation of the solution and a Bayesian quantification of its error. While these methods are well established on Euclidean domains, their extension to manifolds lacks, to our knowledge, any convergence analysis. We extend them to compact manifolds and study their convergence in two settings. First, for known manifolds whose Laplace-Beltrami eigenpairs are available, the prior can be evaluated exactly, but the PDE operator generally cannot be applied to it in closed form. We replace this operator by a standard generalized finite difference discretization on a point cloud sampled from the manifold, and show that the resulting method is consistent at the points of the cloud, at a rate governed by their fill distance. Second, for implicit manifolds, known only t

ALT Adaptive inference systems balance accuracy and inference cost by routing inputs to models with varying capabilities. Although jointly training models and routers is empirically more sample-efficient than training them separately, principled losses for joint training remain underexplored. We address this gap in a classification setting by proposing cost-sensitive hinge losses for jointly training classifiers with single-stage or cascade routers. Single-stage routers directly select a model per input, whereas cascade routers sequentially decide at each stage whether to predict with the current model or defer to the next stage, which uses a more accurate but more costly model. For both settings, we prove linear calibration bounds establishing Bayes consistency: minimizing the hinge risk recovers the Bayes-optimal risk under the target cost-sensitive zero-one loss. Moreover, since routers can require more labeled data to train than classifiers, we propose a novel post-training prediction-

ALT In this work, we formalize the concept of intrinsic dimension of objective landscapes introduced in Li et al. (2018), which measures the structural complexity of learning problems by training along randomly generated subspaces. We define the intrinsic dimension of a set Θ⊂ℝᵖ as the minimal dimension dᵢₙₜ(Θ) such that a random plane of dimension dᵢₙₜ(Θ) in ℝᵖ intersects Θ with probability one. We show that it is a generalized notion of co-dimension that depends on the global geometry of Θ. We study mathematical properties of the intrinsic dimension and develop techniques to estimate the intrinsic dimension of sets Θ that arise in learning problems. More specifically, we derive upper and/or lower bounds where Θ is the set of optimal parameters in student-teacher settings with shallow and deep ReLU networks, including finite and infinite sample cases, and with and without regularization. We discuss how our theory leads to novel types of generalization bounds, thereby showing that the intr

ALT Tabular foundation models use in-context learning to estimate conditional expected outcomes and average treatment effects for specific groups of individuals. Using these estimates to guide scientific decisions requires reliable uncertainty quantification. The challenge is to account simultaneously for two sources of uncertainty: limited-context bias and context variation arising from the small labeled sample supplied at inference. Predictive spread does not directly quantify both sources of uncertainty, and we observe that the ordinary bootstrap may substantially overestimate context variation, whereas subsampling may underestimate it. We introduce TabCUQ, a perturbation-based framework that injects noise into observed outcomes and applies the foundation model to the perturbed contexts. TabCUQ estimates context variation using wild-bootstrap dispersion and accounts for limited-context bias using the maximum deviation between perturbed and original estimates. For a kernel-regression ana

ALT A streaming estimator must compute from the evidence that fits in memory. We determine when storing an accurate mean is easier than computing it. For Gaussian mean estimation over the Euclidean unit ball in ℝᵈ, we establish the minimax rate under squared error with K persistent states, a known horizon, and a public time index: min{1, K⁻²/ᵈ+ᵈ⁄ₙ (1+log(ed)/1+(log K)/d)}. Unrestricted computation followed by K-level encoding has rate min{1,d/n+K⁻²/ᵈ}. At a fixed sufficiently small error, statistically optimal sample size therefore requires order dlog d working bits, while order d bits suffice to store the answer. The lower bound tracks repeated state compression through an entropy potential obtained by Gaussian smoothing. A matching construction combines a multiscale modular counter with workspace reused across coordinate groups. In one dimension, it attains the terminal rate n⁻¹+K⁻². Any fixed confidence requirement that the realized loss not exceed the zero baseline preserves these rate

ALT We study satisficing regret minimization in bandits with knapsacks (BwK), where a learner sequentially selects randomized policies, i.e., distributions over arms, to obtain mean rewards above a satisficing threshold as frequently as possible while respecting finite resource budgets. The performance is measured by satisficing regret, the cumulative deficits of the selected policies' mean rewards below this threshold. Resource constraints introduce two difficulties: an algorithm can incur linear satisficing regret even with small standard regret as deficits to the threshold cannot be compensated, and random consumption can cause early resource depletion even when policies are feasible in expectation. To address these challenges, we propose SELECTwK, a general algorithmic template that uses a BwK learning oracle algorithm with sub-linear expected (standard) regret upper bound to search for and test candidate satisficing policies (i.e., policies whose mean rewards exceed satisficing thresh

ALT Parameter estimation and classification in binary logistic regression deteriorate severely when the number of minority class observations (n₁) is smaller than the feature dimension (p). Under such rank-deficient rarity (n₁ < p), maximum likelihood estimators suffer from quasi-complete separation, penalized likelihood methods exhibit severe intercept bias, and convex minority oversampling techniques such as SMOTE collapse into an (n₁ - 1)-dimensional affine subspace. In this paper, we propose a novel Self-Calibrated Zero-Complement estimator that shifts the paradigm from minority interpolation to majority fringe extrapolation. Grounded in Stein's Lemma, we prove that the conditional expectation shift of the abundant majority class (Y=0) induced by logistic truncation uniquely identifies the true hyperplane normal boldsymbolβ after Mahalanobis whitening. We further establish that extrapolating synthetic minority instances outward from the high-quantile density fringe of the majority dist

ALT Industrial systems continuously generate time series that vary in sampling frequency, scale, dynamics, and operating regime. Consequently a single time-series foundation model may not be able to represent all regimes effectively and retraining a large model for every new dataset or regime shift is computationally expensive. Modular foundation models address this issue by reusing specialised encoders trained on diverse datasets and by allowing the expansion of capabilities through the addition of new experts. This approach requires compact experts to minimize storage cost and a proper selection mechanism of experts for unseen data. We propose a modular foundation model framework that combines end-to-end Bayesian encoder training with uncertainty based routing. Using Singular Bayesian Neural Networks (SBNNs), we build a collection of compact Bayesian encoders whose epistemic uncertainty guides label-free selection. At the selected ranks, our Bayesian collection uses 63.8% less storage th

ALT Neural networks are powerful memorizers which are often able to interpolate their training data even when the labels have been randomized. While there exists an extensive literature on the memorization capacity of fully connected networks, in practice more specific architectures with problem-dependent properties are preferred, such as convolutional networks for image classification. In this paper we consider convolutional networks which are invariant with respect to a given arbitrary finite group of orthogonal transformations, which may include translations, reflections, and/or rotations of images and invariances of higher order tensors. We prove upper bounds on the size of an invariant convolutional neural network with threshold activations that is required to interpolate a given dataset with two classes. Our bounds are formulated in terms of a complexity measure that takes the invariant structure into account and is instance-specific, i.e., it is related to the mutual arrangement of

ALT Human judgment calls made throughout the data science life cycle can change the conclusions drawn from data. Independent analysts often make different choices when given the same data, leading to often unacceptable variations in results and contributing to the ongoing reproducibility crisis across scientific disciplines. The rapid adoption of agentic tools threatens to amplify this crisis. Several approaches have been developed to address this variation by following the same strategy: repeating an analysis under different combinations of analytical decisions and inspecting the distribution of outcomes. Examples include multiverse analysis, specification curve analysis, and vibration of effects. More broadly, the Predictability-Computability-Stability (PCS) framework assesses how documented judgment calls across the data science life cycle affect conclusions, with predictability serving as a reality-check that a model captures signal. Each of these approaches results in a "multiverse" o

ALT Synthetic data are a promising way to maintain privacy while disseminating research data, thereby enabling compliance with the open science principles. Popular methods for generating synthetic data include sequential synthesis, where each variable is generated conditionally on the preceding ones in a given order. This order, also called the visit sequence of the synthesizer, is known to affect the quality of synthetic data. This raises the question of the criteria that should guide its choice. In this work, we present a novel method to perform such selection based on a goodness of fit criterion, and show that it outperforms alternative structure-based methods, both in terms of synthesis quality and time complexity. The proposed method also has the advantage of being applicable to any type of data and any class of conditional distribution models.

ALT Allowing for small fraction of errors in retrieved stored binary patterns, Hopfield-Hebbian neural network (NN) associative memory dynamics [14] achieves linearly scaled capacity. The famous proportionality constant αc≈ 0.14 is obtained in a large dimensional scenario as the ratio of the number of reliably stored patterns m and the patterns' length n under the assumption that uniformly randomly chosen binary patterns are independent among themselves. While such an idealized scenario is now well understood, more practical scenarios with correlated/noisy datasets remaining challenging even for classical statistical mechanics replica approaches. We consider precisely such practical settings where each archetype pattern is replaced with M correlated examples with parameter r governing the quality of correlation. We develop a powerful Random Duality Theory (RDT) based framework to analyze correlated data and already on the first level of RDT lifting obtain capacity characterizations that re

ALT Many problems in computational statistics and machine learning can be formulated as approximating a target distribution by minimizing a discrepancy over a restricted class of candidate distributions. This minimum-discrepancy viewpoint appears, for example, in numerical integration, sampling, generative modelling, and statistical inference, but these instances are often studied separately in the literature. This thesis develops the first unified treatment of these many problems through the minimization of the maximum mean discrepancy (MMD), a kernel-based discrepancy between probability measures. The central theme is that MMD is not only a statistically meaningful distance between probability distributions, but also a tractable optimization objective to design practical and efficient algorithms. This thesis summarizes the recent theoretical and empirical advancements of such minimum-MMD based algorithms.

ALT Feature extraction from unstructured data typically relies on opaque embeddings unsuited to interpretable decision-making. Interpretable features, on the other hand, typically require costly, and often infeasible, hand-engineering. Large language models (LLMs) offer a natural bridge, as they can read raw data and describe it in interpretable terms. One could simply ask an LLM to extract features from a large sample of the dataset, but context limitations and diluted attention can lead to unsatisfactory results. We introduce a method that automatically discovers interpretable, decision-relevant features from unstructured data. Our key idea is to perform pairwise comparisons between items, asking an LLM to articulate, in interpretable terms, the features distinguishing each pair. We treat these explanations, or rationales, as data: we develop a statistical choice model that combines these rationales across the comparisons to identify relevant features and estimate per-item, per-feature s

ALT Deep Gaussian processes (DGPs) are flexible surrogate models for computer experiments with nonstationary response surfaces, but inference over their latent representations remains computationally challenging. Existing methods typically rely on variational inference or Markov chain Monte Carlo (MCMC), which impose variational-family assumptions or require costly posterior sampling. We propose the latent maximum-a-posteriori deep Gaussian process (LaMP-DGP), which uses gradient-based optimization to obtain MAP estimates of the latent representations and enables efficient probabilistic prediction through linked-GP calculations. To scale LaMP-DGP to large datasets, we develop a mini-batch optimization algorithm and, under a polynomial mini-batch bias-decay condition, establish a stationarity bound for latent-state optimization that motivates a growing batch-size schedule. Across numerical experiments and real-data analyses, LaMP-DGP provides consistently competitive predictive performance