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Pragya Sur

Pragya Sur contributes to research discovery and scholarly infrastructure.

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math.ST Machine Learning Methodology Statistics Theory Information Theory math.IT

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preprint2026arXiv

Self-Distillation is Optimal Among Spectral Shrinkage Estimators in Spiked Covariance Models

Self-distillation has emerged as a promising technique for improving model performance in modern machine learning systems. We develop the statistical foundations of self-distillation in spiked covariance models, by introducing and analyzing a broad class of estimators, namely spectral shrinkage estimators. We establish that for spiked covariance matrices with $s$ spikes, $s$-step self-distillation achieves optimal performance among spectral shrinkage estimators, outperforming well-known estimators in statistics and machine learning. Moreover, we show that $s$ steps are necessary for optimality: any $(s-k)$-step distilled estimator is strictly suboptimal for $1 \leq k \leq s$. For the special subclass of isotropic covariances, we show that optimally tuned Ridge regression performs best among spectral shrinkage estimators. We also study a federated approach where multiple data centers share spectral shrinkage estimators and a common server seeks to aggregate them to achieve optimal performance. In this case, we find that the best local rule again takes the form of self-distillation, though it differs from the optimal rule when data are hosted centrally on a single server. Together, our results elucidate why self-distillation improves predictive performance and provide a broader statistical framework connecting it with classical shrinkage-based methods.

preprint2025arXiv

Characterizing Finite-Dimensional Posterior Marginals in High-Dimensional GLMs via Leave-One-Out

We investigate Bayes posterior distributions in high-dimensional generalized linear models (GLMs) under the proportional asymptotics regime, where the number of features and samples diverge at a comparable rate. Specifically, we characterize the limiting behavior of finite-dimensional marginals of the posterior. We establish that the posterior does not contract in this setting. Yet, the finite-dimensional posterior marginals converge to Gaussian tilts of the prior, where the mean of the Gaussian depends on the true signal coordinates of interest. Notably, the effect of the prior survives even in the limit of large samples and dimensions. We further characterize the behavior of the posterior mean and demonstrate that the posterior mean can strictly outperform the maximum likelihood estimate in mean-squared error in natural examples. Importantly, our results hold regardless of the sparsity level of the underlying signal. On the technical front, we introduce leave-one-out strategies for studying these marginals that may be of independent interest for analyzing low-dimensional functionals of high-dimensional signals in other Bayesian inference problems.

preprint2025arXiv

Spectrum-Aware Debiasing: A Modern Inference Framework with Applications to Principal Components Regression

Debiasing is a fundamental concept in high-dimensional statistics. While degrees-of-freedom adjustment is the state-of-the-art technique in high-dimensional linear regression, it is limited to i.i.d. samples and sub-Gaussian covariates. These constraints hinder its broader practical use. Here, we introduce Spectrum-Aware Debiasing--a novel method for high-dimensional regression. Our approach applies to problems with structured dependencies, heavy tails, and low-rank structures. Our method achieves debiasing through a rescaled gradient descent step, deriving the rescaling factor using spectral information of the sample covariance matrix. The spectrum-based approach enables accurate debiasing in much broader contexts. We study the common modern regime where the number of features and samples scale proportionally. We establish asymptotic normality of our proposed estimator (suitably centered and scaled) under various convergence notions when the covariates are right-rotationally invariant. Such designs have garnered recent attention due to their crucial role in compressed sensing. Furthermore, we devise a consistent estimator for its asymptotic variance. Our work has two notable by-products: first, we use Spectrum-Aware Debiasing to correct bias in principal components regression (PCR), providing the first debiased PCR estimator in high dimensions. Second, we introduce a principled test for checking alignment between the signal and the eigenvectors of the sample covariance matrix. This test is independently valuable for statistical methods developed using approximate message passing, leave-one-out, or convex Gaussian min-max theorems. We demonstrate our method through simulated and real data experiments. Technically, we connect approximate message passing algorithms with debiasing and provide the first proof of the Cauchy property of vector approximate message passing (V-AMP).

preprint2023arXiv

The Asymptotic Distribution of the MLE in High-dimensional Logistic Models: Arbitrary Covariance

We study the distribution of the maximum likelihood estimate (MLE) in high-dimensional logistic models, extending the recent results from Sur (2019) to the case where the Gaussian covariates may have an arbitrary covariance structure. We prove that in the limit of large problems holding the ratio between the number $p$ of covariates and the sample size $n$ constant, every finite list of MLE coordinates follows a multivariate normal distribution. Concretely, the $j$th coordinate $\hat β_j$ of the MLE is asymptotically normally distributed with mean $α_\star β_j$ and standard deviation $σ_\star/τ_j$; here, $β_j$ is the value of the true regression coefficient, and $τ_j$ the standard deviation of the $j$th predictor conditional on all the others. The numerical parameters $α_\star > 1$ and $σ_\star$ only depend upon the problem dimensionality $p/n$ and the overall signal strength, and can be accurately estimated. Our results imply that the MLE's magnitude is biased upwards and that the MLE's standard deviation is greater than that predicted by classical theory. We present a series of experiments on simulated and real data showing excellent agreement with the theory.

preprint2023arXiv

Universality in block dependent linear models with applications to nonparametric regression

Over the past decade, characterizing the exact asymptotic risk of regularized estimators in high-dimensional regression has emerged as a popular line of work. This literature considers the proportional asymptotics framework, where the number of features and samples both diverge, at a rate proportional to each other. Substantial work in this area relies on Gaussianity assumptions on the observed covariates. Further, these studies often assume the design entries to be independent and identically distributed. Parallel research investigates the universality of these findings, revealing that results based on the i.i.d.~Gaussian assumption extend to a broad class of designs, such as i.i.d.~sub-Gaussians. However, universality results examining dependent covariates so far focused on correlation-based dependence or a highly structured form of dependence, as permitted by right rotationally invariant designs. In this paper, we break this barrier and study a dependence structure that in general falls outside the purview of these established classes. We seek to pin down the extent to which results based on i.i.d.~Gaussian assumptions persist. We identify a class of designs characterized by a block dependence structure that ensures the universality of i.i.d.~Gaussian-based results. We establish that the optimal values of the regularized empirical risk and the risk associated with convex regularized estimators, such as the Lasso and ridge, converge to the same limit under block dependent designs as they do for i.i.d.~Gaussian entry designs. Our dependence structure differs significantly from correlation-based dependence, and enables, for the first time, asymptotically exact risk characterization in prevalent nonparametric regression problems in high dimensions. Finally, we illustrate through experiments that this universality becomes evident quite early, even for relatively moderate sample sizes.

preprint2022arXiv

Multi-Study Boosting: Theoretical Considerations for Merging vs. Ensembling

Cross-study replicability is a powerful model evaluation criterion that emphasizes generalizability of predictions. When training cross-study replicable prediction models, it is critical to decide between merging and treating the studies separately. We study boosting algorithms in the presence of potential heterogeneity in predictor-outcome relationships across studies and compare two multi-study learning strategies: 1) merging all the studies and training a single model, and 2) multi-study ensembling, which involves training a separate model on each study and ensembling the resulting predictions. In the regression setting, we provide theoretical guidelines based on an analytical transition point to determine whether it is more beneficial to merge or to ensemble for boosting with linear learners. In addition, we characterize a bias-variance decomposition of estimation error for boosting with component-wise linear learners. We verify the theoretical transition point result in simulation and illustrate how it can guide the decision on merging vs. ensembling in an application to breast cancer gene expression data.

preprint2020arXiv

Abstracting Fairness: Oracles, Metrics, and Interpretability

It is well understood that classification algorithms, for example, for deciding on loan applications, cannot be evaluated for fairness without taking context into account. We examine what can be learned from a fairness oracle equipped with an underlying understanding of ``true'' fairness. The oracle takes as input a (context, classifier) pair satisfying an arbitrary fairness definition, and accepts or rejects the pair according to whether the classifier satisfies the underlying fairness truth. Our principal conceptual result is an extraction procedure that learns the underlying truth; moreover, the procedure can learn an approximation to this truth given access to a weak form of the oracle. Since every ``truly fair'' classifier induces a coarse metric, in which those receiving the same decision are at distance zero from one another and those receiving different decisions are at distance one, this extraction process provides the basis for ensuring a rough form of metric fairness, also known as individual fairness. Our principal technical result is a higher fidelity extractor under a mild technical constraint on the weak oracle's conception of fairness. Our framework permits the scenario in which many classifiers, with differing outcomes, may all be considered fair. Our results have implications for interpretablity -- a highly desired but poorly defined property of classification systems that endeavors to permit a human arbiter to reject classifiers deemed to be ``unfair'' or illegitimately derived.

preprint2020arXiv

Representation via Representations: Domain Generalization via Adversarially Learned Invariant Representations

We investigate the power of censoring techniques, first developed for learning {\em fair representations}, to address domain generalization. We examine {\em adversarial} censoring techniques for learning invariant representations from multiple "studies" (or domains), where each study is drawn according to a distribution on domains. The mapping is used at test time to classify instances from a new domain. In many contexts, such as medical forecasting, domain generalization from studies in populous areas (where data are plentiful), to geographically remote populations (for which no training data exist) provides fairness of a different flavor, not anticipated in previous work on algorithmic fairness. We study an adversarial loss function for $k$ domains and precisely characterize its limiting behavior as $k$ grows, formalizing and proving the intuition, backed by experiments, that observing data from a larger number of domains helps. The limiting results are accompanied by non-asymptotic learning-theoretic bounds. Furthermore, we obtain sufficient conditions for good worst-case prediction performance of our algorithm on previously unseen domains. Finally, we decompose our mappings into two components and provide a complete characterization of invariance in terms of this decomposition. To our knowledge, our results provide the first formal guarantees of these kinds for adversarial invariant domain generalization.

preprint2018arXiv

A modern maximum-likelihood theory for high-dimensional logistic regression

Every student in statistics or data science learns early on that when the sample size largely exceeds the number of variables, fitting a logistic model produces estimates that are approximately unbiased. Every student also learns that there are formulas to predict the variability of these estimates which are used for the purpose of statistical inference; for instance, to produce p-values for testing the significance of regression coefficients. Although these formulas come from large sample asymptotics, we are often told that we are on reasonably safe grounds when $n$ is large in such a way that $n \ge 5p$ or $n \ge 10p$. This paper shows that this is far from the case, and consequently, inferences routinely produced by common software packages are often unreliable. Consider a logistic model with independent features in which $n$ and $p$ become increasingly large in a fixed ratio. Then we show that (1) the MLE is biased, (2) the variability of the MLE is far greater than classically predicted, and (3) the commonly used likelihood-ratio test (LRT) is not distributed as a chi-square. The bias of the MLE is extremely problematic as it yields completely wrong predictions for the probability of a case based on observed values of the covariates. We develop a new theory, which asymptotically predicts (1) the bias of the MLE, (2) the variability of the MLE, and (3) the distribution of the LRT. We empirically also demonstrate that these predictions are extremely accurate in finite samples. Further, an appealing feature is that these novel predictions depend on the unknown sequence of regression coefficients only through a single scalar, the overall strength of the signal. This suggests very concrete procedures to adjust inference; we describe one such procedure learning a single parameter from data and producing accurate inference

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