arXiv subjects

Pragya Sur

Publications and source records attributed to Pragya Sur.

At least 19 recordsLinked to original sources

arXiv2026

DAIF: A Data-Driven Intermediate Fusion Framework for Multimodal Supervised Learning via Approximate Message Passing

Multimodal supervised learning seeks to leverage multiple heterogeneous data sources to improve predictive performance. A central challenge is determining the fusion granularity across modalities: over-integration may amplify noise while under-integration fails to exploit cross-modal dependence. Existing approaches rely on pre-specified fusion architectures, from early to late fusion, that may not adapt to the underlying dependence structure among modalities. We propose DAIF, a data adaptive intermediate fusion framework that combines random matrix theory and non-parametric dependence measures to learn fusion structure directly from data. We operate under a Bayesian multimodal factor model where the prior on the latent factors determines the cross-modal dependence. Our method clusters modalities based on estimated intermodal dependence, then performs clusterwise empirical Bayes estimation of the priors. These estimated priors are used to construct denoisers within an approximate message passing (AMP) framework, yielding denoised low-dimensional features that borrow strength across related modalities while preserving modality-specific signal. The resulting embeddings are used for downstream supervised prediction. We evaluate the framework through simulations under varying dependence structures and signal regimes, comparing against several benchmark methods, and demonstrate its practical utility on two multimodal datasets, namely a trimodal TEA-seq dataset (Swanson et al., 2021) and TCGA-BRCA dataset (Goldman et al., 2020). In the first example, we predict the expression level of a T-cell differentiation marker protein and in the second case we analyze patient survival prediction based on multimodal information. Our method competes with or outperforms the state-of-the-art techniques in both prediction problems, demonstrating its versatility across diverse supervised learning tasks.

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arXiv2026

Operationalizing Individual Fairness via Gradient Descent and Bradley-Terry Models

Individual fairness, the notion that "similar individuals should be treated similarly," provides a strong and flexible fairness guarantee for algorithmic decision makers. However, a barrier to implementing individual fairness in practice is the difficulty of learning the similarity metric over individuals. In this work, we present an algorithm for learning a Mahalanobis similarity metric from triplet queries of the form "is individual $i$ more similar to individual $j$ or $k$?" We work in the standard Bradley-Terry model for pairwise comparisons. Our algorithm consists of a spectral initialization step followed by gradient descent. We provide extensive theoretical guarantees on our algorithm, showing that it converges quickly to the ground truth metric despite the non-convexity of the loss in our model. Because our focus is on fairness, we also show that individual fairness with respect to an estimated metric is sufficient to achieve similar fairness with respect to the true metric. We also discuss potential applications of our work to AI model tuning. Finally, we present experimental results that demonstrate the convergence of our algorithm and the fairness performance of downstream fair predictors trained on our estimated metric.

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arXiv2026

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.

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arXiv2026

Multi-layer Cross-attention is Provably Optimal for Multi-modal In-context Learning

Recent progress has rapidly advanced our understanding of the mechanisms underlying in-context learning in modern attention-based neural networks. However, existing results focus exclusively on unimodal data; in contrast, the theoretical underpinnings of in-context learning for multi-modal data remain poorly understood. We introduce a mathematically tractable framework for studying multi-modal learning and explore when transformer-like architectures can recover Bayes-optimal performance in-context. To model multi-modal problems, we assume the observed data arises from a latent factor model. Our first result comprises a negative take on expressibility: we prove that single-layer, linear self-attention fails to recover the Bayes-optimal predictor uniformly over the task distribution. To address this limitation, we introduce a novel, linearized cross-attention mechanism, which we study in the regime where both the number of cross-attention layers and the context length are large. We show that this cross-attention mechanism is provably Bayes optimal when optimized using gradient flow. Our results underscore the benefits of depth for in-context learning and establish the provable utility of cross-attention for multi-modal distributions.

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arXiv2025

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.

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arXiv2025

Preventing Model Collapse Under Overparametrization: Optimal Mixing Ratios for Interpolation Learning and Ridge Regression

Model collapse occurs when generative models degrade after repeatedly training on their own synthetic outputs. We study this effect in overparameterized linear regression in a setting where each iteration mixes fresh real labels with synthetic labels drawn from the model fitted in the previous iteration. We derive precise generalization error formulae for minimum-$\ell_2$-norm interpolation and ridge regression under this iterative scheme. Our analysis reveals intriguing properties of the optimal mixing weight that minimizes long-term prediction error and provably prevents model collapse. For instance, in the case of min-$\ell_2$-norm interpolation, we establish that the optimal real-data proportion converges to the reciprocal of the golden ratio for fairly general classes of covariate distributions. Previously, this property was known only for ordinary least squares, and additionally in low dimensions. For ridge regression, we further analyze two popular model classes -- the random-effects model and the spiked covariance model -- demonstrating how spectral geometry governs optimal weighting. In both cases, as well as for isotropic features, we uncover that the optimal mixing ratio should be at least one-half, reflecting the necessity of favoring real-data over synthetic. We study three additional settings: (i) where real data is fixed and fresh labels are not obtained at each iteration, (ii) where covariates vary across iterations but fresh real labels are available each time, and (iii) where covariates vary with time but only a fraction of them receive fresh real labels at each iteration. Across these diverse settings, we characterize when model collapse is inevitable and when synthetic data improves learning. We validate our theoretical results with extensive simulations.

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arXiv2025

Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators

Approximate Message Passing (AMP) algorithms enable precise characterization of certain classes of random objects in the high-dimensional limit, and have found widespread applications in fields such as signal processing, statistics, and communications. In this work, we introduce Multi-Environment Generalized Long AMP, a novel AMP framework that applies to transfer learning problems with multiple data sources and distribution shifts. We rigorously establish state evolution for multi-environment GLAMP. We demonstrate the utility of this framework by precisely characterizing the risk of three Lasso-based transfer learning estimators for the first time: the Stacked Lasso, the Model Averaging Estimator, and the Second Step Estimator. We also demonstrate the remarkable finite sample accuracy of our theory via extensive simulations.

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arXiv2025

Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling

We study the fundamental problem of calibrating a linear binary classifier of the form $\sigma(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $\sigma$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ and the true linear weight $w_\star$. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.

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arXiv2024

Generalization error of min-norm interpolators in transfer learning

This paper establishes the generalization error of pooled min-$\ell_2$-norm interpolation in transfer learning, where data from diverse distributions are available. Min-norm interpolators arise naturally as implicit regularized limits of modern machine learning algorithms. Prior work has characterized their out-of-distribution risk when samples from the test distribution are unavailable during training. In many applications, however, limited test samples may be available at training time, yet properties of min-norm interpolation in this regime remain poorly understood. We address this gap by characterizing the bias and variance of pooled min-$\ell_2$-norm interpolation under both covariate shift and model shift. Our results yield several important implications. In certain cases under model shift, we show that adding data always hurts when the signal-to-noise ratio (SNR) is low. At higher SNR levels, transfer learning is beneficial provided the shift-to-signal ratio falls below a threshold that we characterize explicitly. Under covariate shift, we find that when the source sample size is small relative to the dimension, greater heterogeneity between domains reduces risk, and vice versa. While our model shift results are initially established for Gaussian designs, we extend them to more general designs through a universality argument. To illustrate the broader applicability of our technical tools beyond interpolation learning, we characterize the risk of a bias-corrected estimator that uses the pooled interpolator as an initialization and corrects the resulting bias with target data. On the technical side, we develop a novel anisotropic local law and a Lindeberg-swapping argument, yielding tools that may be of independent interest in random matrix theory and universality analysis. Finally, we supplement our theory with simulations demonstrating the finite-sample efficacy of our results.

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arXiv2024

HEDE: Heritability estimation in high dimensions by Ensembling Debiased Estimators

Estimating heritability remains a significant challenge in statistical genetics. Diverse approaches have emerged over the years that are broadly categorized as either random effects or fixed effects heritability methods. In this work, we focus on the latter. We propose HEDE, an ensemble approach to estimate heritability or the signal-to-noise ratio in high-dimensional linear models where the sample size and the dimension grow proportionally. Our method ensembles post-processed versions of the debiased lasso and debiased ridge estimators, and incorporates a data-driven strategy for hyperparameter selection that significantly boosts estimation performance. We establish rigorous consistency guarantees that hold despite adaptive tuning. Extensive simulations demonstrate our method's superiority over existing state-of-the-art methods across various signal structures and genetic architectures, ranging from sparse to relatively dense and from evenly to unevenly distributed signals. Furthermore, we discuss the advantages of fixed effects heritability estimation compared to random effects estimation. Our theoretical guarantees hold for realistic genotype distributions observed in genetic studies, where genotypes typically take on discrete values and are often well-modeled by sub-Gaussian distributed random variables. We establish our theoretical results by deriving uniform bounds, built upon the convex Gaussian min-max theorem, and leveraging universality results. Finally, we showcase the efficacy of our approach in estimating height and BMI heritability using the UK Biobank.

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arXiv2024

ROTI-GCV: Generalized Cross-Validation for right-ROTationally Invariant Data

Two key tasks in high-dimensional regularized regression are tuning the regularization strength for accurate predictions and estimating the out-of-sample risk. It is known that the standard approach -- $k$-fold cross-validation -- is inconsistent in modern high-dimensional settings. While leave-one-out and generalized cross-validation remain consistent in some high-dimensional cases, they become inconsistent when samples are dependent or contain heavy-tailed covariates. As a first step towards modeling structured sample dependence and heavy tails, we use right-rotationally invariant covariate distributions -- a crucial concept from compressed sensing. In the proportional asymptotics regime where the number of features and samples grow comparably, which is known to better reflect the empirical behavior in moderately sized datasets, we introduce a new framework, ROTI-GCV, for reliably performing cross-validation under these challenging conditions. Along the way, we propose new estimators for the signal-to-noise ratio and noise variance. We conduct experiments that demonstrate the accuracy of our approach in a variety of synthetic and semi-synthetic settings.

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arXiv2024

Predictive Inference in Multi-environment Scenarios

We address the challenge of constructing valid confidence intervals and sets in problems of prediction across multiple environments. We investigate two types of coverage suitable for these problems, extending the jackknife and split-conformal methods to show how to obtain distribution-free coverage in such non-traditional, potentially hierarchical data-generating scenarios. We demonstrate a novel resizing method to adapt to problem difficulty, which applies both to existing approaches for predictive inference and the methods we develop; this reduces prediction set sizes using limited information from the test environment, a key to the methods' practical performance, which we evaluate through neurochemical sensing and species classification datasets. Our contributions also include extensions for settings with non-real-valued responses, a theory of consistency for predictive inference in these general problems, and insights on the limits of conditional coverage.

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arXiv2023

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.

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arXiv2023

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).

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arXiv2023

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.

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arXiv2022

A New Central Limit Theorem for the Augmented IPW Estimator: Variance Inflation, Cross-Fit Covariance and Beyond

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.

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arXiv2022

A Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models

We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau envelopes of the loss $\ell$. We use our finite-sample bound to directly recover the "optimistic rate" of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand's well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation.

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arXiv2022

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.

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