A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-ℓ1\ell_1-Norm Interpolated Classifiers

This paper establishes a precise high-dimensional asymptotic theory for boosting on separable data, taking statistical and computational perspectives. We consider a high-dimensional setting where the number of features (weak learners) $p$ scales with the sample size $n$, in an overparametrized regime. Under a class of statistical models, we provide an exact analysis of the generalization error of boosting when the algorithm interpolates the training data and maximizes the empirical $\ell_1$-margin. Further, we explicitly pin down the relation between the boosting test error and the optimal Bayes error, as well as the proportion of active features at interpolation (with zero initialization). In turn, these precise characterizations answer certain questions raised in \cite{breiman1999prediction, schapire1998boosting} surrounding boosting, under assumed data generating processes. At the heart of our theory lies an in-depth study of the maximum-$\ell_1$-margin, which can be accurately described by a new system of non-linear equations; to analyze this margin, we rely on Gaussian comparison techniques and develop a novel uniform deviation argument. Our statistical and computational arguments can handle (1) any finite-rank spiked covariance model for the feature distribution and (2) variants of boosting corresponding to general $\ell_q$-geometry, $q \in [1, 2]$. As a final component, via the Lindeberg principle, we establish a universality result showcasing that the scaled $\ell_1$-margin (asymptotically) remains the same, whether the covariates used for boosting arise from a non-linear random feature model or an appropriately linearized model with matching moments.Comment: 68 pages, 4 figure

Near-optimal multiple testing in Bayesian linear models with finite-sample FDR control

In high dimensional variable selection problems, statisticians often seek to design multiple testing procedures controlling the false discovery rate (FDR) and simultaneously discovering more relevant variables. Model-X methods, such as Knockoffs and conditional randomization tests, achieve the first goal of finite-sample FDR control under the assumption of known covariates distribution. However, it is not clear whether these methods can concurrently achieve the second goal of maximizing the number of discoveries. In fact, designing procedures to discover more relevant variables with finite-sample FDR control is a largely open question, even in the arguably simplest linear models. In this paper, we derive near-optimal testing procedures in high dimensional Bayesian linear models with isotropic covariates. We propose a Model-X multiple testing procedure, PoEdCe, which provably controls the frequentist FDR from finite samples even under model misspecification, and conjecturally achieves near-optimal power when the data follow the Bayesian linear model with a known prior. PoEdCe has three important ingredients: Posterior Expectation, distilled Conditional randomization test (dCRT), and the Benjamini-Hochberg procedure with e-values (eBH). The optimality conjecture of PoEdCe is based on a heuristic calculation of its asymptotic true positive proportion (TPP) and false discovery proportion (FDP), which is supported by methods from statistical physics as well as extensive numerical simulations. Furthermore, when the prior is unknown, we show that an empirical Bayes variant of PoEdCe still has finite-sample FDR control and achieves near-optimal power.Comment: 45 pages, 5 figure

Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning

Wasserstein distributionally robust optimization has recently emerged as a powerful framework for robust estimation, enjoying good out-of-sample performance guarantees, well-understood regularization effects, and computationally tractable reformulations. In such framework, the estimator is obtained by minimizing the worst-case expected loss over all probability distributions which are close, in a Wasserstein sense, to the empirical distribution. In this paper, we propose a Wasserstein distributionally robust estimation framework to estimate an unknown parameter from noisy linear measurements, and we focus on the task of analyzing the squared error performance of such estimators. Our study is carried out in the modern high-dimensional proportional regime, where both the ambient dimension and the number of samples go to infinity at a proportional rate which encodes the under/over-parametrization of the problem. Under an isotropic Gaussian features assumption, we show that the squared error can be recovered as the solution of a convex-concave optimization problem which, surprinsingly, involves at most four scalar variables. Importantly, the precise quantification of the squared error allows to accurately and efficiently compare different ambiguity radii and to understand the effect of the under/over-parametrization on the estimation error. We conclude the paper with a list of exciting research directions enabled by our results.Comment: This paper was previously titled "The Performance of Wasserstein Distributionally Robust M-Estimators in High Dimensions

The curse of overparametrization in adversarial training: Precise analysis of robust generalization for random features regression

Successful deep learning models often involve training neural network architectures that contain more parameters than the number of training samples. Such overparametrized models have been extensively studied in recent years, and the virtues of overparametrization have been established from both the statistical perspective, via the double-descent phenomenon, and the computational perspective via the structural properties of the optimization landscape. Despite the remarkable success of deep learning architectures in the overparametrized regime, it is also well known that these models are highly vulnerable to small adversarial perturbations in their inputs. Even when adversarially trained, their performance on perturbed inputs (robust generalization) is considerably worse than their best attainable performance on benign inputs (standard generalization). It is thus imperative to understand how overparametrization fundamentally affects robustness. In this paper, we will provide a precise characterization of the role of overparametrization on robustness by focusing on random features regression models (two-layer neural networks with random first layer weights). We consider a regime where the sample size, the input dimension and the number of parameters grow in proportion to each other, and derive an asymptotically exact formula for the robust generalization error when the model is adversarially trained. Our developed theory reveals the nontrivial effect of overparametrization on robustness and indicates that for adversarially trained random features models, high overparametrization can hurt robust generalization.Comment: 86 pages (main file: 25 pages and supplementary: 61 pages). To appear in the Annals of Statistic