A PRECISE HIGH-DIMENSIONAL ASYMPTOTIC THEORY FOR BOOSTING AND MINIMUM-l1-NORM INTERPOLATED CLASSIFIERS

• 成果类型: Article
• 署名作者: Liang, Tengyuan; Sur, Pragya
• 署名单位: University of Chicago; Harvard University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/22-AOS2170
• 发表日期: 2022
• 页码: 1669-1695
• 关键词: logistic-regression; generalization error; robust regression; margin; EXISTENCE; CONVERGENCE; Consistency; algorithms; prediction; models

摘要

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 l(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 (Neural Comput. 11 (1999) 1493-1517; Ann. Statist. 26 (1998) 1651-1686) surrounding boosting, under assumed data generating processes. At the heart of our theory lies an in-depth study of the maximum-l(1)-margin, which can be accurately described by a new system of nonlinear 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 l(q)-geometry, q is an element of [1, 2]. As a final component, via the Lindeberg principle, we establish a universality result showcasing that the scaled l(1)-margin (asymptotically) remains the same, whether the covariates used for boosting arise from a nonlinear random feature model or an appropriately linearized model with matching moments.