统计与数据科学系系列学术报告之三百八十五

时间：2023年02月28日（周二）15:00-16:30

地点：李达三楼104室

主持人：复旦大学管理学院统计与数据科学系蒋斐宇青年副研究员

题目：Universality of regularized regression estimators in high dimensions

报告人：Qiyang Han

Assistant professor of Statistics

Rutgers University

个人简介：

Qiyang Han is an assistant professor of Statistics at Rutgers University. He received a Ph.D. in Statistics in 2018 from University of Washington under the supervision of Professor Jon A. Wellner, and a B.Sc in Mathematics in 2013 from Fudan University. His research expands broadly in mathematical statistics and high dimensional probability, with a particular focus on empirical process theory and its applications to nonparametric and high dimensional statistics. He is a recipient of the NSF CAREER award in 2022, and the Bernoulli Society New Researcher Award in 2023.

摘要：

The Convex Gaussian Min-Max Theorem (CGMT) has emerged as a prominent theoretical tool for analyzing the precise stochastic behavior of various statistical estimators in the so-called high dimensional proportional regime, where the sample size and the signal dimension are of the same order. However, a well recognized limitation of the existing CGMT machinery rests in its stringent requirement on the exact Gaussianity of the design matrix, therefore rendering the obtained precise high dimensional asymptotics largely a specific Gaussian theory in various important statistical models.

This work provides a structural universality framework for a broad class of regularized regression estimators that is particularly compatible with the CGMT machinery. Here universality means that if a `structure' is satisfied by the regression estimator μ ̂_G for a standard Gaussian design G, then it will also be satisfied by μ ̂_A for a general non-Gaussian design A with independent entries. In particular, we show that with a good enough l_∞ bound for the regression estimator μ ̂_A, any `structural property' that can be detected via the CGMT for μ ̂_G also holds for μ ̂_A under a general design A with independent entries.

As a proof of concept, we demonstrate our new universality framework in three key examples of regularized regression estimators: the Ridge, Lasso and regularized robust regression estimators, where new universality properties of risk asymptotics and/or distributions of regression estimators and other related quantities are proved. As a major statistical implication of the Lasso universality results, we validate inference procedures using the degrees-of-freedom adjusted debiased Lasso under general design and error distributions.