Qiyang Han, Rutgers University
[in-person]

OR

Zoom Meeting: https://usc.zoom.us/j/92190922303?pwd=Q3VlN3lhWFpWd2xSOUxNL0Y4aWViUT09
Meeting ID: 921 9092 2303
Passcode: 009563

Abstract: 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 $\hat{\mu}_G$ for a standard Gaussian design $G$, then it will also be satisfied by $\hat{\mu}_A$ for a general non-Gaussian design $A$ with independent entries. In particular, we show that with a good enough $\ell_\infty$ bound for the regression estimator $\hat{\mu}_A$, any `structural property' that can be detected via the CGMT for $\hat{\mu}_G$ also holds for $\hat{\mu}_A$ 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. This talk is based on joint work with Yandi Shen (Chicago).