Universality of regularized regression estimators in high dimensions

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 paper 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 ${\hat{\mathit{\mu }}}_{\mathit{G}}$ for a standard Gaussian design G, then it will also be satisfied by ${\hat{\mathit{\mu }}}_{\mathit{A}}$ 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 ${\hat{\mathit{\mu }}}_{\mathit{A}}$, any “structural property” that can be detected via the CGMT for ${\hat{\mathit{\mu }}}_{\mathit{G}}$ also holds for ${\hat{\mathit{\mu }}}_{\mathit{A}}$ 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. We also provide a counterexample, showing that universality properties for regularized regression estimators do not extend to general isotropic designs.

The proof of our universality results relies on new comparison inequalities for the optimum of a broad class of cost functions and Gordon’s max–min (or min–max) costs, over arbitrary structure sets subject to ${\ell }_{\infty }$ constraints. These results may be of independent interest and broader applicability.