Universality of regularized regression estimators in high dimensions
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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. In particular, we show that with a good enough ℓ_∞ bound for the regression estimator μ̂_A, any `structural property' that can be detected via the CGMT for μ̂_G (under a standard Gaussian design G) also holds for μ̂_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.
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