  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 $\ell_\infty$ bound
for the regression estimator $\hat{\mu}_A$, any `structural property' that can
be detected via the CGMT for $\hat{\mu}_G$ (under a standard Gaussian design
$G$) 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. We also provide a counterexample, showing that universality
properties for regularized regression estimators do not extend to general
isotropic designs.
