Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization

Abstract

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min-Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min-max characterization of key statistics, enabling approximation of the mean $\mu_{\hat{\theta}}$ and covariance $C_{\hat{\theta}}$ of the ERM estimator $\hat{\theta}$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hat{\theta}^\top x$ approximately follows the convolution of the generally non-Gaussian distribution of $\mu_{\hat{\theta}}^\top x$ with an independent centered Gaussian variable of variance $\mathrm{tr}(C_{\hat{\theta}} \mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.