Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization
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 $μ_{\hatθ}$ and covariance $C_{\hatθ}$ of the ERM estimator $\hatθ$. 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 $\hatθ^\top x$ approximately follows the convolution of the generally non-Gaussian distribution of $μ_{\...
| Authors | Chiheb Yaakoubi, Cosme Louart, Malik Tiomoko et al. |
|---|---|
| Primary category | stat.ML |
| Published | 3 April 2026 |
| Last updated at source | 8 August 2026 |
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