Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization
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| Format: | Preprint |
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2026
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| _version_ | 1866918426746814464 |
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| author | Yaakoubi, Chiheb Louart, Cosme Tiomoko, Malik Liao, Zhenyu |
| author_facet | Yaakoubi, Chiheb Louart, Cosme Tiomoko, Malik Liao, Zhenyu |
| contents | 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 $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03146 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization Yaakoubi, Chiheb Louart, Cosme Tiomoko, Malik Liao, Zhenyu Machine Learning 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 $μ_{\hatθ}^\top x$ with an independent centered Gaussian variable of variance $\text{Tr}(C_{\hatθ}\mathbb{E}[xx^\top])$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $μ_{\hatθ}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights. |
| title | Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2604.03146 |