[
    {
        "id": "osp-16919",
        "type": "article-journal",
        "title": "Asymptotic Analysis of Empirical Risk Minimization on Entry-wise i.i.d. Heavy-Tailed Data",
        "author": [
            {
                "family": "Takanami",
                "given": "Kaito"
            },
            {
                "family": "Takahashi",
                "given": "Takashi"
            },
            {
                "family": "Kabashima",
                "given": "Yoshiyuki"
            }
        ],
        "URL": "https://omanscience.com/en/articles/asymptotic-analysis-of-empirical-risk-minimization-on-entry-wise-i-i-d-heavy-tailed-data",
        "language": "en",
        "issued": {
            "date-parts": [
                [
                    2026
                ]
            ]
        },
        "abstract": "Many real-world datasets exhibit unusually large values far more frequently than predicted by Gaussian models. Heavy-tailed distributions capture this behavior, yet evaluating learning performance under them remains challenging because rare, large feature entries retain non-vanishing effects even in high dimensions. Even in the canonical setting of empirical risk minimization for linear regression with entry-wise i.i.d. symmetric $α$-stable data, a precise asymptotic characterization of prediction has been lacking. In this work, we introduce a functional order parameter that describes the random effective problem associated with each coefficient. Using the replica method, we fully characterize the generalization error in the proportional high-dimensional limit where the sample size and feature dimension diverge at a fixed ratio. Additionally, this analysis establishes a heavy-tail universality law, scaling laws relating typical errors to prediction reliability, and the Bayes-optimal prediction error. In addition to characterizing the effects of extreme entries on the learning process, our method applies broadly to other systems with persistent local heterogeneity."
    }
]