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Non-asymptotic estimates of the minimal risk in statistic...
[Submitted on 22 Jun 2026] · 2026-06-23 · via cs.LG updates on arXiv.org

Authors:Liming Wu (1), Sen Yang (2) ((1) Laboratoire de Mathématiques Blaise Pascal, CNRS-UMR 6620, Université Clermont Auvergne, (2) IASM, Harbin Institute of Technology)

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Abstract:In this paper we prove some concentration inequalities for two types of error probabilities in the Empirical Risk Principle (ERP) in statistical learning, which provide a lower bound and an upper bound for the minimal risk (in terms of the minimal empirical risk) with non-asymptotic high confidence. The usual boundedness condition of the empirical risk function is relaxed to the Gaussian or exponential integrability condition. The confidence of the lower bound of the minimal risk is shown to be independent of the number of training parameters and the dimension of the input vectors, allowing one to detect the deficiency of a learning machine efficiently; and the confidence of the upper bound of the minimal risk is proved to be high provided that the sample size $n$ is much greater than the box dimension of the parameter set $\Theta$ in the Orlicz metric $d_{\psi_1}$ associated with the risk functions. Our work is based on Talagrand's concentration inequalities (the sharp versions by Bousquet and Klein-Rio), transport-entropy inequalities and the recent progress in the theory of empirical processes and statistical learning.

Submission history

From: Sen Yang [view email]
[v1] Mon, 22 Jun 2026 13:09:28 UTC (33 KB)

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