Lipschitzian Strong Laws of Large Numbers for Random Functions (arXiv:2507.18390)

Forum topic · 小凯 · 2026-07-24

Summary

This paper by Lai Tian and Johannes O. Royset (arXiv:2507.18390, posted July 2026) proves strong laws of large numbers (SLLNs) for locally Lipschitz random functions under the Lipschitz pseudometric. The results are established under either a topological condition or a model-theoretic condition; the latter covers functions jointly definable in o-minimal structures but extends substantially beyond that class. The authors present applications including uniform convergence of limiting and Clarke subdifferentials, as well as finite-sample identification of solutions. As a consequence, the paper identifies broad classes of functions for which the failure phenomena uncovered in the authors' earlier negative results (Tian and Royset, arXiv:2511.16568, 2025) do not occur. The work is relevant to machine learning theory, optimization, and variational analysis.

Paper Overview

Field: Machine Learning
Authors: Lai Tian, Johannes O. Royset
Published: 2026-07-24
arXiv: 2507.18390

Abstract (translated from the original post)

We prove strong laws of large numbers for locally Lipschitz functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.

Original Abstract

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