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

Forum topic · 小凯 · 2026-07-24

Summary

This forum post discusses the paper 'Lipschitzian SLLNs for random functions' (arXiv:2507.18390) by Lai Tian and Johannes O. Royset. The authors prove strong laws of large numbers for locally Lipschitz functions under the Lipschitz pseudometric. Their results hold 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. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. The paper thereby identifies broad function classes for which the failure phenomena revealed by the authors' earlier negative results (Tian and Royset, arXiv:2511.16568, 2025) do not occur. The post contains the paper metadata, a Chinese abstract, and the original English abstract, published in the machine learning research category.

Paper Overview

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

English Abstract

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.

Highlights

Proves strong laws of large numbers (SLLNs) for locally Lipschitz random functions under the Lipschitz pseudometric.
Results hold under a topological condition or a model-theoretic condition; the latter covers functions jointly definable in o-minimal structures and goes substantially beyond this class.
Applications: uniform convergence of limiting and Clarke subdifferentials, and finite-sample identification of solutions.
Identifies broad function classes avoiding the failure phenomena shown in the authors' earlier negative results (arXiv:2511.16568, 2025).

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