Lipschitzian SLLNs for random functions

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.

Optimization and Control (math.OC); Machine Learning (cs.LG); Statistics Theory (math.ST)

arXiv:2607.20411 [math.OC]

(or arXiv:2607.20411v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2607.20411 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.