Lipschitzian SLLNs for random functions

2026/07/22 by Lai Tian, Johannes O. Royset · 1 voice
#math.OC #cs.LG #math.ST #stat.TH

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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.

Citations

• Finding a stationary point of a stochastic convex problem

Discussions

• Most people will skim past the definitional move here: proving SLLNs in the Lipschitz pseudometric rather than the usual one means you can get uniform convergence of subdifferentials without assuming [bsky, 0 points, 0 comments]

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