Failure of uniform laws of large numbers for subdifferentials and beyond
Optimization and Control
2026-03-17 v2 Statistics Theory
Machine Learning
Statistics Theory
Abstract
We provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under natural assumptions. Our constructions are univariate random Lipschitz functions and bivariate random convex functions with two smooth pieces. Consequently, they resolve the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325 (2007), 1390-1399] in the negative. They also demonstrate the failure of certain graphical and pointwise laws for subdifferentials, revealing fundamental barriers to the consistency of sample-average approximation and subdifferential approximation.
Cite
@article{arxiv.2511.16568,
title = {Failure of uniform laws of large numbers for subdifferentials and beyond},
author = {Lai Tian and Johannes O. Royset},
journal= {arXiv preprint arXiv:2511.16568},
year = {2026}
}
Comments
17 pages, 2 figures; Section 2.3 now includes new discussion of SAA and subdifferential approximation