Failure of uniform laws of large numbers for subdifferentials and beyond

TL;DR

This paper presents counterexamples demonstrating that uniform laws of large numbers do not generally hold for subdifferentials, highlighting fundamental limitations in their statistical consistency.

Contribution

It provides the first explicit counterexamples showing the failure of uniform laws of large numbers for subdifferentials under natural conditions.

Findings

Counterexamples for univariate random Lipschitz functions

Counterexamples for bivariate convex functions with smooth pieces

Demonstrates failure of graphical and pointwise laws for subdifferentials

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.