Failure of uniform laws of large numbers for subdifferentials and beyond

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.