Failure of uniform laws of large numbers for subdifferentials and beyond
By: Lai Tian, Johannes O. Royset
Potential Business Impact:
Math rules break for bumpy shapes.
We provide counterexamples showing that uniform laws of large numbers do not hold for subdifferentials under natural assumptions. Our results apply to random Lipschitz functions and random convex functions with a finite number of smooth pieces. Consequently, they resolve the questions posed by Shapiro and Xu [J. Math. Anal. Appl., 325(2), 2007] in the negative and highlight the obstacles nonsmoothness poses to uniform results.
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