# Failure of uniform laws of large numbers for subdifferentials and beyond

arXiv: https://arxiv.org/abs/2511.16568 · HTML: https://mlsanity.com/paper/2511.16568/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 20 Nov 2025.

## Builds on (most cited first, 6 of 6)

- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Uniform Convergence of Gradients for Non-Convex Learning and Optimization](https://mlsanity.com/paper/1810.11059/) (2018)
- [Graphical Convergence of Subgradients in Nonconvex Optimization and Learning](https://mlsanity.com/paper/1810.07590/) (2018)
- [On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning](https://mlsanity.com/paper/2405.10289/) (2024)
- [A Strong Law of Large Numbers for Random Monotone Operators](https://mlsanity.com/paper/1910.04405/) (2019)
- [Consistency of sample-based stationary points for infinite-dimensional stochastic optimization](https://mlsanity.com/paper/2306.17032/) (2023)

## Built on by (most cited first, 3 of 3)

- [Finding a stationary point of a stochastic convex problem](https://mlsanity.com/paper/2607.06883/) (2026)
- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)
- [Random tilts to find stationary points in stochastic convex optimization](https://mlsanity.com/paper/2609.17798/) (2026)

## Abstract (arXiv)

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.