mlsanityData to 14 Oct 2026

Failure of uniform laws of large numbers for subdifferentials and beyond

arXiv 2511.16568 ↗First posted 20 Nov 2025math.OC math.ST stat.MLcited by 3

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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.

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

6 of 29 references.

References · 6

  1. 2016The Landscape of Empirical Risk for Non-convex Lossescited by 86
  2. 2018Uniform Convergence of Gradients for Non-Convex Learning and Optimizationcited by 38
  3. 2018Graphical Convergence of Subgradients in Nonconvex Optimization and Learningcited by 17
  4. 2024On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learningcited by 3
  5. 2019A Strong Law of Large Numbers for Random Monotone Operatorscited by 2
  6. 2023Consistency of sample-based stationary points for infinite-dimensional stochastic optimizationcited by 2

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026Finding a stationary point of a stochastic convex problemcited by 2
  2. 2026Lipschitzian SLLNs for random functionscited by 0
  3. 2026Random tilts to find stationary points in stochastic convex optimizationcited by 0