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Lipschitzian SLLNs for random functions

arXiv 2607.20411 ↗First posted 22 Jul 2026math.OC cs.LG math.STcited by 0

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Abstract (arXiv)

We prove strong laws of large numbers for random functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.

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References · 18

  1. 2018Stochastic subgradient method converges on tame functionscited by 124
  2. 2016The Landscape of Empirical Risk for Non-convex Lossescited by 86
  3. 2019Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learningcited by 72
  4. 2018Uniform Convergence of Gradients for Non-Convex Learning and Optimizationcited by 38
  5. 2018Gradient Sampling Methods for Nonsmooth Optimizationcited by 23
  6. 2018Graphical Convergence of Subgradients in Nonconvex Optimization and Learningcited by 17
  7. 2020A Stochastic Subgradient Method for Nonsmooth Nonconvex Multi-Level Composition Optimizationcited by 16
  8. 2019Pathological subgradient dynamicscited by 9
  9. 2022Subgradient sampling for nonsmooth nonconvex minimizationcited by 6
  10. 2024On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learningcited by 3
  11. 2025Deep Learning as the Disciplined Construction of Tame Objectscited by 3
  12. 2025Failure of uniform laws of large numbers for subdifferentials and beyondcited by 3
6 more
  1. 2023Consistency of sample-based stationary points for infinite-dimensional stochastic optimizationcited by 2
  2. 2024The gradient's limit of a definable family of functions admits a variational stratificationcited by 2
  3. 2026Finding a stationary point of a stochastic convex problemcited by 2
  4. 2025Uniform Convergence Beyond Glivenko-Cantellicited by 1
  5. 2026On rates of convergence for sample average approximations without smoothnesscited by 1
  6. 2026Graphical stability of set-valued integrals under measure perturbationscited by 1

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