# Lipschitzian SLLNs for random functions

arXiv: https://arxiv.org/abs/2607.20411 · HTML: https://mlsanity.com/paper/2607.20411/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 75 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 18 of 18)

- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Uniform Convergence of Gradients for Non-Convex Learning and Optimization](https://mlsanity.com/paper/1810.11059/) (2018)
- [Gradient Sampling Methods for Nonsmooth Optimization](https://mlsanity.com/paper/1804.11003/) (2018)
- [Graphical Convergence of Subgradients in Nonconvex Optimization and Learning](https://mlsanity.com/paper/1810.07590/) (2018)
- [A Stochastic Subgradient Method for Nonsmooth Nonconvex Multi-Level Composition Optimization](https://mlsanity.com/paper/2001.10669/) (2020)
- [Pathological subgradient dynamics](https://mlsanity.com/paper/1910.13604/) (2019)
- [Subgradient sampling for nonsmooth nonconvex minimization](https://mlsanity.com/paper/2202.13744/) (2022)
- [On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning](https://mlsanity.com/paper/2405.10289/) (2024)
- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)
- [Failure of uniform laws of large numbers for subdifferentials and beyond](https://mlsanity.com/paper/2511.16568/) (2025)
- [Consistency of sample-based stationary points for infinite-dimensional stochastic optimization](https://mlsanity.com/paper/2306.17032/) (2023)
- [The gradient's limit of a definable family of functions admits a variational stratification](https://mlsanity.com/paper/2402.08272/) (2024)
- [Finding a stationary point of a stochastic convex problem](https://mlsanity.com/paper/2607.06883/) (2026)
- [Uniform Convergence Beyond Glivenko-Cantelli](https://mlsanity.com/paper/2510.21506/) (2025)
- [On rates of convergence for sample average approximations without smoothness](https://mlsanity.com/paper/2604.25153/) (2026)
- [Graphical stability of set-valued integrals under measure perturbations](https://mlsanity.com/paper/2607.19035/) (2026)

## 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.