# A Strong Law of Large Numbers for Random Monotone Operators

arXiv: https://arxiv.org/abs/1910.04405 · HTML: https://mlsanity.com/paper/1910.04405/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Oct 2019.

## Built on by (most cited first, 2 of 2)

- [Failure of uniform laws of large numbers for subdifferentials and beyond](https://mlsanity.com/paper/2511.16568/) (2025)
- [Convergence of empirical subgradients for optimal transport-based objectives](https://mlsanity.com/paper/2605.28134/) (2026)

## Abstract (arXiv)

Random monotone operators are stochastic versions of maximal monotone operators which play an important role in stochastic nonsmooth optimization. Several stochastic nonsmooth optimization algorithms have been shown to converge to a zero of a mean operator defined as the expectation, in the sense of the Aumann integral, of a random monotone operator. In this note, we prove a strong law of large numbers for random monotone operators where the limit is the mean operator. We apply this result to the empirical risk minimization problem appearing in machine learning. We show that if the empirical risk minimizers converge as the number of data points goes to infinity, then they converge to an expected risk minimizer.