Finding a stationary point of a stochastic convex problem

Felipe Areces; John Duchi; Malo Sommers

Finding a stationary point of a stochastic convex problem

Machine Learning 2026-07-08 v1 Machine Learning Optimization and Control

Authors: Felipe Areces , John Duchi , Malo Sommers

Abstract

We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.

Keywords

stochastic convex optimization stationary point subdifferential convergence guarantee proximal-point methods

Cite

@article{arxiv.2607.06883,
  title  = {Finding a stationary point of a stochastic convex problem},
  author = {Felipe Areces and John Duchi and Malo Sommers},
  journal= {arXiv preprint arXiv:2607.06883},
  year   = {2026}
}