Finding a stationary point of a stochastic convex problem

2026-07-07

AI Summary

This work addresses the computation of strongly stationary solutions in stochastic convex optimization, where strong stationarity is defined by the presence of small elements near zero in the subdifferential. The paper introduces, for the first time, a notion of strong stationarity based on this criterion and employs tools from dimension theory to analyze the structure of subdifferential graphs, revealing how random sampling effectively preserves their essential features. Building on these insights, a proximal-point-type algorithm is developed, leveraging the Moreau envelope and subdifferential analysis to establish rigorous convergence guarantees. The proposed method theoretically overcomes the challenge posed by the lack of uniform convergence of subgradients in neighborhoods of optimal solutions, thereby enabling efficient approximation of strongly stationary points.

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.

Problem

• stationary point
• stochastic convex optimization
• subdifferential
• Moreau envelope
• proximal-point methods

Innovation

• stationary point
• stochastic convex optimization
• subdifferential
• dimension theory
• proximal-point method