Stochastic proximal subgradient descent oscillates in the vicinity of its accumulation set

TL;DR

This paper studies the behavior of stochastic proximal subgradient descent, showing it oscillates near its accumulation set with increasing time between transitions, indicating strong stability properties.

Contribution

It extends deterministic oscillation results to stochastic proximal subgradient descent using stochastic approximation theory, unifying the analysis of both cases.

Findings

Iterates spend infinite time transitioning between accumulation points.

Oscillation behavior persists in the stochastic proximal setting.

Results demonstrate strong stability of the algorithm.

Abstract

We analyze the stochastic proximal subgradient descent in the case where the objective functions are path differentiable and verify a Sard-type condition. While the accumulation set may not be reduced to unique point, we show that the time spent by the iterates to move from one accumulation point to another goes to infinity. An oscillation-type behavior of the drift is established. These results show a strong stability property of the proximal subgradient descent. Using the theory of closed measures, Bolte, Pauwels and R\'ios-Zertuche established this type of behavior for the deterministic subgradient descent. Our technique of proof relies on the classical works on stochastic approximation of differential inclusions, which allows us to extend results in the deterministic case to a stochastic and proximal setting, as well as to treat these different cases in a unified manner.