Analysis of nonsmooth stochastic approximation: the differential inclusion approach

TL;DR

This paper extends the convergence analysis of stochastic approximation methods to nonsmooth, non-convex problems by using a differential inclusion framework, applicable to algorithms like stochastic subgradient and proximal stochastic gradient descent.

Contribution

It adapts the mean-limit approach to nonsmooth problems, providing a unified convergence framework for various stochastic approximation algorithms.

Findings

Established convergence of stochastic subgradient methods in nonsmooth settings.

Extended differential inclusion approach to constrained and unconstrained problems.

Applicable to deep learning and high-dimensional inference with sparsity penalties.

Abstract

In this paper we address the convergence of stochastic approximation when the functions to be minimized are not convex and nonsmooth. We show that the "mean-limit" approach to the convergence which leads, for smooth problems, to the ODE approach can be adapted to the non-smooth case. The limiting dynamical system may be shown to be, under appropriate assumption, a differential inclusion. Our results expand earlier works in this direction by Benaim et al. (2005) and provide a general framework for proving convergence for unconstrained and constrained stochastic approximation problems, with either explicit or implicit updates. In particular, our results allow us to establish the convergence of stochastic subgradient and proximal stochastic gradient descent algorithms arising in a large class of deep learning and high-dimensional statistical inference with sparsity inducing penalties.