Stochastic proximal splitting algorithm for composite minimization

TL;DR

This paper introduces a stochastic proximal splitting algorithm for composite minimization problems where both smooth and nonsmooth parts are accessed via stochastic information, achieving optimal convergence rates.

Contribution

It extends proximal splitting algorithms to handle stochastic access to both components in composite models, including cases where the nonsmooth part is not proximally tractable.

Findings

Achieves $rac{1}{k}$ convergence rate under strong convexity.

Demonstrates effectiveness through numerical tests on sparse representation models.

Addresses gaps in stochastic optimization for composite problems with non-proximally tractable nonsmooth terms.

Abstract

Supported by the recent contributions in multiple branches, the first-order splitting algorithms became central for structured nonsmooth optimization. In the large-scale or noisy contexts, when only stochastic information on the smooth part of the objective function is available, the extension of proximal gradient schemes to stochastic oracles is based on proximal tractability of the nonsmooth component and it has been deeply analyzed in the literature. However, there remained gaps illustrated by composite models where the nonsmooth term is not proximally tractable anymore. In this note we tackle composite optimization problems, where the access only to stochastic information on both smooth and nonsmooth components is assumed, using a stochastic proximal first-order scheme with stochastic proximal updates. We provide $\mathcal{O}\left( \frac{1}{k} \right)$ the iteration complexity (in expectation of squared distance to the optimal set) under the strong convexity assumption on the objective function. Empirical behavior is illustrated by numerical tests on parametric sparse representation models.