A Constant Step Stochastic Douglas-Rachford Algorithm with Application to Non Separable Regularizations

TL;DR

This paper introduces a stochastic version of the Douglas-Rachford algorithm with constant step size, demonstrating its convergence properties and applying it to structured regularization problems.

Contribution

It proposes a novel stochastic Douglas-Rachford algorithm with constant step size and analyzes its convergence behavior under randomness.

Findings

Iterates remain close to solutions with high probability

Convergence is guaranteed for sufficiently small step sizes

Applicable to structured regularization problems

Abstract

The Douglas Rachford algorithm is an algorithm that converges to a minimizer of a sum of two convex functions. The algorithm consists in fixed point iterations involving computations of the proximity operators of the two functions separately. The paper investigates a stochastic version of the algorithm where both functions are random and the step size is constant. We establish that the iterates of the algorithm stay close to the set of solution with high probability when the step size is small enough. Application to structured regularization is considered.