Tight Global Linear Convergence Rate Bounds for Douglas-Rachford Splitting

TL;DR

This paper establishes tight global linear convergence rate bounds for Douglas-Rachford splitting in monotone inclusion problems, improving upon previous bounds by proving the operator is contractive under various assumptions.

Contribution

The paper provides the first tight global linear convergence bounds for Douglas-Rachford splitting in general monotone inclusion problems, extending previous results.

Findings

Previous bounds were not tight for the considered class.

Douglas-Rachford operator is contractive under certain conditions.

New bounds improve understanding of convergence rates.

Abstract

Recently, several authors have shown local and global convergence rate results for Douglas-Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas-Rachford operator is contractive.