Linear Convergence and Metric Selection for Douglas-Rachford Splitting and ADMM

TL;DR

This paper establishes tight global linear convergence rate bounds for Douglas-Rachford splitting and ADMM under strong convexity and smoothness, and provides methods for optimal metric and step-size selection.

Contribution

It derives tight convergence bounds under specific assumptions and introduces heuristic methods for parameter selection when assumptions are not met.

Findings

Convergence rate bounds are tight for the considered problem class.

Optimal metric and step-size selection can improve convergence.

Heuristic parameter selection methods are proposed for broader problem classes.

Abstract

Recently, several convergence rate results for Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) have been presented in the literature. In this paper, we show global linear convergence rate bounds for Douglas-Rachford splitting and ADMM under strong convexity and smoothness assumptions. We further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters. For problems that satisfy the assumptions, we show how to select step-size and metric for the algorithm that optimize the derived convergence rate bounds. For problems with a similar structure that do not satisfy the assumptions, we present heuristic step-size and metric selection methods.