Convergence rates of Forward--Douglas--Rachford splitting method

TL;DR

This paper analyzes the convergence rates of the Forward--Douglas--Rachford splitting method, establishing global sublinear and local linear convergence under certain conditions, with applications in signal processing, inverse problems, and machine learning.

Contribution

It provides the first comprehensive analysis of both global and local convergence rates for FDR, including conditions for local linear convergence and finite identification of smooth manifolds.

Findings

Global convergence rate is o(1/k) in Bregman divergence.

Objective function convergence rate for Forward--Backward is o(1/k) with suitable step-size.

Local linear convergence occurs after finite identification of a smooth manifold.

Abstract

Over the past years, operator splitting methods have become ubiquitous for non-smooth optimization owing to their simplicity and efficiency. In this paper, we consider the Forward--Douglas--Rachford splitting method (FDR) [10,40], and study both global and local convergence rates of this method. For the global rate, we establish an $o(1/k)$ convergence rate in terms of a Bregman divergence suitably designed for the objective function. Moreover, when specializing to the case of Forward--Backward splitting method, we show that convergence rate of the objective function of the method is actually $o(1/k)$ for a large choice of the descent step-size. Then locally, based on the assumption that the non-smooth part of the optimization problem is partly smooth, we establish local linear convergence of the method. More precisely, we show that the sequence generated by FDR method first (i) identifies a smooth manifold in a finite number of iteration, and then (ii) enters a local linear convergence regime, which is for instance characterized in terms of the structure of the underlying active smooth manifold. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from applicative fields including, for instance, signal/image processing, inverse problems and machine learning.