Forward-backward algorithms devised by graphs

TL;DR

This paper introduces a graph-based framework for designing forward-backward algorithms to find zeros of sums of monotone operators, enabling the creation of new methods with minimal computational overhead.

Contribution

It extends existing frameworks to include cocoercive operators and uses graph structures to systematically generate efficient forward-backward algorithms.

Findings

The framework can recover known algorithms.

New algorithms are generated by different graph configurations.

Performance varies with the choice of graphs.

Abstract

In this work, we present a methodology for devising forward-backward methods for finding zeros in the sum of a finite number of maximally monotone operators. We extend the framework and techniques from [SIAM J. Optim., 34 (2024), pp. 1569-1594] to cover the case involving a finite number of cocoercive operators, which should be directly evaluated instead of computing their resolvent. The algorithms are induced by three graphs that determine how the algorithm variables interact with each other and how they are combined to compute each resolvent. The hypotheses on these graphs ensure that the algorithms obtained have minimal lifting and are frugal, meaning that the ambient space of the underlying fixed point operator has minimal dimension and that each resolvent and each cocoercive operator is evaluated only once per iteration. This framework not only allows to recover some known methods, but also to generate new ones, as the forward-backward algorithm induced by a complete graph. We conclude with a numerical experiment showing how the choice of graphs influences the performance of the algorithms.