Convergence Rates for Projective Splitting

TL;DR

This paper analyzes the convergence rates of projective splitting methods for monotone operator inclusions, establishing ergodic and linear rates under various assumptions, thus advancing theoretical understanding of these algorithms.

Contribution

It provides the first detailed convergence rate analysis for projective splitting, including ergodic, strong, and linear convergence results under different conditions.

Findings

Established $O(1/k)$ ergodic convergence rate for convex optimization.

Proved strong convergence and $O(1/\sqrt{k})$ ergodic rate for strongly monotone inclusions.

Achieved linear convergence for inclusions with strong monotonicity and cocoercivity.

Abstract

Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming one of the most flexible operator splitting frameworks available. While weak convergence of the iterates to a solution has been established, there have been few attempts to study convergence rates of projective splitting. The purpose of this paper is to do so under various assumptions. To this end, there are three main contributions. First, in the context of convex optimization, we establish an $O(1/k)$ ergodic function convergence rate. Second, for strongly monotone inclusions, strong convergence is established as well as an ergodic $O(1/\sqrt{k})$ convergence rate for the distance of the iterates to the solution. Finally, for inclusions featuring strong monotonicity and cocoercivity, linear convergence is established.