Robust Convergence Analysis of Three-Operator Splitting

TL;DR

This paper introduces a unified framework using robust control theory to analyze and certify the convergence rates of the three-operator splitting method for optimization, applicable under various assumptions.

Contribution

It provides a novel approach to certify linear and sublinear convergence of TOS using matrix inequalities and robust control, guiding parameter selection for optimal performance.

Findings

Framework certifies convergence rates under diverse conditions.

Matrix inequalities verify sublinear/linear convergence.

Numerical examples demonstrate practical effectiveness.

Abstract

Operator splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-operator splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to verify sublinear/linear convergence of the TOS algorithm and guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. We illustrate our results numerically by solving an input-constrained optimal control problem.