Metric Subregularity and the Proximal Point Method

TL;DR

This paper investigates how metric subregularity influences the linear convergence of proximal point methods in finding zeros of maximal monotone operators, including extensions to multiple operators and randomized algorithms.

Contribution

It establishes metric subregularity as sufficient for linear convergence and generalizes results to multiple operators and randomized proximal methods.

Findings

Metric subregularity guarantees linear convergence of proximal point methods.

Convergence rates are extended to problems involving multiple monotone operators.

Randomized and averaged proximal methods also achieve linear convergence under metric subregularity.

Abstract

We examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators. We begin by showing how metric subregularity is sufficient for linear convergence to a zero of a maximal monotone operator. This result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods.