On the metric resolvent: nonexpansiveness, convergence rates and applications

TL;DR

This paper analyzes the nonexpansive properties and convergence rates of metric resolvent-based fixed-point iterations, extending classical operators and unifying various first-order algorithms under a common framework.

Contribution

It introduces a unified analysis of metric resolvent fixed-point iterations, extending proximity operators to variable metrics and applying this to analyze multiple first-order algorithms.

Findings

Established convergence rate bounds for metric resolvent iterations.

Extended proximity operator and Moreau's decomposition to variable metrics.

Unified analysis framework for various first-order operator splitting algorithms.

Abstract

In this paper, we study the nonexpansive properties of metric resolvent, and present a convergence rate analysis for the associated fixed-point iterations (Banach-Picard and Krasnosel'skii-Mann types). Equipped with a variable metric, we develop the global ergodic and non-ergodic iteration-complexity bounds in terms of both solution distance and objective value. A byproduct of our expositions also extends the proximity operator and Moreau's decomposition identity to arbitrary variable metric. It is further shown that many classes of the first-order operator splitting algorithms, including alternating direction methods of multipliers, primal-dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be analyzed within this unified framework.