Convergence rate analysis for averaged fixed point iterations in the presence of H\"older regularity

TL;DR

This paper analyzes the convergence rates of averaged fixed point iterations in Hilbert spaces under H"older regularity, extending existing results and applying them to several algorithms including Douglas-Rachford.

Contribution

It introduces a generalized H"older regularity condition that ensures sublinear and linear convergence of fixed point iterations, broadening the scope beyond linear regularity.

Findings

Established sublinear and linear convergence under H"older regularity

Provided convergence rate analysis for key algorithms like Krasnoselskii-Mann and Douglas-Rachford

Showed automatic satisfaction of H"older regularity for convex polynomial sets in finite dimensions

Abstract

In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded H\"older regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for Krasnoselskii-Mann iterations, the cyclic projection algorithm, and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a finite dimensional space, we show that the H\"older regularity properties are automatically satisfied, from which sublinear convergence follows.