Sharp convergence rates for averaged nonexpansive maps

TL;DR

This paper derives precise convergence rate estimates for averaged nonexpansive maps in normed spaces, proving the optimality of a recent asymptotic regularity bound and confirming a key conjecture through tight bounds and Markov chain analysis.

Contribution

It provides sharp convergence rate estimates for averaged nonexpansive maps and confirms the optimality of a recent asymptotic regularity bound, settling a longstanding conjecture.

Findings

The asymptotic regularity bound with constant 1/√π is sharp.

Recursive bounds are tight and attained by a constructed nonexpansive map.

Markov chain analysis confirms the optimality of the convergence rate constant.

Abstract

We establish sharp estimates for the convergence rate of the Kranosel'ski\v{\i}-Mann fixed point iteration in general normed spaces, and we use them to show that the asymptotic regularity bound recently proved in [11] (Israel Journal of Mathematics 199(2), 757-772, 2014) with constant $1/\sqrt{\pi}$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in [3] (Proceedings of the 2nd International Conference on Fixed Point Theory and Applications, World Scientific Press, London, 27-66, 1992) which we reinterpret in terms of a nested family of optimal transport problems. We show that these bounds are tight by building a nonexpansive map $T:[0,1]^{\mathbb N}\to[0,1]^{\mathbb N}$ that attains them with equality, settling the main conjecture in [3]. The recursive bounds are in turn reinterpreted as absorption probabilities for an underlying Markov chain which is used to establish the tightness of the constant $1/\sqrt{\pi}$.