Sharp convergence rates for averaged nonexpansive maps

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 with constant $1/\sqrt{\pi}$ is sharp and cannot be improved. To this end we consider the recursive bounds introduced in 3 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 $1/\sqrt{\pi}$.