Convergence Rates with Inexact Non-expansive Operators

Abstract: In this paper, we present a convergence rate analysis for the inexact Krasnosel'skii-Mann iteration built from nonexpansive operators. Our results include two main parts: we first establish global pointwise and ergodic iteration-complexity bounds, and then, under a metric subregularity assumption, we establish local linear convergence for the distance of the iterates to the set of fixed points. The obtained iteration-complexity result can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature, including the Forward-Backward, the Generalized Forward-Backward, Douglas-Rachford, alternating direction method of multipliers (ADMM) and Primal-Dual splitting methods. For these methods, we also develop easily verifiable termination criteria for finding an approximate solution, which can be seen as a generalization of the termination criterion for the classical gradient descent method. We finally develop a parallel analysis for the non-stationary Krasnosel'skii-Mann iteration. The usefulness of our results is illustrated by applying them to a large class of structured monotone inclusion and convex optimization problems. Experiments on some large scale inverse problems in signal and image processing problems are shown.