Distributed Banach-Picard Iteration for Locally Contractive Maps

TL;DR

This paper introduces a distributed algorithm for finding fixed points of averaged locally contractive maps across multiple agents, extending classical iteration methods to networked settings without relying on optimization structures.

Contribution

It develops a novel distributed Banach-Picard iteration for non-optimization-based maps, proving linear convergence using perturbation theory in a networked environment.

Findings

The algorithm converges linearly to the fixed point.

Convergence proof employs perturbation theory of linear operators.

The method applies to non-LC maps via averaging across agents.

Abstract

The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to be the average of individual (not necessarily LC) maps held by a set of agents linked by a communication network. An additional difficulty is that the LC map is not assumed to come from an underlying optimization problem, which prevents exploiting strong global properties such as convexity or Lipschitzianity. Yet, we propose a distributed algorithm and prove its convergence, in fact showing that it maintains the linear rate of the standard Banach-Picard iteration for the average LC map. As another contribution, our proof imports tools from perturbation theory of linear operators, which, to the best of our knowledge, had not been used before in the theory of distributed computation.