Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

TL;DR

This paper introduces two distributed algorithms, D-KM and D-BKM, for finding fixed points of global nonexpansive operators in multi-agent networks, with applications to distributed optimization and linear algebra.

Contribution

It develops novel distributed fixed point algorithms that converge weakly and extend classical methods to block-coordinate and other problems.

Findings

Both algorithms converge weakly to a fixed point.

Algorithms are applied to distributed gradient descent and linear algebra.

Numerical examples validate theoretical results.

Abstract

This paper investigates the problem of finding a fixed point for a global nonexpansive operator under time-varying communication graphs in real Hilbert spaces, where the global operator is separable and composed of an aggregate sum of local nonexpansive operators. Each local operator is only privately accessible to each agent, and all agents constitute a network. To seek a fixed point of the global operator, it is indispensable for agents to exchange local information and update their solution cooperatively. To solve the problem, two algorithms are developed, called distributed Krasnosel'ski\u{\i}-Mann (D-KM) and distributed block-coordinate Krasnosel'ski\u{\i}-Mann (D-BKM) iterations, for which the D-BKM iteration is a block-coordinate version of the D-KM iteration in the sense of randomly choosing and computing only one block-coordinate of local operators at each time for each agent. It is shown that the proposed two algorithms can both converge weakly to a fixed point of the global operator. Meanwhile, the designed algorithms are applied to recover the classical distributed gradient descent (DGD) algorithm, devise a new block-coordinate DGD algorithm, handle a distributed shortest distance problem in the Hilbert space for the first time, and solve linear algebraic equations in a novel distributed approach. Finally, the theoretical results are corroborated by a few numerical examples.