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

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'skiȋ-Mann (D-KM) and distributed block-coordinate Krasnosel'skiȋ-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.

DOT and DOP: Linearly Convergent Algorithms for Finding Fixed Points of Multi-Agent Operators

“…Moreover, no convergence speeds are provided in [16]- [18]. In addition, the same problem as here is also studied in [21] for nonexpansive operators, where the convergence rate is not analyzed, while a linear convergence rate is established here and more general operators are considered, i.e., quasi-nonexpansive operators. Remark 4.…”

“…In more recent years, distributed algorithms have received a growing attention in the fixed point finding problem [16]- [21]. For instance, distributed algorithms were developed in [16], [17] to find a common fixed point for a finite group of paracontraction operators.…”

“…Regarding the Hilbert space, the authors in [19] investigated distributed optimization under random and directed interconnection graphs, and the authors in [20] took into account the common fixed point finding problem for a finite collection of nonexpansive operators, where two distributed algorithms were proposed with a full coordinate updating and a random block-coordinate updating, respectively. Furthermore, a more general scenario, where no common fixed points are assumed for all operators, was investigated in [21], where two distributed algorithms were devised to resolve the problem. Nevertheless, the convergence rate is not analyzed for the proposed algorithms in [21].…”

“…Furthermore, a more general scenario, where no common fixed points are assumed for all operators, was investigated in [21], where two distributed algorithms were devised to resolve the problem. Nevertheless, the convergence rate is not analyzed for the proposed algorithms in [21].…”

“…Compared with the closely related work [21], where no convergence speed is provided, a different algorithm from that in [21] is developed here and shown to be convergent at a linear rate. It should be noted that the problem here is more general than the common fixed point seeking problem [16]- [18], [20], where all local operators are assumed to have at least one common fixed point, while this assumption is dropped here.…”