An Arrow-Hurwicz-Uzawa Type Flow as Least Squares Solver for Network Linear Equations

TL;DR

This paper introduces a distributed continuous-time and discrete-time algorithm based on Arrow-Hurwicz-Uzawa flow for solving least squares linear equations over networks, with proven exponential convergence and robustness to switching networks.

Contribution

It develops a novel distributed least squares solver using Arrow-Hurwicz-Uzawa flow with convergence guarantees and analyzes its performance over various network topologies and switching conditions.

Findings

Exponential convergence of the algorithms to the least squares solution.

Necessary and sufficient graph conditions for convergence.

Robustness of the algorithms under high-frequency switching networks.

Abstract

We study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms. Each node has access to one of the linear equations and holds a dynamic state. The aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying interaction graph. A continuous-time distributed least squares solver over networks is developed in the form of the famous Arrow-Hurwicz-Uzawa flow. A necessary and sufficient condition is established on the graph Laplacian for the continuous-time distributed algorithm to give the least squares solution in the limit, with an exponentially fast convergence rate. The feasibility of different fundamental graphs is discussed including path graph, star graph, etc. Moreover, a discrete-time distributed algorithm is developed by Euler's method, converging exponentially to the least squares solution at the node states with suitable step size and graph conditions. The exponential convergence rate for both the continuous-time and discrete-time algorithms under the established conditions is confirmed by numerical examples. Finally, we investigate the performance of the proposed flow under switching networks, and surprisingly, switching networks at high switching frequencies can lead to approximate least square solvers even if all graphs in the switching signal fail to do so in the absence of structure switching.