Distributed Linear Equations Over Random Networks

Abstract

Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts, or node recreation, etc. In this article, we study the convergence and convergence rate of distributed linear equation protocols over a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ast$</tex-math></inline-formula> -mixing random network, where the temporal and spatial dependencies between the node-to-node communications are allowed. When the network linear equation admits exact solutions, we prove the exponential convergence rate of the distributed projection consensus algorithm in the mean-squared sense. Motivated by the randomized Kaczmarz algorithm, we also propose a distributed randomized projection consensus algorithm, where each node randomly selects one row of local linear equations for projection per iteration, and establish an exponential rate of convergence. When the network linear equation admits no exact solution, we prove that a distributed gradient-descent-like algorithm with diminishing step-sizes can drive all nodes’ states to a least-squares solution at a sublinear rate. These results collectively illustrate that distributed computations may overcome communication correlations if the prototype algorithms enjoy certain contractive properties or are designed with suitable parameters.