Distributed Linear Equations over Random Networks

TL;DR

This paper analyzes the convergence behavior of distributed algorithms for solving linear equations over random networks with stochastic communication links, providing theoretical guarantees and rates for various scenarios.

Contribution

It establishes convergence rates for distributed linear equation algorithms over -mixing random networks, including exact and approximate solutions, with novel bounds and randomized methods.

Findings

Mean-squared exponential convergence for exact solutions

Exponential convergence of a randomized projection algorithm

Sublinear convergence to least-squares solutions with diminishing step-sizes

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 paper, we study the convergence and convergence rate of distributed linear equation protocols over a $\ast$-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 mean-squared exponential convergence rate of the distributed projection consensus algorithm, while the lower and upper bound estimations of the convergence rate are also given for independent and identically distributed (i.i.d.) random graphs. 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 convergence rate. 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.