Exponential convergence of a distributed algorithm for solving linear algebraic equations

Abstract

In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form Ax = b assuming that the equation has at least one solution. The equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix [ A b ], the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on N(t) were derived under which the algorithm can cause all agents' estimates to converge exponentially fast to the same solution to Ax = b. These conditions were also shown to be necessary for exponential convergence, provided the data about [ A b ] available to the agents is "non-redundant". The aim of this paper is to relax this "non-redundant" assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

Distributed Computation of Linear Matrix Equations: An Optimization Perspective

“…• We employ different substitutional decomposition methods to reformulate the original computation matrix equations as distributed constrained optimization problems with different constraints in the standard structures. Note that the decompositions are new compared with those in the distributed computation of the linear algebraic equation of the form Ax = b in [11]- [14], [17], [18].…”

“…. , n}, t ≥ 0, X i (t) and Y i (t) are the estimates of solutions to problem (18) by (1). (19) is a primal-dual algorithm, whose primal variables are X i and Y i and dual variables are Λ 1 i , Λ 2 i , and Λ 3 i .…”

“…Specifically, [12] proposed a discrete-time distributed algorithm for a solvable linear equation and presented the necessary and sufficient conditions for exponential convergence of the algorithm, while [15] developed a continuous-time algorithm with an exponential convergence rate for a nonsingular and square A and extended the algorithm to the case where A is of full row rank with bounds on the convergence rate. Furthermore, [18] constructed a distributed algorithm and derived the necessary and sufficient conditions on a time-dependent graph for an exponential convergence rate. Additionally, [11] considered distributed computation for a June 12, 2018 DRAFT least squares solution to the linear equations that may have no exact solutions, by providing approximate least squares solutions, while [16] dealt with the problem for the least squares solutions with different graphs and appropriate step-sizes.…”

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

“…Each agent recursively updates its estimate of the solution using the current estimates from its neighbors. Recently several solutions under different sufficient conditions have been proposed [28]- [30], and in particular in [30], the sequence of the neighbor relationship graphs G(k) is required to be repeated jointly strongly connected. We show that a much weaker condition is sufficient to solve the problem almost surely, namely the algorithm in [30] works if there exists a fixed length such that any subsequence of {G(k)} at this length is jointly strongly connected with positive probability.…”

Distributed least squares solver for network linear equations

“…By exchanging their states with neighboring nodes over an underlying interaction graph, all nodes collaboratively solve the linear equations. Various distributed algorithms based on distributed control and optimization have been developed for solving the linear equations which have exact solutions, among which discrete-time algorithms are given in Mou et al (2015); Liu et al (2018Liu et al ( , 2017; Lu and Tang (2018); Wang et al (2019a) and continuous-time algorithms are presented in Anderson et al (2016); Shi et al (2017). However, most of these existing algorithms can only produce least square solutions for over-determined linear equations in the approximate sense (Mou et al, 2015) or for limited graph structures (Wang and Elia, 2012;Shi et al, 2017).…”