Decentralized gradient algorithm for solution of a linear equation

TL;DR

This paper introduces a decentralized gradient algorithm enabling multiple agents connected by a graph to collaboratively solve linear equations efficiently, with proven exponential convergence and extensions to non-square matrices.

Contribution

It presents a novel decentralized gradient method for solving linear equations with convergence guarantees, applicable to arbitrary connected graphs and extended to full row rank matrices.

Findings

Agents' estimates converge exponentially to the solution.

The algorithm works for arbitrary connected graph structures.

Convergence rate bounds are established.

Abstract

The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. The equation for $x_i$ includes additive terms weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the $i$-th and $j$-th agents are adjacent. The results are extended to the case where $A$ is not square but has full row rank, and bounds are given on the convergence rate.