Network Flows that Solve Least Squares for Linear Equations

TL;DR

This paper introduces a distributed continuous-time algorithm for solving least-squares linear equations over networks, providing convergence guarantees and rates for fixed and switching graph topologies.

Contribution

It proposes a novel first-order distributed algorithm with proven convergence and explicit rates for both unique and non-unique solutions in networked systems.

Findings

Convergence is achieved with nonintegrable, diminishing step sizes.

Explicit convergence rates are established for various step size choices.

Numerical examples demonstrate the impact of step size on convergence speed.

Abstract

This paper presents a first-order {distributed continuous-time algorithm} for computing the least-squares solution to a linear equation over networks. Given the uniqueness of the solution, with nonintegrable and diminishing step size, convergence results are provided for fixed graphs. The exact rate of convergence is also established for various types of step size choices falling into that category. For the case where non-unique solutions exist, convergence to one such solution is proved for constantly connected switching graphs with square integrable step size, and for uniformly jointly connected switching graphs under the boundedness assumption on system states. Validation of the results and illustration of the impact of step size on the convergence speed are made using a few numerical examples.