Scalable Distributed Least Squares Algorithm for Linear Algebraic Equations via Periodic Scheduling

TL;DR

This paper introduces a scalable distributed least-squares algorithm that uses periodic scheduling to efficiently solve linear algebraic equations in networks with bandwidth constraints, ensuring convergence and practical tracking.

Contribution

It presents a novel discrete-time distributed algorithm with a cyclic scheduling protocol that enhances scalability and communication efficiency for solving least-squares problems.

Findings

Agents' solutions converge exponentially under fixed step size.

The algorithm handles time-varying observations with bounded tracking error.

Simulations validate scalability and effectiveness.

Abstract

In this work, we propose a novel discrete-time distributed algorithm for finding least-squares solutions of linear algebraic equations with a scheduling protocol to further enhance its scalability. Each agent in the network is assumed to know some rows of the coefficient matrix and the corresponding entries in the observation vector. Unlike typical distributed algorithms, our approach considers communication bandwidth limits, allowing agents to transmit only a portion of their ``guessed" solution, independent of its dimension. A cyclic scheduling protocol determines which portion is transmitted at each iteration. Assuming a small fixed step size and a diagonalizable algorithm matrix, we prove that agents' ``guessed" solutions converge exponentially to a least squares solution. For cases where the observation vectors are time-varying, a modified algorithm guarantees practical convergence, with tracking error bounded by the single-step variation in the observation vector. Simulations and comparisons with state-of-the-art algorithms validate our algorithm's feasibility and scalability.