Scalable Distributed Least Squares Algorithm for Linear Algebraic Equations via Periodic Scheduling
Shenyu Liu
TL;DR
This work tackles scalable distributed LS solution of LAEs $A x = b$ under bandwidth constraints by proposing a continuous-time foundation (CT-D-LS) and a discrete-time scalable extension (DT-SD-LS) with cyclic scheduling that transmits only fixed-size portions of each agent's state. It proves exponential convergence to LS solutions for static problems and provides a tracking guarantee when observations vary over time, with tracking error bounded by the per-step variation in $b$. Simulations demonstrate favorable scalability and competitive performance against state-of-the-art distributed LS methods, especially as problem size grows. The approach promises practical impact for large-scale, bandwidth-limited distributed estimation and regression tasks.
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.