Accuracy Analysis for Distributed Weighted Least-Squares Estimation in Finite Steps and Loopy Networks

TL;DR

This paper provides a theoretical analysis of the accuracy of a distributed weighted least-squares estimation algorithm in cyclic networks, showing exponential accuracy improvement with loop-free depth and faster convergence than iterative matrix inversion methods.

Contribution

It offers bounds on estimation error and covariance for cyclic graphs, demonstrating the algorithm's high accuracy and rapid convergence in such networks.

Findings

Accuracy improves exponentially with loop-free depth.

The algorithm converges faster than iterative matrix inversion methods.

It may be preferable for cyclic networks depending on loop-free depth.

Abstract

Distributed parameter estimation for large-scale systems is an active research problem. The goal is to derive a distributed algorithm in which each agent obtains a local estimate of its own subset of the global parameter vector, based on local measurements as well as information received from its neighbours. A recent algorithm has been proposed, which yields the optimal solution (i.e., the one that would be obtained using a centralized method) in finite time, provided the communication network forms an acyclic graph. If instead, the graph is cyclic, the only available alternative algorithm, which is based on iterative matrix inversion, achieving the optimal solution, does so asymptotically. However, it is also known that, in the cyclic case, the algorithm designed for acyclic graphs produces a solution which, although non optimal, is highly accurate. In this paper we do a theoretical study of the accuracy of this algorithm, in communication networks forming cyclic graphs. To this end, we provide bounds for the sub-optimality of the estimation error and the estimation error covariance, for a class of systems whose topological sparsity and signal-to-noise ratio satisfy certain condition. Our results show that, at each node, the accuracy improves exponentially with the so-called loop-free depth. Also, although the algorithm no longer converges in finite time in the case of cyclic graphs, simulation results show that the convergence is significantly faster than that of methods based on iterative matrix inversion. Our results suggest that, depending on the loop-free depth, the studied algorithm may be the preferred option even in applications with cyclic communication graphs.