Convergence of a Distributed Least Squares

TL;DR

This paper extends the theoretical understanding of distributed least squares algorithms in sensor networks, demonstrating convergence and regret bounds without requiring traditional assumptions like independence or Gaussianity, even under weak excitation conditions.

Contribution

It generalizes classical LS results to distributed settings, establishing convergence and regret bounds without independence or Gaussian assumptions, and introduces the weakest excitation condition for network-wide estimation.

Findings

Convergence of distributed LS estimator proven under weak excitation.

No need for independence, stationarity, or Gaussian assumptions.

Distributed estimation effective even when individual sensors cannot estimate alone.

Abstract

In this paper, we consider a least-squares (LS)-based distributed algorithm build on a sensor network to estimate an unknown parameter vector of a dynamical system, where each sensor in the network has partial information only but is allowed to communicate with its neighbors. Our main task is to generalize the well-known theoretical results on the traditional LS to the current distributed case by establishing both the upper bound of the accumulated regrets of the adaptive predictor and the convergence of the distributed LS estimator, with the following key features compared with the existing literature on distributed estimation: Firstly, our theory does not need the previously imposed independence, stationarity or Gaussian property on the system signals, and hence is applicable to stochastic systems with feedback. Secondly, the cooperative excitation condition introduced and used in this paper for the convergence of the distributed LS estimate is the weakest possible one, which shows that even if any individual sensor cannot estimate the unknown parameter by the traditional LS, the whole network can still fulfill the estimation task by the distributed LS.