Distributed adaptive estimation for stochastic large regression models

TL;DR

This paper introduces a distributed recursive least squares algorithm for stochastic large regression models with infinitely many parameters, analyzing convergence and prediction error without independence assumptions.

Contribution

It proposes a novel distributed recursive least squares method for infinite-dimensional regression models, incorporating cooperative excitation and advanced stochastic analysis techniques.

Findings

Almost sure convergence under cooperative excitation condition

Asymptotic upper bound of accumulated regret established

Handles correlated feedback signals without independence assumptions

Abstract

This paper studies the distributed adaptiveestimation problems for stochastic large regression modelswith an infinite number of parameters. By constructing a re-cursive local cost function, we propose a novel distributedrecursive least squares algorithm to estimate the unknownsystem parameters, where the growth rate of regressors'dimension is characterized by a non-decreasing positivefunction. The almost sure convergence of the proposedalgorithm is established under a cooperative excitationcondition, which incorporates the temporal information andthe spatial information to reflect the cooperative effectamong multiple agents. Moreover, we analyze the predic-tion error by establishing the asymptotic upper boundof the accumulated regret without any excitation condi-tions. The main difficulty of theoretical analysis lies in howto analyze properties of the product of non-independentand non-stationary random matrices, whose dimensionschange over time simultaneously. Some techniques, suchas stochastic Lyapunov function, double-array martingaletheory and algebraic graph theory, are employed to dealwith the above issue. Our theoretical results are derivedwithout imposing independence or stationarity assump-tions on the regression vectors, thereby not excluding thecorrelated feedback signals.