Distributed Least Mean Square Estimation With Communication Noises Over Random Graphs

For the online distributed estimation problem of time-varying parameters, we study a linear regression model with measurement noises over time-varying random graphs. We propose a distributed normalized least mean square (LMS) algorithm, where each node updates its own estimate by the least mean square term, and sums the differences between its own estimate and the estimates of its neighbors with additive and multiplicative communication noises by the consensus term. By the algebraic graph theory and the stochastic analysis techniques, we obtain sufficient conditions for the boundedness of the tracking error. For a sequence of general random graphs, if the random graphs and the regression matrices satisfy the stochastic spatio-temporal persistence of excitation condition, then the mean-square tracking error is bounded by choosing appropriate constant gains. Furthermore, for conditional balanced graphs and Markovian switching graphs, we give sufficient conditions such that the persistence of excitation condition holds. Finally, we illustrate the effectiveness of the theoretical results through a numerical example.