Stability analysis of the consensus-based distributed LMS algorithm

We deal with consensus-based online estimation and tracking of (non-) stationary signals using ad hoc wireless sensor networks (WSNs). A distributed (D-) least-mean square (LMS) like algorithm is developed, which offers simplicity and flexibility, while it solely relies on single-hop communications among sensors. Starting from a pertinent squared-error cost, we apply the alternating-direction method of multipliers to minimize it in a distributed fashion; and utilize stochastic approximation tools to eliminate the need for a complete statistical characterization of the processes of interest. By resorting to stochastic averaging and perturbed Lyapunov techniques, we further establish that local estimates are exponentially convergent to the true parameter of interest when observations are noise free and linearly related to it. This convergence result is necessary for bounding the estimation error in the presence of noise, and holds not only when regressors are white across time but even when they exhibit temporal correlations. Numerical tests confirm the merits of the novel D-LMS algorithm and its stability analysis.