Asymptotic Efficiency of Distributed Random Sampling Algorithm

In this paper, we focus on estimating the distribution of underlying parameter over random networks through reconstructing the empirical distribution of initial samples, which can be viewed as a particular average consensus problem. A class of quantized communication protocol, in which neighbors exchange information randomly selected based on the current estimate, is considered. To improve the convergence rate of this distributed random sampling algorithm, we introduce the Polyak average scheme, and show that asymptotic efficiency can be achieved through the averaging technique under proper conditions. The results show that the minimum limit covariance matrix of estimation error can be reached, i.e., the proposed algorithm achieves the highest possible rate of convergence. Finally, we provide a numerical simulation to validate the theoretical results of this work.