Distributed Lossy Averaging

In this paper, an information theoretic formulation of the distributed averaging problem previously studied in computer science and control is presented. We assume a network with m nodes each observing a white Gaussian noise (WGN) source. The nodes communicate and perform local processing with the goal of computing the average of the sources to within a prescribed mean squared error distortion. The network rate distortion function R* (D) for a two-node network with correlated Gaussian sources is established. A general cutset lower bound on R*(D) is established and shown to be achievable to within a factor of 2 via a centralized protocol over a star network. A lower bound on the network rate distortion function for distributed weighted-sum protocols, which is larger in order than the cutset bound by a factor of log m, is established. An upper bound on the network rate distortion function for gossip-base weighted-sum protocols, which is only log log m larger in order than the lower bound for a complete graph network, is established. The results suggest that using distributed protocols results in a factor of log m increase in order relative to centralized protocols.