Distributed lossy averaging

An information theoretic formulation of distributed averaging is presented. We assume a network with m nodes each observing an i.i.d. 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 2-node network with correlated Gaussian sources is established. A general cutset lower bound on R* (D) with independent Gaussian sources is established and shown to be achievable to within a factor of 2 via a centralized protocol. A lower bound on the network rate distortion function for distributed weighted-sum protocols that is larger than the cutset bound by a factor of log m is established. An upper bound on the expected network rate distortion function for gossip-based weighted-sum protocols that is only a factor of log log m larger than this lower bound is established. The results suggest that using distributed protocols results in a factor of log m increase in communication relative to centralized protocols.