  Distributed averaging, or distributed average consensus, is a common method
for computing the sample mean of the data dispersed among the nodes of a
network in a decentralized manner. By iteratively exchanging messages with
neighbors, the nodes of the network can converge to an agreement on the sample
mean of their initial states. In real-world scenarios, these messages are
subject to bandwidth and power constraints, which motivates the design of a
lossy compression strategy. Few prior works consider the rate allocation
problem from the perspective of constrained optimization, which provides a
principled method for the design of lossy compression schemes, allows for the
relaxation of certain assumptions, and offers performance guarantees. We show
for Gaussian-distributed initial states with entropy-coded scalar quantization
and vector quantization that the coding rates for distributed averaging can be
optimized through generalized geometric programming. In the absence of side
information from past states, this approach finds a rate allocation over nodes
and iterations that minimizes the aggregate coding rate required to achieve a
target mean square error within a finite run time. Our rate allocation is
compared to some of the prior art through numerical simulations. The results
motivate the incorporation of side-information through differential or
predictive coding to improve rate-distortion performance.
