On the Convergence Rate of Average Consensus and Distributed Optimization over Unreliable Networks

We consider the problems of reaching average consensus and solving consensus-based optimization over unreliable communication networks wherein packets may be dropped accidentally during transmission. Existing work either assumes that the link failures affect the communication on both directions or that the message senders {\em know exactly}, in each iteration, how many of their outgoing links are functioning properly. In this paper, we consider directed links, and we {\em do not} require each node know its current outgoing degree. First, we propose and characterize the convergence rate of reaching average consensus. Then we apply our robust consensus update to the classical distributed dual averaging method wherein the consensus update is used as the information aggregation primitive. We show that the local iterates converge to a common optimum of the global objective at rate $O(\frac{1}{\sqrt{t}})$, where $t$ is the number of iterations, matching the failure-free performance of the distributed dual averaging method.