A gossip method for optimal consensus on a binary state from binary actions

In this paper, we study the problem of distributed hypothesis testing in cooperative networks of agents over a given undirected graph. All the agents try to reach consensus on the state of nature based on their private signals and the binary actions of their neighbors. This is a challenging problem because the exchanged information of the agents regarding their observations used for making decisions is highly compressed. We propose a set of gossip-type methods for which two communicating agents reach the optimal local consensus with probability one by a few exchanges of binary actions at every time slot. We prove that the decision of each agent converges in probability to the optimal decision held by a fictitious fusion center. We also provide theoretical results on how the edge selection probability effects the expected time at which a consensus of all the agents is reached. Simulation results that demonstrate the communication cost and the convergence time of the method are provided.