#### Distributed Learning of Average Belief Over Networks Using Sequential Observations

##### Kaiqing Zhang, Yang Liu, Ji Liu, Mingyan Liu, Tamer Başar

This paper addresses the problem of distributed learning of average belief with sequential observations, in which a network of $n>1$ agents aim to reach a consensus on the average value of their beliefs, by exchanging information only with their neighbors. Each agent has sequentially arriving samples of its belief in an online manner. The neighbor relationships among the $n$ agents are described by a graph which is possibly time-varying, whose vertices correspond to agents and whose edges depict neighbor relationships. Two distributed online algorithms are introduced for undirected and directed graphs, which are both shown to converge to the average belief almost surely. Moreover, the sequences generated by both algorithms are shown to reach consensus with an $O(1/t)$ rate with high probability, where $t$ is the number of iterations. For undirected graphs, the corresponding algorithm is modified for the case with quantized communication and limited precision of the division operation. It is shown that the modified algorithm causes all $n$ agents to either reach a quantized consensus or enter a small neighborhood around the average of their beliefs. Numerical simulations are then provided to corroborate the theoretical results.

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