Stochastic Approximate Consensus with Samples of Group Average
Résumé
Reaching agreement despite noise in communication is a fundamental problem in multi-agent systems. Here we study this problem under an idealized model, where it is assumed that agents can sense the general tendency in the system. More specifically, we consider~$n$ agents, each being associated with a real-valued number. In each round, each agent receives a noisy measurement of the average value, and then updates its value, which is in turn perturbed by random drift. We assume that both noises in measurements and drift follow Gaussian distributions. What should be the updating policy of agents if their goal is to minimize the expected deviation of the agents' values from the average value? We prove that a distributed weighted-average algorithm optimally minimizes this deviation for each agent, and for any round. Interestingly, this optimality holds even in the centralized setting, where a master agent can gather all the agents' measurements and instruct a move to each agent.
We find this result surprising since it can be shown that the total measurements obtained by all agents contain strictly more information about the deviation of Agent~$i$ from the average value, than the information contained in the measurements obtained by Agent~$i$. Although this information is relevant for Agent~$i$, it is not processed by it when running a weighted-average algorithm. Nevertheless, the weighted-average algorithm is optimal, since by running it, other agents manage to fully process this information in a way that perfectly benefits Agent~$i$.
Finally, we also analyze the drift of the center of mass and show that no distributed algorithm can achieve as small of a drift as can be achieved by the best centralized algorithm. In light of this, we also show that the drift associated with our weighted-average algorithm incurs a small overhead over the best possible drift in the centralized setting.
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