Distributed Alignment Processes With Samples of Group Average

This article studies a stochastic alignment problem assuming that agents can sense the general tendency of the system. More specifically, we consider $n$ agents, each being associated with a real number. In each round, each agent receives a noisy measurement of the system's average value and then updates its value. This value is then perturbed by random drift. We assume that both noise and drift are Gaussian. We prove that a distributed weighted-average algorithm optimally minimizes the deviation of each agent from the average value, and for every 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 one. We find this result surprising since it can be shown that the set of measurements obtained by all agents contains strictly more information about the deviation of Agent $i$ from the average value, than the information contained in the measurements obtained by Agent $i$ alone. Although this information is relevant for Agent $i$, it is not processed by it when running a weighted-average algorithm. Finally, we also analyze the drift of the center of mass and show that no distributed algorithm can achieve drift that is as small as the one that can be achieved by the best centralized algorithm.