This paper addresses tracking of a moving target in a multi-agent network. The target follows a linear dynamics corrupted by an adversarial noise, i.e., the noise is not generated from a statistical distribution. The location of the target at each time induces a global time-varying loss function, and the global loss is a sum of local losses, each of which is associated to one agent. Agents noisy observations could be nonlinear. We formulate this problem as a distributed online optimization where agents communicate with each other to track the minimizer of the global loss. We then propose a decentralized version of the Mirror Descent algorithm and provide the non-asymptotic analysis of the problem. Using the notion of dynamic regret, we measure the performance of our algorithm versus its offline counterpart in the centralized setting. We prove that the bound on dynamic regret scales inversely in the network spectral gap, and it represents the adversarial noise causing deviation with respect to the linear dynamics. Our result subsumes a number of results in the distributed optimization literature. Finally, in a numerical experiment, we verify that our algorithm can be simply implemented for multi-agent tracking with nonlinear observations.
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