Individual Regret Bounds for the Distributed Online Alternating Direction Method of Multipliers

We consider a distributed online optimization problem where, at each time, a group of agents choose their individual states, after which an individual cost function is revealed to each of them. The whole network then faces a regret according to the cumulative sum of costs incurred by the agents' chosen states, and each agent faces an individual regret according to the cumulative sum of costs incurred by the agent's state estimation, perceived as the whole network's chosen state. In order to tackle the minimization of the individual regret using only local information, we assume that the group of agents communicate over a fixed undirected connected graph. We then propose an online version of the alternating direction method of multipliers algorithm, distributed over the communication graph, which allows each agent to drive its individual average regret over time to zero.