Decentralized online optimization with global objectives and local communication

We consider a decentralized online convex optimization problem in a static undirected network of agents, where each agent controls only a coordinate (or a part) of the global decision vector. For such a problem, we propose a decentralized variant of Nesterov's primal-dual algorithm with dual averaging. To mitigate the disagreements on the primal-vector updates, the agents implement a generalization of the local information-exchange dynamics recently proposed by Li and Marden [1]. We show that the regret has sublinear growth of O (√T) with the time horizon T when the stepsize is of the form 1/√t and the objective functions are Lipschitzcontinuous convex functions with Lipschitz gradients. We prove an analogous bound on the expected regret for the stochastic variant of the algorithm.