Coordinate Dual Averaging for Decentralized Online Optimization with Nonseparable Global Objectives

We consider a decentralized online convex optimization problem in a network of agents, where each agent controls only a coordinate (or a part) of the global decision vector. For such a problem, we propose two decentralized variants (ODA-C and ODA-PS) of Nesterov's primal-dual algorithm with dual averaging. In ODA-C, 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] over a static undirected graph. In ODA-PS, the agents implement the broadcast-based push-sum dynamics [2] over a time-varying sequence of uniformly connected digraphs. We show that the regret bounds in both cases have sublinear growth of O(T), with the time horizon T, when the stepsize is of the form 1t and the objective functions are Lipschitz-continuous convex functions with Lipschitz gradients. We also implement the proposed algorithms on a sensor network to complement our theoretical analysis.