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 <inline-formula><tex-math notation="LaTeX">$O(\sqrt{T})$</tex-math></inline-formula>, with the time horizon <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula>, when the stepsize is of the form <inline-formula><tex-math notation="LaTeX">$1/\sqrt{t}$</tex-math></inline-formula> 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.