A saddle point algorithm for networked online convex optimization

This paper considers an online convex optimization problem in a distributed setting, where a connected network collectively solves a learning problem while only exchanging information between neighboring nodes. We formulate two expressions to describe distributed regret and present a variant of the Arrow-Hurwicz saddle point algorithm to solve the distributed regret minimization problem. Using Lagrange multipliers to penalize the discrepancy between them, only neighboring nodes exchange decision values and Lagrange multipliers. We show that decisions made with this saddle point algorithm lead to vanishing regret of the order of O(1/√T) where T is the final iteration time, and further depends on the smoothness of the cost functions and the size and connectivity of the network. Using a recursive least squares example, we find that the numerical results corroborate our theoretical findings.