Distributed online second-order dynamics for convex optimization over switching connected graphs

This paper studies the regret of a family of distributed algorithms for online convex unconstrained optimization. A team of agents cooperate in a decision making process enabled by local interactions and the knowledge of each agent about the local cost functions associated with its decisions in previous rounds. We propose a class of online, second-order distributed coordination algorithms that combine subgradient descent on the local objectives revealed in the previous round and proportional-integral feedback on the disagreement among neighboring agents. The communication network is given by a time-varying sequence of connected graphs, and the local objectives can be adversarially adaptive to the agents’ behavior. The goal of each agent is to incur a cumulative cost over time with respect to the sum of local objectives across the network that is competitive with the best fixed and centralized decision in hindsight. For this, we establish the classical logarithmic regret bound under strong convexity of the local objectives.