Online Distributed Optimization With Nonconvex Objective Functions Under Unbalanced Digraphs: Dynamic Regret Analysis

In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents. Each agent's objective function is time-varying and nonconvex and the agents exchange information with their neighbors through a time-varying directed graph. Particularly, here the graph is not assumed to be balanced. To handle this problem, an online distributed algorithm is proposed based on the projection-free strategy and the push-sum protocol. The algorithm's performance is measured using dynamic regrets whose offline benchmark is to find a stationary point of the objective function at each time. Under mild assumptions on the graph and objective functions, we prove that if the deviation of the objective function sequence is sublinear with the square root of the time horizon, and the deviation of the objective function gradient sequence is sublinear with the time horizon, then the dynamic regret increases sublinearly. Finally, simulation experiments are presented to verify the effectiveness of the theoretical results.