Zeroth-Order Methods for Online Distributed Optimization with Strongly Pseudoconvex Cost Functions

Abstract

This paper studies an online distributed optimization problem over multi-agent systems. In this problem, the goal of agents is to cooperatively minimize the sum of locally dynamic cost functions. Different from most existing works on distributed optimization, here we consider the case where the cost function is strongly pseudoconvex and real gradients of objective functions are not available. To handle this problem, an online zeroth-order stochastic optimization algorithm involving the single-point gradient estimator is proposed. Under the algorithm, each agent only has access to the information associated with its own cost function and the estimate of the gradient, and exchange local state information with its immediate neighbors via a time-varying digraph. The performance of the algorithm is measured by the expectation of dynamic regret. Under mild assumptions on graphs, we prove that if the cumulative deviation of minimizer sequence grows within a certain rate, then the expectation of dynamic regret grows sublinearly. Finally, a simulation example is given to illustrate the validity of our results.