Randomized Gradient-Free Mirror Descent Algorithm for Distributed Online Optimization

An online distributed optimization problem over a multi-agent system is concerned in this paper. Each agent can only evaluate the function value of its local objective function. A distributed zeroth-order mirror descent algorithm is proposed by adopting a two-point gradient estimator in the mirror descent scheme. Specifically, we employ gradient-free techniques to adapt the algorithm to scenarios in the absence of derivative information. The proposed algorithm utilizes a two-point gradient estimation technique, ensuring precise convergence to the optimal function value. It is proved that an average regularized regret of
$O(1/\sqrt{T})$
convergence rate is achieved under the proposed algorithm, which is the best known T-rate of gradient-free algorithms in offline settings. Finally, the effectiveness of the algorithm is validated through numerical experiments.