Distributed Zeroth-Order Gradient Tracking for Weakly Convex Optimization Over Unbalanced Graphs

Distributed weakly convex optimization is a significant class of problems in signal and information processing, with wide-ranging applications such as sparse dictionary learning, low-rank matrix completion, and robust phase retrieval. Most existing distributed algorithms for solving this type of problem are designed based on exact gradient information. However, it is challenging to obtain this information as closed-form analytical expressions are often unavailable in certain circumstances. In this article, we propose a gradient estimation scheme for distributed weakly convex optimization problems, estimating the gradient information using finite differences in orthogonal random directions. This approach is more general and has better estimation effectiveness than existing methods based on stochastic vectors. Furthermore, we design a projected zeroth-order gradient tracking algorithm, which effectively solves the considered problem over an unbalanced communication topology. We also demonstrate that the proposed algorithm converges to a stationary point with a rate of (Formula presented) from the perspective of the Moreau envelope. Finally, we provide two examples to verify the effectiveness of our algorithm.