Linear convergence of zeroth-order algorithm for distributed weakly convex optimization with sharpness property

This paper establishes a linear convergence rate for a distributed zeroth-order algorithm in weakly convex optimization over a time-varying graph. We utilize a more general gradient/subgradient estimation scheme with orthogonal directions to estimate gradient/subgradient information in the proposed algorithm, which is more effective than the conventional methods based on stochastic vectors. Furthermore, by utilizing the geometrically diminishing step size and the difference factor, we demonstrate that the proposed zeroth-order algorithm linearly converges to the optimal solution. Finally, we provide a numerical example to verify the correctness of our theoretical findings and illustrate the effectiveness of the proposed algorithm.