Distributed Stochastic Frank-Wolfe for Constrained Composite Minimization

This article considers distributed composite minimization with set constraints and nonsmooth terms represented by indicator functions. In this setup, projecting onto set constraints is difficult, while indicator functions admit efficient proximal operations. Problems of this form frequently arise in the context of semidefinite programming (SDP) and its applications, such as clustering and kernel learning. However, existing distributed algorithms for composite minimization are designed exclusively for the unconstrained setting. One straightforward approach to handle set constraints involves employing projection operations, which are computationally prohibitive for some set constraints. Hence, it is essential to develop a projection-free scheme to circumvent projection operations over set constraints, such as the Frank-Wolfe (FW) algorithm. This article develops a novel distributed stochastic FW algorithm for constrained composite minimization. By combining the recursive momentum, gradient tracking and Nesterov smoothing techniques, the proposed distributed algorithm achieves comparable convergence results and complexity guarantees to centralized stochastic algorithms. Numerical studies are provided to demonstrate the efficacy of our theoretical findings.