Distributed Momentum-Based Frank-Wolfe Algorithm for Stochastic Optimization

Abstract

This paper considers distributed stochastic optimization, in which a number of agents cooperate to optimize a global objective function through local computations and information exchanges with neighbors over a network. Stochastic optimization problems are usually tackled by variants of projected stochastic gradient descent. However, projecting a point onto a feasible set is often expensive. The Frank-Wolfe (FW) method has well-documented merits in handling convex constraints, but existing stochastic FW algorithms are basically developed for centralized settings. In this context, the present work puts forth a distributed stochastic Frank-Wolfe solver, by judiciously combining Nesterov's momentum and gradient tracking techniques for stochastic convex and nonconvex optimization over networks. It is shown that the convergence rate of the proposed algorithm is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(k^{-\frac{1}{2}})$</tex> for convex optimization, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathcal{O}(1/\log_{2}(k))$</tex> for nonconvex optimization. The efficacy of the algorithm is demonstrated by numerical simulations against a number of competing alternatives.