Distributed Momentum-based Frank-Wolfe Algorithm for Stochastic Optimization

TL;DR

This paper introduces a distributed stochastic Frank-Wolfe algorithm that combines momentum and gradient tracking, effectively handling convex and nonconvex optimization over networks without expensive projections.

Contribution

It develops a novel distributed stochastic Frank-Wolfe method with convergence guarantees for both convex and nonconvex problems, leveraging momentum and gradient tracking techniques.

Findings

Convergence rate of O(k^{-1/2}) for convex optimization.

Convergence rate of O(1/log_2(k)) for nonconvex optimization.

Numerical simulations demonstrate superior performance over competing methods.

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 $\mathcal{O}(k^{-\frac{1}{2}})$ for convex optimization, and $\mathcal{O}(1/\mathrm{log}_2(k))$ for nonconvex optimization. The efficacy of the algorithm is demonstrated by numerical simulations against a number of competing alternatives.