Variance-reduced and projection-free stochastic optimization

Abstract

The Frank-Wolfe optimization algorithm has recently regained popularity for machine learning applications due to its projection-free property and its ability to handle structured constraints. However, in the stochastic learning setting, it is still relatively understudied compared to the gradient descent counterpart. In this work, leveraging a recent variance reduction technique, we propose two stochastic Frank-Wolfe variants which substantially improve previous results in terms of the number of stochastic gradient evaluations needed to achieve 1 - e accuracy. For example, we improve from O(1/ϵ) to O(ln1/ϵ) if the objective function is smooth and strongly convex, and from 0(1/ϵ2) to O(1/ϵ15) if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a mulliclass classification application.