Variance-Reduced and Projection-Free Stochastic Optimization

TL;DR

This paper introduces variance-reduced, projection-free stochastic optimization algorithms based on Frank-Wolfe, significantly reducing the number of gradient evaluations needed for high-accuracy solutions in machine learning tasks.

Contribution

It proposes two novel stochastic Frank-Wolfe variants utilizing variance reduction, achieving improved theoretical convergence rates over previous methods.

Findings

Reduced gradient evaluations from O(1/ε) to O(ln(1/ε)) for strongly convex functions.

Reduced evaluations from O(1/ε²) to O(1/ε^{1.5}) for Lipschitz functions.

Experimental validation on real datasets confirms theoretical improvements.

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 $1-\epsilon$ accuracy. For example, we improve from $O(\frac{1}{\epsilon})$ to $O(\ln\frac{1}{\epsilon})$ if the objective function is smooth and strongly convex, and from $O(\frac{1}{\epsilon^2})$ to $O(\frac{1}{\epsilon^{1.5}})$ if the objective function is smooth and Lipschitz. The theoretical improvement is also observed in experiments on real-world datasets for a multiclass classification application.