Boosting Frank-Wolfe by Chasing Gradients

TL;DR

This paper introduces an enhanced Frank-Wolfe algorithm that accelerates convergence by better aligning descent directions with the negative gradient, achieving faster rates and improved computational efficiency.

Contribution

The paper proposes a novel subroutine that chases the negative gradient to speed up Frank-Wolfe while maintaining its projection-free nature.

Findings

Convergence rate improved from O(1/t) to exponential decay.

Method outperforms state-of-the-art in CPU time and iteration efficiency.

Significant practical speedups demonstrated in experiments.

Abstract

The Frank-Wolfe algorithm has become a popular first-order optimization algorithm for it is simple and projection-free, and it has been successfully applied to a variety of real-world problems. Its main drawback however lies in its convergence rate, which can be excessively slow due to naive descent directions. We propose to speed up the Frank-Wolfe algorithm by better aligning the descent direction with that of the negative gradient via a subroutine. This subroutine chases the negative gradient direction in a matching pursuit-style while still preserving the projection-free property. Although the approach is reasonably natural, it produces very significant results. We derive convergence rates $\mathcal{O}(1/t)$ to $\mathcal{O}(e^{-\omega t})$ of our method and we demonstrate its competitive advantage both per iteration and in CPU time over the state-of-the-art in a series of computational experiments.