Acceleration of Frank-Wolfe Algorithms with Open-Loop Step-Sizes

TL;DR

This paper investigates the convergence behavior of Frank-Wolfe algorithms when using open-loop step-sizes, revealing conditions under which they outperform traditional line-search methods and demonstrating their competitiveness with momentum-based variants.

Contribution

It provides the first theoretical analysis of open-loop step-size rules in Frank-Wolfe algorithms, explaining their faster convergence in certain scenarios like kernel herding.

Findings

FW with open-loop step-sizes converges faster in specific settings.

Open-loop FW can outperform line-search FW in kernel herding.

Open-loop FW is competitive with momentum-based variants.

Abstract

Frank-Wolfe algorithms (FW) are popular first-order methods for solving constrained convex optimization problems that rely on a linear minimization oracle instead of potentially expensive projection-like oracles. Many works have identified accelerated convergence rates under various structural assumptions on the optimization problem and for specific FW variants when using line-search or short-step, requiring feedback from the objective function. Little is known about accelerated convergence regimes when utilizing open-loop step-size rules, a.k.a. FW with pre-determined step-sizes, which are algorithmically extremely simple and stable. Not only is FW with open-loop step-size rules not always subject to the same convergence rate lower bounds as FW with line-search or short-step, but in some specific cases, such as kernel herding in infinite dimensions, it has been empirically observed that FW with open-loop step-size rules enjoys to faster convergence rates than FW with line-search or short-step. We propose a partial answer to this unexplained phenomenon in kernel herding, characterize a general setting for which FW with open-loop step-size rules converges non-asymptotically faster than with line-search or short-step, and derive several accelerated convergence results for FW with open-loop step-size rules. Finally, we demonstrate that FW with open-loop step-sizes can compete with momentum-based open-loop FW variants.