Projection-free Online Learning over Strongly Convex Sets

TL;DR

This paper improves the regret bounds for projection-free online learning algorithms over strongly convex sets, achieving better performance than previous methods by refining step-size strategies and introducing a strongly convex variant.

Contribution

It introduces a refined step-size rule for OFW and a strongly convex variant, leading to improved regret bounds over general and strongly convex sets.

Findings

OFW achieves $O(T^{2/3})$ regret over convex sets.

Strongly convex variant of OFW achieves $O( oot T)$ regret over strongly convex sets.

Enhanced algorithms outperform previous projection-free methods in regret bounds.

Abstract

To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of $O(T^{3/4})$, which is worse than the regret of projection-based algorithms, where $T$ is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of $O(T^{2/3})$ for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of $O(T^{2/3})$ over general convex sets and a better regret bound of $O(\sqrt{T})$ over strongly convex sets.