Projection-free Online Learning over Strongly Convex Sets
This work addresses the performance gap between projection-free and projection-based online learning algorithms for researchers in optimization and machine learning, offering incremental improvements in regret bounds for specific constraint sets.
The paper tackles the problem of projection-free online learning over strongly convex sets, showing that a modified Online Frank-Wolfe algorithm achieves a regret bound of O(T^{2/3}) for general convex losses and O(√T) for strongly convex losses over strongly convex sets.
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.