  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.
