  To efficiently solve distributed online learning problems with complicated
constraints, previous studies have proposed several distributed projection-free
algorithms. The state-of-the-art one achieves the $O({T}^{3/4})$ regret bound
with $O(\sqrt{T})$ communication complexity. In this paper, we further exploit
the strong convexity of loss functions to improve the regret bound and
communication complexity. Specifically, we first propose a distributed
projection-free algorithm for strongly convex loss functions, which enjoys a
better regret bound of $O(T^{2/3}\log T)$ with smaller communication complexity
of $O(T^{1/3})$. Furthermore, we demonstrate that the regret of distributed
online algorithms with $C$ communication rounds has a lower bound of
$\Omega(T/C)$, even when the loss functions are strongly convex. This lower
bound implies that the $O(T^{1/3})$ communication complexity of our algorithm
is nearly optimal for obtaining the $O(T^{2/3}\log T)$ regret bound up to
polylogarithmic factors. Finally, we extend our algorithm into the bandit
setting and obtain similar theoretical guarantees.
