  In this paper, we revisit the online non-monotone continuous DR-submodular
maximization problem over a down-closed convex set, which finds wide real-world
applications in the domain of machine learning, economics, and operations
research. At first, we present the Meta-MFW algorithm achieving a $1/e$-regret
of $O(\sqrt{T})$ at the cost of $T^{3/2}$ stochastic gradient evaluations per
round. As far as we know, Meta-MFW is the first algorithm to obtain
$1/e$-regret of $O(\sqrt{T})$ for the online non-monotone continuous
DR-submodular maximization problem over a down-closed convex set. Furthermore,
in sharp contrast with ODC algorithm \citep{thang2021online}, Meta-MFW relies
on the simple online linear oracle without discretization, lifting, or rounding
operations. Considering the practical restrictions, we then propose the
Mono-MFW algorithm, which reduces the per-function stochastic gradient
evaluations from $T^{3/2}$ to 1 and achieves a $1/e$-regret bound of
$O(T^{4/5})$. Next, we extend Mono-MFW to the bandit setting and propose the
Bandit-MFW algorithm which attains a $1/e$-regret bound of $O(T^{8/9})$. To the
best of our knowledge, Mono-MFW and Bandit-MFW are the first sublinear-regret
algorithms to explore the one-shot and bandit setting for online non-monotone
continuous DR-submodular maximization problem over a down-closed convex set,
respectively. Finally, we conduct numerical experiments on both synthetic and
real-world datasets to verify the effectiveness of our methods.
