  In this paper, we study fundamental problems of maximizing DR-submodular
continuous functions that have real-world applications in the domain of machine
learning, economics, operations research and communication systems. It captures
a subclass of non-convex optimization that provides both theoretical and
practical guarantees. Here, we focus on minimizing regret for online arriving
non-monotone DR-submodular functions over different types of convex sets:
hypercube, down-closed and general convex sets.
  First, we present an online algorithm that achieves a $1/e$-approximation
ratio with the regret of $O(T^{2/3})$ for maximizing DR-submodular functions
over any down-closed convex set. Note that, the approximation ratio of $1/e$
matches the best-known guarantee for the offline version of the problem.
Moreover, when the convex set is the hypercube, we propose a tight
1/2-approximation algorithm with regret bound of $O(\sqrt{T})$. Next, we give
an online algorithm that achieves an approximation guarantee (depending on the
search space) for the problem of maximizing non-monotone continuous
DR-submodular functions over a \emph{general} convex set (not necessarily
down-closed). To best of our knowledge, no prior algorithm with approximation
guarantee was known for non-monotone DR-submodular maximization in the online
setting. Finally we run experiments to verify the performance of our algorithms
on problems arising in machine learning domain with the real-world datasets.
