  In this paper, we consider an online optimization process, where the
objective functions are not convex (nor concave) but instead belong to a broad
class of continuous submodular functions. We first propose a variant of the
Frank-Wolfe algorithm that has access to the full gradient of the objective
functions. We show that it achieves a regret bound of $O(\sqrt{T})$ (where $T$
is the horizon of the online optimization problem) against a
$(1-1/e)$-approximation to the best feasible solution in hindsight. However, in
many scenarios, only an unbiased estimate of the gradients are available. For
such settings, we then propose an online stochastic gradient ascent algorithm
that also achieves a regret bound of $O(\sqrt{T})$ regret, albeit against a
weaker $1/2$-approximation to the best feasible solution in hindsight. We also
generalize our results to $\gamma$-weakly submodular functions and prove the
same sublinear regret bounds. Finally, we demonstrate the efficiency of our
algorithms on a few problem instances, including non-convex/non-concave
quadratic programs, multilinear extensions of submodular set functions, and
D-optimal design.
