  In this paper, we consider an online optimization problem over $T$ rounds
where at each step $t\in[T]$, the algorithm chooses an action $x_t$ from the
fixed convex and compact domain set $\mathcal{K}$. A utility function
$f_t(\cdot)$ is then revealed and the algorithm receives the payoff $f_t(x_t)$.
This problem has been previously studied under the assumption that the
utilities are adversarially chosen monotone DR-submodular functions and
$\mathcal{O}(\sqrt{T})$ regret bounds have been derived. We first characterize
the class of strongly DR-submodular functions and then, we derive regret bounds
for the following new online settings: $(1)$ $\{f_t\}_{t=1}^T$ are monotone
strongly DR-submodular and chosen adversarially, $(2)$ $\{f_t\}_{t=1}^T$ are
monotone submodular (while the average $\frac{1}{T}\sum_{t=1}^T f_t$ is
strongly DR-submodular) and chosen by an adversary but they arrive in a
uniformly random order, $(3)$ $\{f_t\}_{t=1}^T$ are drawn i.i.d. from some
unknown distribution $f_t\sim \mathcal{D}$ where the expected function
$f(\cdot)=\mathbb{E}_{f_t\sim\mathcal{D}}[f_t(\cdot)]$ is monotone
DR-submodular. For $(1)$, we obtain the first logarithmic regret bounds. In
terms of the second framework, we show that it is possible to obtain similar
logarithmic bounds with high probability. Finally, for the i.i.d. model, we
provide algorithms with $\tilde{\mathcal{O}}(\sqrt{T})$ stochastic regret
bound, both in expectation and with high probability. Experimental results
demonstrate that our algorithms outperform the previous techniques in the
aforementioned three settings.
