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.

翻译：在本文中, 我们考虑了一个在线优化问题, 超过$T 圆圈 。 在这样的假设下, 每个步骤 $[T], 算法从固定的 convex 和压缩域选择一个动作 $x_ t$, 设置为$\ mathcal{K} 美元。 一个工具函数 $f_ t (cdot), 然后算法得到 $f_ t (x_ t) 的支付 。 这个问题在以下假设下已经研究 : 公共事业是对抗性地选择的 单调 DR- subdal 函数和 $\ mathal{O} (sqqrt{t{t} 从固定的 固定域域域中选择一个动作 $xxx_ t$_ axxx=modal_ a modral_ demodal_ tamodral_ a model.