We study contextual combinatorial bandits with probabilistically triggered arms (C$^2$MAB-T) under a variety of smoothness conditions that capture a wide range of applications, such as contextual cascading bandits and contextual influence maximization bandits. Under the triggering probability modulated (TPM) condition, we devise the C$^2$-UCB-T algorithm and propose a novel analysis that achieves an $\tilde{O}(d\sqrt{KT})$ regret bound, removing a potentially exponentially large factor $O(1/p_{\min})$, where $d$ is the dimension of contexts, $p_{\min}$ is the minimum positive probability that any arm can be triggered, and batch-size $K$ is the maximum number of arms that can be triggered per round. Under the variance modulated (VM) or triggering probability and variance modulated (TPVM) conditions, we propose a new variance-adaptive algorithm VAC$^2$-UCB and derive a regret bound $\tilde{O}(d\sqrt{T})$, which is independent of the batch-size $K$. As a valuable by-product, our analysis technique and variance-adaptive algorithm can be applied to the CMAB-T and C$^2$MAB setting, improving existing results there as well. We also include experiments that demonstrate the improved performance of our algorithms compared with benchmark algorithms on synthetic and real-world datasets.

翻译：我们研究了基于概率触发臂的情境组合赌博机（C$^2$MAB-T）在多种平滑性条件下的问题，这些条件涵盖了广泛的应用场景，例如情境级联赌博机与情境影响最大化赌博机。在触发概率调制（TPM）条件下，我们设计了C$^2$-UCB-T算法，并提出了一种新颖的分析方法，实现了$\tilde{O}(d\sqrt{KT})$的遗憾界，去除了可能指数级大的因子$O(1/p_{\min})$，其中$d$为情境维度，$p_{\min}$为任意臂可被触发的最小正概率，批大小$K$为每轮最多可触发的臂数。在方差调制（VM）或触发概率和方差调制（TPVM）条件下，我们提出了一种新的方差自适应算法VAC$^2$-UCB，并推导出与批大小$K$无关的$\tilde{O}(d\sqrt{T})$遗憾界。作为有价值的副产品，我们的分析技术与方差自适应算法可应用于CMAB-T和C$^2$MAB设置，从而改进现有结果。我们还通过实验表明，在合成数据集和真实世界数据集上，我们的算法相比基准算法具有更优的性能。