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, we find 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.

本文针对情境组合赌博问题，提出了一种具有触发概率和方差自适应的算法，并给出了相应的遗憾界。在合成和真实世界数据集上的实验结果表明算法的优越性。