We study combinatorial multi-armed bandit with probabilistically triggered arms (CMAB-T) and semi-bandit feedback. We resolve a serious issue in the prior CMAB-T studies where the regret bounds contain a possibly exponentially large factor of $1/p^*$, where $p^*$ is the minimum positive probability that an arm is triggered by any action. We address this issue by introducing triggering probability moderated (TPM) bounded smoothness conditions into the general CMAB-T framework, and show that many applications such as influence maximization bandit and combinatorial cascading bandit satisfy such TPM conditions. As a result, we completely remove the factor of $1/p^*$ from the regret bounds, achieving significantly better regret bounds for influence maximization and cascading bandits than before. Finally, we provide lower bound results showing that the factor $1/p^*$ is unavoidable for general CMAB-T problems, suggesting that TPM conditions are crucial in removing this factor.

该论文研究具有概率触发臂的组合多臂赌博机(CMAB-T)和半弱馈反的问题，将触发概率调制(TPM)条件引入CMAB-T框架中以达到更好的遗憾界限，并发现许多应用程序如影响最大化赌博机和组合级联赌博机满足TPM条件。此外，本文提供了下限结果，表明这种条件对于消除$1/p^*$因子是至关重要的。

概率触发手臂组合半赌博算法的遗憾上界优化及应用