  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 a triggering
probability modulated (TPM) bounded smoothness condition into the general
CMAB-T framework, and show that many applications such as influence
maximization bandit and combinatorial cascading bandit satisfy this TPM
condition. 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 the TPM condition is crucial in removing this factor.
