Analysis of Thompson Sampling for Combinatorial Multi-armed Bandit with Probabilistically Triggered Arms

We analyze the regret of combinatorial Thompson sampling (CTS) for the combinatorial multi-armed bandit with probabilistically triggered arms under the semi-bandit feedback setting. We assume that the learner has access to an exact optimization oracle but does not know the expected base arm outcomes beforehand. When the expected reward function is Lipschitz continuous in the expected base arm outcomes, we derive $O(\sum_{i =1}^m \log T / (p_i \Delta_i))$regretboundforCTS,where$m$denotesthenumberofbasearms,$p_i$denotestheminimumnon-zerotriggeringprobabilityofbasearm$i$and$\Delta_i$denotestheminimumsuboptimalitygapofbasearm$i$. We also compare CTS with combinatorial upper confidence bound (CUCB) via numerical experiments on a cascading bandit problem.