We consider stochastic multi-armed bandit problems with complex actions over a set of basic arms, where the decision maker plays a complex action rather than a basic arm in each round. The reward of the complex action is some function of the basic arms' rewards, and the feedback observed may not necessarily be the reward per-arm. For instance, when the complex actions are subsets of the arms, we may only observe the maximum reward over the chosen subset. Thus, feedback across complex actions may be coupled due to the nature of the reward function. We prove a frequentist regret bound for Thompson sampling in a very general setting involving parameter, action and observation spaces and a likelihood function over them. The bound holds for discretely-supported priors over the parameter space and without additional structural properties such as closed-form posteriors, conjugate prior structure or independence across arms. The regret bound scales logarithmically with time but, more importantly, with an improved constant that non-trivially captures the coupling across complex actions due to the structure of the rewards. As applications, we derive improved regret bounds for classes of complex bandit problems involving selecting subsets of arms, including the first nontrivial regret bounds for nonlinear MAX reward feedback from subsets.

考虑了具有复杂操作的随机多臂赌博问题，其中决策者在每轮中进行复杂操作而不仅仅是基本臂。复杂操作的奖励是基本臂奖励的某些函数，并且观察到的反馈可能不是每个臂的奖励。作者在一般情况下证明了一种频率后悔上限和 Thompson 抽样算法的 相容性，并应用于包括选择臂子集的一类复杂赌博问题中。

Thompson采样用于复杂的赌博机问题