Thompson Sampling for Complex Bandit Problems
This work addresses complex decision-making problems in machine learning, such as subset selection with coupled feedback, offering incremental theoretical improvements for bandit algorithms.
The paper tackles stochastic multi-armed bandit problems with complex actions, where rewards are functions of basic arms and feedback may be coupled, by proving a frequentist regret bound for Thompson sampling in a general setting without requiring conjugate priors or independence. The result includes an improved constant that captures coupling effects and provides the first nontrivial regret bounds for nonlinear MAX reward feedback from subsets.
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.