  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.

Machine Learning;Machine Learning