  We consider a finite-horizon multi-armed bandit (MAB) problem in a Bayesian
setting, for which we propose an information relaxation sampling framework.
With this framework, we define an intuitive family of control policies that
include Thompson sampling (TS) and the Bayesian optimal policy as endpoints.
Analogous to TS, which, at each decision epoch pulls an arm that is best with
respect to the randomly sampled parameters, our algorithms sample entire future
reward realizations and take the corresponding best action. However, this is
done in the presence of "penalties" that seek to compensate for the
availability of future information.
  We develop several novel policies and performance bounds for MAB problems
that vary in terms of improving performance and increasing computational
complexity between the two endpoints. Our policies can be viewed as natural
generalizations of TS that simultaneously incorporate knowledge of the time
horizon and explicitly consider the exploration-exploitation trade-off. We
prove associated structural results on performance bounds and suboptimality
gaps. Numerical experiments suggest that this new class of policies perform
well, in particular in settings where the finite time horizon introduces
significant exploration-exploitation tension into the problem. Finally,
inspired by the finite-horizon Gittins index, we propose an index policy that
builds on our framework that particularly outperforms the state-of-the-art
algorithms in our numerical experiments.
