  We introduce and study a new class of stochastic bandit problems, referred to
as predictive bandits. In each round, the decision maker first decides whether
to gather information about the rewards of particular arms (so that their
rewards in this round can be predicted). These measurements are costly, and may
be corrupted by noise. The decision maker then selects an arm to be actually
played in the round. Predictive bandits find applications in many areas; e.g.
they can be applied to channel selection problems in radio communication
systems. In this paper, we provide the first theoretical results about
predictive bandits, and focus on scenarios where the decision maker is allowed
to measure at most one arm per round. We derive asymptotic instance-specific
regret lower bounds for these problems, and develop algorithms whose regret
match these fundamental limits. We illustrate the performance of our algorithms
through numerical experiments. In particular, we highlight the gains that can
be achieved by using reward predictions, and investigate the impact of the
noise in the corresponding measurements.
