  Despite rapid progress in theoretical reinforcement learning (RL) over the
last few years, most of the known guarantees are worst-case in nature, failing
to take advantage of structure that may be known a priori about a given RL
problem at hand. In this paper we address the question of whether worst-case
lower bounds for regret in online learning of Markov decision processes (MDPs)
can be circumvented when information about the MDP, in the form of predictions
about its optimal $Q$-value function, is given to the algorithm. We show that
when the predictions about the optimal $Q$-value function satisfy a reasonably
weak condition we call distillation, then we can improve regret bounds by
replacing the set of state-action pairs with the set of state-action pairs on
which the predictions are grossly inaccurate. This improvement holds for both
uniform regret bounds and gap-based ones. Further, we are able to achieve this
property with an algorithm that achieves sublinear regret when given arbitrary
predictions (i.e., even those which are not a distillation). Our work extends a
recent line of work on algorithms with predictions, which has typically focused
on simple online problems such as caching and scheduling, to the more complex
and general problem of reinforcement learning.
