  This paper studies systematic exploration for reinforcement learning with
rich observations and function approximation. We introduce a new model called
contextual decision processes, that unifies and generalizes most prior
settings. Our first contribution is a complexity measure, the Bellman rank,
that we show enables tractable learning of near-optimal behavior in these
processes and is naturally small for many well-studied reinforcement learning
settings. Our second contribution is a new reinforcement learning algorithm
that engages in systematic exploration to learn contextual decision processes
with low Bellman rank. Our algorithm provably learns near-optimal behavior with
a number of samples that is polynomial in all relevant parameters but
independent of the number of unique observations. The approach uses Bellman
error minimization with optimistic exploration and provides new insights into
efficient exploration for reinforcement learning with function approximation.
