  Value function approximation has demonstrated phenomenal empirical success in
reinforcement learning (RL). Nevertheless, despite a handful of recent progress
on developing theory for RL with linear function approximation, the
understanding of general function approximation schemes largely remains
missing. In this paper, we establish a provably efficient RL algorithm with
general value function approximation. We show that if the value functions admit
an approximation with a function class $\mathcal{F}$, our algorithm achieves a
regret bound of $\widetilde{O}(\mathrm{poly}(dH)\sqrt{T})$ where $d$ is a
complexity measure of $\mathcal{F}$ that depends on the eluder dimension [Russo
and Van Roy, 2013] and log-covering numbers, $H$ is the planning horizon, and
$T$ is the number interactions with the environment. Our theory generalizes
recent progress on RL with linear value function approximation and does not
make explicit assumptions on the model of the environment. Moreover, our
algorithm is model-free and provides a framework to justify the effectiveness
of algorithms used in practice.
