  Reinforcement learning (RL) algorithms based on high-dimensional function
approximation have achieved tremendous empirical success in large-scale
problems with an enormous number of states. However, most analysis of such
algorithms gives rise to error bounds that involve either the number of states
or the number of features. This paper considers the situation where the
function approximation is made either using the kernel method or the two-layer
neural network model, in the context of a fitted Q-iteration algorithm with
explicit regularization. We establish an $\tilde{O}(H^3|\mathcal
{A}|^{\frac14}n^{-\frac14})$ bound for the optimal policy with $Hn$ samples,
where $H$ is the length of each episode and $|\mathcal {A}|$ is the size of
action space. Our analysis hinges on analyzing the $L^2$ error of the
approximated Q-function using $n$ data points. Even though this result still
requires a finite-sized action space, the error bound is independent of the
dimensionality of the state space.
