An $L^2$ Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation
This provides theoretical guarantees for RL in high-dimensional settings, addressing a key bottleneck for scalability, though it is incremental as it builds on existing fitted Q-iteration methods.
The paper tackles the problem of analyzing reinforcement learning algorithms with high-dimensional function approximation, establishing an error bound for the optimal policy that is independent of state space dimensionality, specifically achieving an $ ilde{O}(H^3|\mathcal{A}|^{ rac14}n^{- rac14})$ bound with $Hn$ samples.
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.