Model Selection for Generic Reinforcement Learning

Abstract

We address the problem of model selection for the finite horizon episodic Reinforcement Learning (RL) problem where the transition kernel $P^{\ast }$ belongs to a family of models ${\mathcal{P}}^{\ast }$ with finite metric entropy. In the model selection framework, instead of ${\mathcal{P}}^{\ast }$, we are given $M$ nested families of transition kernels $\cP_1 \subset \cP_2 \subset \ldots \subset \cP_M$. We propose and analyze a novel algorithm, namely \emph{Adaptive Reinforcement Learning (General)} (\texttt{ARL-GEN}) that adapts to the smallest such family where the true transition kernel $P^{\ast }$ lies. \texttt{ARL-GEN} uses the Upper Confidence Reinforcement Learning (\texttt{UCRL}) algorithm with value targeted regression as a blackbox and puts a model selection module at the beginning of each epoch. Under a mild separability assumption on the model classes, we show that \texttt{ARL-GEN} obtains a regret of $\Tilde{\mathcal{O}}(d_{\mathcal{E}}^*H^2+\sqrt{d_{\mathcal{E}}^* \mathbb{M}^* H^2 T})$,withhighprobability,where$H$isthehorizonlength,$T$ is the total number of steps, $d_{\mathcal{E}}^{\ast }$ is the Eluder dimension and ${\mathbb{M}}^{\ast }$ is the metric entropy corresponding to ${\mathcal{P}}^{\ast }$. Note that this regret scaling matches that of an oracle that knows ${\mathcal{P}}^{\ast }$ in advance. We show that the cost of model selection for \texttt{ARL-GEN} is an additive term in the regret having a weak dependence on $T$. Subsequently, we remove the separability assumption and consider the setup of linear mixture MDPs, where the transition kernel $P^{\ast }$ has a linear function approximation. With this low rank structure, we propose novel adaptive algorithms for model selection, and obtain (order-wise) regret identical to that of an oracle with knowledge of the true model class.