  We address the problem of model selection for the finite horizon episodic
Reinforcement Learning (RL) problem where the transition kernel $P^*$ belongs
to a family of models $\mathcal{P}^*$ with finite metric entropy. In the model
selection framework, instead of $\mathcal{P}^*$, 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 $P^*$ 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
$\Tilde{\mathcal{O}}(d_{\mathcal{E}}^*H^2+\sqrt{d_{\mathcal{E}}^* \mathbb{M}^*
H^2 T})$, with high probability, where $H$ is the horizon length, $T$ is the
total number of steps, $d_{\mathcal{E}}^*$ is the Eluder dimension and
$\mathbb{M}^*$ is the metric entropy corresponding to $\mathcal{P}^*$. Note
that this regret scaling matches that of an oracle that knows $\mathcal{P}^*$
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 $P^*$ 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.
