Model Selection in Reinforcement Learning with General Function Approximations

We consider model selection for classic Reinforcement Learning (RL) environments -- Multi Armed Bandits (MABs) and Markov Decision Processes (MDPs) -- under general function approximations. In the model selection framework, we do not know the function classes, denoted by $\mathcal{F}$ and $\mathcal{M}$, where the true models -- reward generating function for MABs and and transition kernel for MDPs -- lie, respectively. Instead, we are given $M$ nested function (hypothesis) classes such that true models are contained in at-least one such class. In this paper, we propose and analyze efficient model selection algorithms for MABs and MDPs, that \emph{adapt} to the smallest function class (among the nested $M$ classes) containing the true underlying model. Under a separability assumption on the nested hypothesis classes, we show that the cumulative regret of our adaptive algorithms match to that of an oracle which knows the correct function classes (i.e., $\cF$ and $\cM$) a priori. Furthermore, for both the settings, we show that the cost of model selection is an additive term in the regret having weak (logarithmic) dependence on the learning horizon $T$.