Provably Efficient Reinforcement Learning for Discounted MDPs with Feature Mapping

Modern tasks in reinforcement learning are always with large state and action spaces. To deal with them efficiently, one often uses predefined feature mapping to represents states and actions in a low dimensional space. In this paper, we study reinforcement learning with feature mapping for discounted Markov Decision Processes (MDPs). We propose a novel algorithm which makes use of the feature mapping and obtains a $\tilde O(d\sqrt{T}/(1-\gamma)^2)$ regret, where $d$ is the dimension of the feature space, $T$ is the time horizon and $\gamma$ is the discount factor of the MDP. To the best of our knowledge, this is the first polynomial regret bound without accessing to a generative model or making strong assumptions such as ergodicity of the MDP. By constructing a special class of MDPs, we also show that for any algorithms, the regret is lower bounded by $\Omega(d\sqrt{T}/(1-\gamma)^{1.5})$. Our upper and lower bound results together suggest that the proposed reinforcement learning algorithm is near-optimal up to a $(1-\gamma)^{-0.5}$ factor.