Minimax Optimal Q Learning with Nearest Neighbors

Understand

Analyzing the Markov decision process (MDP) with continuous state spaces is generally challenging.

• A recent interesting work \cite{shah2018q} solves MDP with bounded continuous state space by a nearest neighbor $Q$ learning approach, which has a sample complexity of $\tilde{O}(\frac{1}{\epsilon^{d+3}(1-\gamma)^{d+7}})$ for $\epsilon$-accurate $Q$ function estimation with discount factor $\gamma$.
• In this paper, we propose two new nearest neighbor $Q$ learning methods, one for the offline setting and the other for the online setting.
• We show that the sample complexities of these two methods are $\tilde{O}(\frac{1}{\epsilon^{d+2}(1-\gamma)^{d+2}})$ and $\tilde{O}(\frac{1}{\epsilon^{d+2}(1-\gamma)^{d+3}})$ for offline and online methods respectively, which significantly improve over existing results and have minimax optimal dependence over $\epsilon$.