Minimax Optimal Q Learning with Nearest Neighbors

TL;DR

This paper introduces two new nearest neighbor Q-learning algorithms for continuous state MDPs, achieving improved sample complexity bounds that are minimax optimal and more efficient than previous methods.

Contribution

The paper proposes offline and online nearest neighbor Q-learning methods with significantly improved sample complexities and better resource utilization, extending applicability to unbounded state spaces.

Findings

Sample complexity for offline method: (rac{1}{\u03b5^{d+2}(1-\u03b3)^{d+2}})

Sample complexity for online method: (rac{1}{\u03b5^{d+2}(1-\u03b3)^{d+3}})

Methods are minimax optimal and more computationally efficient.

Abstract

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 $\epsilon$. We achieve such improvement by utilizing the samples more efficiently. In particular, the method in \cite{shah2018q} clears up all samples after each iteration, thus these samples are somewhat wasted. On the other hand, our offline method does not remove any samples, and our online method only removes samples with time earlier than $\beta t$ at time $t$ with $\beta$ being a tunable parameter, thus our methods significantly reduce the loss of information. Apart from the sample complexity, our methods also have additional advantages of better computational complexity, as well as suitability to unbounded state spaces.