Q-learning with Nearest Neighbors

We consider model-free reinforcement learning for infinite-horizon discounted Markov Decision Processes (MDPs) with a continuous state space and unknown transition kernel, when only a single sample path under an arbitrary policy of the system is available. We consider the Nearest Neighbor Q-Learning (NNQL) algorithm to learn the optimal Q function using nearest neighbor regression method. As the main contribution, we provide tight finite sample analysis of the convergence rate. In particular, for MDPs with a d-dimensional state space and the discounted factor gamma is an element of (0, 1), given an arbitrary sample path with "covering time" L, we establish that the algorithm is guaranteed to output an E-accurate estimate of the optimal Q-function using (O) over tilde (L/(epsilon(3)(1 - gamma)(7))) samples. For instance, for a wellbehaved MDP, the covering time of the sample path under the purely random policy scales as (O) over tilde (1/epsilon(d)), so the sample complexity scales as (O) over tilde (1/epsilon(d+3)). Indeed, we establish a lower bound that argues that the dependence of (Omega) over tilde (1/epsilon(d+2)) is necessary.