Learning to Control in Metric Space with Optimal Regret

We study online reinforcement learning for finite-horizon deterministic control systems with arbitrary state and action spaces. Suppose the transition dynamics and reward function is unknown, but the state and action space is endowed with a metric that characterizes the proximity between different states and actions. We provide a surprisingly simple upper-confidence reinforcement learning algorithm that uses a function approximation oracle to estimate optimistic Q functions from experiences. We show that the regret of the algorithm after K episodes is o(DLK)^{\frac{d}{d+1}}H where D is the diameter of the state-action space, L is a smoothness parameter, and d is the doubling dimension of the state-action space with respect to the given metric. We also establish a near-matching regret lower bound. The proposed method can be adapted to work for more structured transition systems, including the finite-state case and the case where value functions are linear combinations of features, where the method also achieve the optimal regret.