We consider the problem of undiscounted reinforcement learning in continuous state space. Regret bounds in this setting usually hold under various assumptions on the structure of the reward and transition function. Under the assumption that the rewards and transition probabilities are Lipschitz, for 1-dimensional state space a regret bound of O(T3/4)O(T3/4) after any T steps has been given by Ortner and Ryabko (2012). Here we improve upon this result by using non-parametric kernel density estimation for estimating the transition probability distributions, and obtain regret bounds that depend on the smoothness of the transition probability distributions. In particular, under the assumption that the transition probability functions are smoothly differentiable, the regret bound is shown to be O(T2/3)O(T2/3) asymptotically for reinforcement learning in 1-dimensional state space. Finally, we also derive improved regret bounds for higher dimensional state space.
L. K., Ortner, R., and Ryabko, D., “Improved Regret Bounds for Undiscounted Continuous Reinforcement Learning.”, Proceedings of The 32nd International Conference on Machine Learning. pp. 524–532, 2015.