No-Regret Reinforcement Learning with Value Function Approximation: a Kernel Embedding Approach
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We consider the regret minimisation problem in reinforcement learning (RL) in the episodic setting. In many real-world RL environments, the state and action spaces are continuous or very large. Existing approaches establish regret guarantees by either a low-dimensional representation of the probability transition model or a functional approximation of Q functions. However, the understanding of function approximation schemes for state value functions largely remains missing. In this paper, we propose an online model-based RL algorithm, namely the CME-RL, that learns representations of transition distributions as embeddings in a reproducing kernel Hilbert space while carefully balancing the exploitation-exploration tradeoff. We demonstrate the efficiency of our algorithm by proving a frequentist (worst-case) regret bound that is of order Õ(Hγ_N√(N))[ Õ(·) hides only absolute constant and poly-logarithmic factors], where H is the episode length, N is the total number of time steps and γ_N is an information theoretic quantity relating the effective dimension of the state-action feature space. Our method bypasses the need for estimating transition probabilities and applies to any domain on which kernels can be defined. It also brings new insights into the general theory of kernel methods for approximate inference and RL regret minimization.
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