Tractable and Provably Efficient Distributional Reinforcement Learning with General Value Function Approximation

Distributional reinforcement learning improves performance by effectively capturing environmental stochasticity, but a comprehensive theoretical understanding of its effectiveness remains elusive. In this paper, we present a regret analysis for distributional reinforcement learning with general value function approximation in a finite episodic Markov decision process setting. We first introduce a key notion of Bellman unbiasedness for a tractable and exactly learnable update via statistical functional dynamic programming. Our theoretical results show that approximating the infinite-dimensional return distribution with a finite number of moment functionals is the only method to learn the statistical information unbiasedly, including nonlinear statistical functionals. Second, we propose a provably efficient algorithm, SF-LSVI, achieving a regret bound of O (dE H^3{2}K) where H is the horizon, K is the number of episodes, and dE is the eluder dimension of a function class.