A Finite Sample Analysis for Quantile Temporal Difference Learning in Distributional Reinforcement Learning
We establish a global finite-sample guarantee for synchronous quantile temporal-difference learning (QTD) in tabular distributional reinforcement learning. The proof separates two stability mechanisms. A global comparison argument, based on the order monotonicity of reward cumulative distribution functions and the $W_\infty$ contraction of the distributional Bellman operator, brings an arbitrarily initialized iterate into a local neighborhood. Inside that neighborhood, we linearize the QTD mean field. Its Jacobian is a nonsingular $M$-matrix, and the associated positive semigroup permits a var
