A Finite Sample Analysis for Quantile Temporal Difference Learning in Distributional Reinforcement Learning

Abstract: 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 variance-sensitive martingale analysis. For stepsizes $α_t=c(t+1)<sup>{-a}$ with $a\in(1/2,1)$, the leading last-iterate fluctuation is of order $\widetilde O\bigl(T<sup>{-a/2}/\sqrt{1-γ}\bigr)$ and has no polynomial dependence on the number of quantiles. The deterministic transient and the required burn-in can still depend on the smallest Bellman-target density, which is of order $m<sup>{-1}$ in the worst case. The result therefore distinguishes sharply between the local stochastic fluctuation and the global sample complexity.