A Sharp Finite-Iteration Theory for Asynchronous Categorical Distributional Temporal-Difference Learning

We study finite-iteration behavior of asynchronous categorical distributional temporal-difference methods, covering scalar categorical TD (CTD) in the Cramér geometry and multivariate signed-categorical TD (MTD) in the maximum mean discrepancy (MMD) geometry. We establish high-probability guarantees under i.i.d. sampling and under a continuing Markovian trajectory. For CTD, the Cramér sample complexity to the true return law has leading term O(ρ-1(1-γ)-2 epsilon-2) in the i.i.d. and O(μ-1(1-γ)-2 epsilon-2+μ-1t mix) in the Markovian setting, without using variance reduction or data dropping. For MTD, the MMD sample complexity has leading terms O(ρ-1(1-γ)-1-c epsilon-2) and O(μ-1(1-γ)-1-c epsilon-2+μ-1t mix), where c in(0,2) is the homogeneity exponent of the MMD kernel, and they reduce to the CTD rates when c=1. These are, to our knowledge, the first finite-iteration rates for MTD. For undiscounted fixed-horizon policy evaluation, the same analysis applies to fixed-horizon versions of CTD and MTD under i.i.d. and episodic sampling, and the rates match the discounted ones with the effective horizon (1-γ)-1 replaced by the horizon H in the leading term. Matching minimax lower bounds show that the leading terms, and the implied 1-Wasserstein rates, are optimal under i.i.d., Markovian, and episodic sampling. Together, these results provide a unified non-asymptotic analysis of asynchronous categorical distributional TD across scalar, multivariate, discounted, and fixed-horizon settings, with rates that are minimax optimal in their leading terms.