Near Minimax-Optimal Distributional Temporal Difference Algorithms and The Freedman Inequality in Hilbert Spaces

Distributional reinforcement learning (DRL) has achieved empirical success in various domains. One of the core tasks in the field of DRL is distributional policy evaluation, which involves estimating the return distribution $\eta^\pi$ for a given policy $\pi$. The distributional temporal difference (TD) algorithm has been accordingly proposed, which is an extension of the temporal difference algorithm in the classic RL literature. In the tabular case, \citet{rowland2018analysis} and \citet{rowland2023analysis} proved the asymptotic convergence of two instances of distributional TD, namely categorical temporal difference algorithm (CTD) and quantile temporal difference algorithm (QTD), respectively. In this paper, we go a step further and analyze the finite-sample performance of distributional TD. To facilitate theoretical analysis, we propose a non-parametric distributional TD algorithm (NTD). For a $\gamma$-discounted infinite-horizon tabular Markov decision process, we show that for NTD we need $\tilde{O}\left(\frac{1}{\varepsilon^{2p}(1-\gamma)^{2p+1}}\right)$ iterations to achieve an $\varepsilon$-optimal estimator with high probability, when the estimation error is measured by the $p$-Wasserstein distance. This sample complexity bound is minimax optimal (up to logarithmic factors) in the case of the $1$-Wasserstein distance. To achieve this, we establish a novel Freedman's inequality in Hilbert spaces, which would be of independent interest. In addition, we revisit CTD, showing that the same non-asymptotic convergence bounds hold for CTD in the case of the $p$-Wasserstein distance.