Statistical Efficiency of Distributional Temporal Difference Learning and Freedman's Inequality in Hilbert Spaces

• URL: http://arxiv.org/abs/2403.05811v4
• Date: Thu, 16 Jan 2025 03:31:46 GMT
• Title: Statistical Efficiency of Distributional Temporal Difference Learning and Freedman's Inequality in Hilbert Spaces
• Authors: Yang Peng, Liangyu Zhang, Zhihua Zhang,
• Abstract summary: In this paper, we focus on the non-asymptotic statistical rates of distributional temporal difference learning. We show that for NTD with a generative model, we need $tildeO(varepsilon-2 mu_pi,min-1 (1-gamma)-3+t_mixmu_pi,min-1 (1-gamma)-1)$ sample complexity bounds in the case of the $1$-Wasserstein distance. We establish a novel Freedman's inequality
• Score: 24.03281329962804
• License:
• Abstract: Distributional reinforcement learning (DRL) has achieved empirical success in various domains. One core task in DRL is distributional policy evaluation, which involves estimating the return distribution $\eta^\pi$ for a given policy $\pi$. Distributional temporal difference learning has been accordingly proposed, which extends the classic temporal difference learning (TD) in RL. In this paper, we focus on the non-asymptotic statistical rates of distributional TD. To facilitate theoretical analysis, we propose non-parametric distributional TD (NTD). For a $\gamma$-discounted infinite-horizon tabular Markov decision process, we show that for NTD with a generative model, we need $\tilde{O}(\varepsilon^{-2}\mu_{\min}^{-1}(1-\gamma)^{-3})$ interactions with the environment to achieve an $\varepsilon$-optimal estimator with high probability, when the estimation error is measured by the $1$-Wasserstein. This sample complexity bound is minimax optimal up to logarithmic factors. In addition, we revisit categorical distributional TD (CTD), showing that the same non-asymptotic convergence bounds hold for CTD in the case of the $1$-Wasserstein distance. We also extend our analysis to the more general setting where the data generating process is Markovian. In the Markovian setting, we propose variance-reduced variants of NTD and CTD, and show that both can achieve a $\tilde{O}(\varepsilon^{-2} \mu_{\pi,\min}^{-1}(1-\gamma)^{-3}+t_{mix}\mu_{\pi,\min}^{-1}(1-\gamma)^{-1})$ sample complexity bounds in the case of the $1$-Wasserstein distance, which matches the state-of-the-art statistical results for classic policy evaluation. To achieve the sharp statistical rates, we establish a novel Freedman's inequality in Hilbert spaces. This new Freedman's inequality would be of independent interest for statistical analysis of various infinite-dimensional online learning problems.

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