{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/estimation-and-inference-in-distributional","title":"Estimation and Inference in Distributional Reinforcement Learning","arxiv_id":"2309.17262","date":"2023-09-29","proceeding":null,"authors":["Liangyu Zhang","Yang Peng","Jiadong Liang","Wenhao Yang","Zhihua Zhang"],"abstract":"In this paper, we study distributional reinforcement learning from the perspective of statistical efficiency. We investigate distributional policy evaluation, aiming to estimate the complete return distribution (denoted $\\eta^\\pi$) attained by a given policy $\\pi$. We use the certainty-equivalence method to construct our estimator $\\hat\\eta^\\pi$, given a generative model is available. In this circumstance we need a dataset of size $\\widetilde O\\left(\\frac{|\\mathcal{S}||\\mathcal{A}|}{\\varepsilon^{2p}(1-\\gamma)^{2p+2}}\\right)$ to guarantee the $p$-Wasserstein metric between $\\hat\\eta^\\pi$ and $\\eta^\\pi$ less than $\\varepsilon$ with high probability. This implies the distributional policy evaluation problem can be solved with sample efficiency. Also, we show that under different mild assumptions a dataset of size $\\widetilde O\\left(\\frac{|\\mathcal{S}||\\mathcal{A}|}{\\varepsilon^{2}(1-\\gamma)^{4}}\\right)$ suffices to ensure the Kolmogorov metric and total variation metric between $\\hat\\eta^\\pi$ and $\\eta^\\pi$ is below $\\varepsilon$ with high probability. Furthermore, we investigate the asymptotic behavior of $\\hat\\eta^\\pi$. We demonstrate that the ``empirical process'' $\\sqrt{n}(\\hat\\eta^\\pi-\\eta^\\pi)$ converges weakly to a Gaussian process in the space of bounded functionals on Lipschitz function class $\\ell^\\infty(\\mathcal{F}_{\\text{W}})$, also in the space of bounded functionals on indicator function class $\\ell^\\infty(\\mathcal{F}_{\\text{KS}})$ and bounded measurable function class $\\ell^\\infty(\\mathcal{F}_{\\text{TV}})$ when some mild conditions hold. Our findings give rise to a unified approach to statistical inference of a wide class of statistical functionals of $\\eta^\\pi$.","url_abs":"https://arxiv.org/abs/2309.17262v2","url_pdf":"https://arxiv.org/pdf/2309.17262v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"estimation-and-inference-in-distributional","repo_url":"https://github.com/zhangliangyu32/estimationandinferencedistributionalrl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"distributional-reinforcement-learning","task_name":"Distributional Reinforcement Learning"},{"task_slug":"reinforcement-learning","task_name":"Reinforcement Learning"},{"task_slug":"reinforcement-learning-2","task_name":"reinforcement-learning"}],"methods":[{"method_slug":"gaussian-process","method_name":"Gaussian Process"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2309.17262","atlas_url":"https://app.syntology.ai/?focus=2309.17262","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2309.17262"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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