{"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/a-cramer-distance-perspective-on-non-crossing","title":"A Cramér Distance perspective on Quantile Regression based Distributional Reinforcement Learning","arxiv_id":"2110.00535","date":"2021-10-01","proceeding":"NeurIPS 2021 12","authors":["Alix Lhéritier","Nicolas Bondoux"],"abstract":"Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR) based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the quantiles have been shown to improve the performance of QR-DQN for uncertainty-based exploration strategies. The contribution of this work is in the setting of fixed quantile levels and is twofold. First, we prove that the Cram\\'er distance yields a projection that coincides with the 1-Wasserstein one and that, under non-crossing constraints, the squared Cram\\'er and the quantile regression losses yield collinear gradients, shedding light on the connection between these important elements of DRL. Second, we propose a low complexity algorithm to compute the Cram\\'er distance.","url_abs":"https://arxiv.org/abs/2110.00535v2","url_pdf":"https://arxiv.org/pdf/2110.00535v2.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":"a-cramer-distance-perspective-on-non-crossing","repo_url":"https://github.com/alherit/cr-dqn","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"distributional-reinforcement-learning","task_name":"Distributional Reinforcement Learning"},{"task_slug":"reinforcement-learning-1","task_name":"Reinforcement Learning (RL)"},{"task_slug":"quantile-regression","task_name":"quantile regression"},{"task_slug":"regression-1","task_name":"regression"},{"task_slug":"reinforcement-learning-2","task_name":"reinforcement-learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2110.00535","atlas_url":"https://app.syntology.ai/?focus=2110.00535","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2110.00535"}},"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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