{"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/revisiting-the-softmax-bellman-operator","title":"Revisiting the Softmax Bellman Operator: New Benefits and New Perspective","arxiv_id":"1812.00456","date":"2018-12-02","proceeding":null,"authors":["Zhao Song","Ronald E. Parr","Lawrence Carin"],"abstract":"The impact of softmax on the value function itself in reinforcement learning (RL) is often viewed as problematic because it leads to sub-optimal value (or Q) functions and interferes with the contraction properties of the Bellman operator. Surprisingly, despite these concerns, and independent of its effect on exploration, the softmax Bellman operator when combined with Deep Q-learning, leads to Q-functions with superior policies in practice, even outperforming its double Q-learning counterpart. To better understand how and why this occurs, we revisit theoretical properties of the softmax Bellman operator, and prove that $(i)$ it converges to the standard Bellman operator exponentially fast in the inverse temperature parameter, and $(ii)$ the distance of its Q function from the optimal one can be bounded. These alone do not explain its superior performance, so we also show that the softmax operator can reduce the overestimation error, which may give some insight into why a sub-optimal operator leads to better performance in the presence of value function approximation. A comparison among different Bellman operators is then presented, showing the trade-offs when selecting them.","url_abs":"https://arxiv.org/abs/1812.00456v2","url_pdf":"https://arxiv.org/pdf/1812.00456v2.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":"revisiting-the-softmax-bellman-operator","repo_url":"https://github.com/zhao-song/Softmax-DQN","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"revisiting-the-softmax-bellman-operator","repo_url":"https://github.com/Wadaboa/flatland-challenge","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[{"task_slug":"atari-games","task_name":"Atari Games"},{"task_slug":"q-learning","task_name":"Q-Learning"},{"task_slug":"reinforcement-learning","task_name":"Reinforcement Learning"},{"task_slug":"reinforcement-learning-1","task_name":"Reinforcement Learning (RL)"}],"methods":[{"method_slug":"double-q-learning","method_name":"Double Q-learning"},{"method_slug":"q-learning","method_name":"Q-Learning"},{"method_slug":"softmax","method_name":"Softmax"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1812.00456","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1812.00456"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+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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