{"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/p-mean-regret-for-stochastic-bandits","title":"p-Mean Regret for Stochastic Bandits","arxiv_id":"2412.10751","date":"2024-12-14","proceeding":null,"authors":["Anand Krishna","Philips George John","Adarsh Barik","Vincent Y. F. Tan"],"abstract":"In this work, we extend the concept of the $p$-mean welfare objective from social choice theory (Moulin 2004) to study $p$-mean regret in stochastic multi-armed bandit problems. The $p$-mean regret, defined as the difference between the optimal mean among the arms and the $p$-mean of the expected rewards, offers a flexible framework for evaluating bandit algorithms, enabling algorithm designers to balance fairness and efficiency by adjusting the parameter $p$. Our framework encompasses both average cumulative regret and Nash regret as special cases. We introduce a simple, unified UCB-based algorithm (Explore-Then-UCB) that achieves novel $p$-mean regret bounds. Our algorithm consists of two phases: a carefully calibrated uniform exploration phase to initialize sample means, followed by the UCB1 algorithm of Auer, Cesa-Bianchi, and Fischer (2002). Under mild assumptions, we prove that our algorithm achieves a $p$-mean regret bound of $\\tilde{O}\\left(\\sqrt{\\frac{k}{T^{\\frac{1}{2|p|}}}}\\right)$ for all $p \\leq -1$, where $k$ represents the number of arms and $T$ the time horizon. When $-1<p<0$, we achieve a regret bound of $\\tilde{O}\\left(\\sqrt{\\frac{k^{1.5}}{T^{\\frac{1}{2}}}}\\right)$. For the range $0< p \\leq 1$, we achieve a $p$-mean regret scaling as $\\tilde{O}\\left(\\sqrt{\\frac{k}{T}}\\right)$, which matches the previously established lower bound up to logarithmic factors (Auer et al. 1995). This result stems from the fact that the $p$-mean regret of any algorithm is at least its average cumulative regret for $p \\leq 1$. In the case of Nash regret (the limit as $p$ approaches zero), our unified approach differs from prior work (Barman et al. 2023), which requires a new Nash Confidence Bound algorithm. Notably, we achieve the same regret bound up to constant factors using our more general method.","url_abs":"https://arxiv.org/abs/2412.10751v1","url_pdf":"https://arxiv.org/pdf/2412.10751v1.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":"p-mean-regret-for-stochastic-bandits","repo_url":"https://github.com/philips-george/p-mean-regret-stochastic-bandits","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"fairness","task_name":"Fairness"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2412.10751","atlas_url":"https://app.syntology.ai/?focus=2412.10751","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}