[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124864-en":3,"doc-seo-124864-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124864,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","MOTS - Minimax Optimal Thompson Sampling","Thompson sampling is widely used for online decision problems due to simple implementation and strong empirical results, yet its ability to attain the minimax lower bound for K-armed bandits remains unresolved. This paper introduces MOTS, a variant that adaptively clips the sampling instance of the chosen arm at each time step. The work proves minimax optimal regret O(sqrt(pKT)) for finite horizons and asymptotically optimal regret for Gaussian rewards as the horizon grows.","MOTS: Minimax Optimal Thompson Sampling  \nTianyuan Jin 1 Pan Xu 2 Jieming Shi 3 Xiaokui Xiao 1 Quanquan Gu 2  \nAbstract  \nThompson sampling is one of the most widely used algorithms for many online decision problems, due to its simplicity in implementation and superior empirical performance over other stateof-the-art methods. Despite its popularity and empirical success, it has remained an open problem whether Thompson sampling can match the minimax lower bound 􀀊( pKT ) for K-armed bandit problems, where T is the total time horizon.  \nIn this paper, we solve this long open problem by proposing a variant of Thompson sampling called MOTS that adaptively clips the sampling instance of the chosen arm at each time step. We prove that this simple variant of Thompson sampling achieves the minimax optimal regret bound O ( pKT ) for finite time horizon T, as well as the asymptotic optimal regret bound for Gaussian rewards when T approaches infinity. To our knowledge, MOTS is the first Thompson sampling type algorithm that achieves the minimax optimality for multi-armed bandit problems.  \n1. Introduction  \nThe Multi-Armed Bandit (MAB) problem is a sequential decision process which is typically described as a game between the agent and the environment with K arms. The game proceeds in T time steps. In each time step t = 1; : : : ; T, the agent plays an arm At 2 f1; 2 ; 􀀁 􀀁 􀀁 ; Kg based on the observation of the previous t 􀀀 1 time steps, and then observes a reward rt that is independently generated from a 1-subGaussian distribution with mean value 􀀖At , where 􀀖 1 ; 􀀖2 ; 􀀁 􀀁 􀀁 ; 􀀖 K 2 R are unknown. The goal of the agent is to maximize the cumulative reward over T time steps. The performance of a strategy for MAB is measured by the expected cumulative difference over T time steps between  \n1 School of Computing, National University of Singapore, Singapore 2Department of Computer Science, University of California, Los Angeles, USA 3Department of Computing, The Hong Kong Polytechnic University, Hong Kong. Correspondence to: Xiaokui  \nXiao \u003C[xkxiao@nus.edu.sg](xkxiao@nus.edu.sg) >, Quanquan Gu \u003C[qgu@cs.ucla.edu](qgu@cs.ucla.edu) >.  \nProceedings of the 38 th International Conference on Machine Learning, PMLR 139, 2021 . Copyright 2021 by the author(s) .  \nplaying the best arm and playing the arm according to the strategy, which is also called the regret of a bandit strategy. Formally, the regret R􀀖 (T ) is defined as follows  \nR􀀖 (T ) = T 􀀁 i2f1m;2a; 􀀁 ;Kg 􀀖i 􀀀 E􀀖 \" Xt1 rt \\#: (1)  \nFor a fixed time horizon T, the problem-independent lower bound (Auer et al., 2002b) states that any strategy has at least a regret in the order of 􀀊( pKT ), which is called the minimax optimal regret. On the other hand, for a fixed model (i.e., 􀀖 1 ; : : : ; 􀀖 K are fixed), Lai & Robbins (1985) proved that any strategy must have at least C (􀀖) log(T )(1 􀀀 o(1))  \nregret when the horizon T approaches infinity, where C (􀀖) is a constant depending on the model. Therefore, a strategy with a regret upper-bounded by C (􀀖) log(T )(1 􀀀 o(1)) is asymptotically optimal.  \nThis paper studies the earliest bandit strategy, Thompson sampling (TS) (Thompson, 1933) . It has been observed in practice that TS often achieves a smaller regret than many upper confidence bound (UCB)-based algorithms (Chapelle & Li, 2011 ; Wang & Chen, 2018) . In addition, TS is simple and easy to implement. Despite these advantages, the theoretical analysis of TS algorithms has not been established until the past decade. In particular, in the seminal work by Agrawal & Goyal (2012), they provided the first finitetime analysis of TS. Kaufmann et al. (2012) and Agrawal & Goyal (2013) showed that the regret bound of TS is asymptotically optimal when using Beta priors. Subsequently, Agrawal & Goyal (2017) showed that TS with Beta priors achieves an O ( p KT log T ) problem-independent regret bound while maintaining the asymptotic optimality. In addition, they proved that TS with Gaussian priors can achieve an improv","cbCaioCIlZi7OVin","https://ap.wps.com/l/cbCaioCIlZi7OVin","pdf",3136214,1,10,"English","en",105,"# Abstract\n# Introduction\n## Multi-Armed Bandit Setup and Regret\n## Thompson Sampling and Known Bounds\n## Minimax Optimality and Main Contributions\n## Comparison Table of TS Variants","[{\"question\":\"What problem does MOTS address in Thompson sampling for bandits?\",\"answer\":\"It addresses whether Thompson sampling can achieve the minimax optimal regret bound O(sqrt(pKT)) for K-armed bandit problems over a finite horizon.\"},{\"question\":\"How does MOTS modify standard Thompson sampling?\",\"answer\":\"MOTS adaptively clips the sampling instance of the chosen arm at each time step based on past pulls.\"},{\"question\":\"What theoretical regret guarantees are proved for MOTS?\",\"answer\":\"MOTS is proved to achieve minimax optimal regret O(sqrt(pKT)) for finite horizons and asymptotically optimal regret for Gaussian rewards as T approaches infinity.\"}]","MOTS - Minimax Optimal Thompson Sampling | PDF",1785895097,25,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"mots-minimax-optimal-thompson-sampling","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/mots-minimax-optimal-thompson-sampling/124864/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does MOTS address in Thompson sampling for bandits?","Question",{"text":75,"@type":76},"It addresses whether Thompson sampling can achieve the minimax optimal regret bound O(sqrt(pKT)) for K-armed bandit problems over a finite horizon.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does MOTS modify standard Thompson sampling?",{"text":80,"@type":76},"MOTS adaptively clips the sampling instance of the chosen arm at each time step based on past pulls.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical regret guarantees are proved for MOTS?",{"text":84,"@type":76},"MOTS is proved to achieve minimax optimal regret O(sqrt(pKT)) for finite horizons and asymptotically optimal regret for Gaussian rewards as T approaches infinity.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":21,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]