[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86080-en":3,"doc-seo-86080-105":29,"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":20,"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":13,"seo_description":14,"update_tm":27,"read_time":28},86080,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Bandit PCA with Minimax Optimal Regret","Bandit PCA with bandit feedback studies an online principal component analysis game where, at each round, an adversary chooses a symmetric gain matrix with spectrum in [0,1] and rank at most r. The learner selects a unit vector and observes only the scalar quadratic reward, minimizing regret to the best fixed direction in hindsight. The paper improves prior upper and lower bounds, achieving minimax regret of order r√dT up to polylog factors, and proposes a multiscale mirror-descent method. It also proves a lower bound via an adaptive adversary and links to quantum tomography.","Bandit PCA with Minimax Optimal Regret  \nMoïse Blanchard  \nGeorgia Tech [mblanchard41@gatech.edu](mblanchard41@gatech.edu)  \nDmitrii Ostrovskii  \nGeorgia Tech [ostrov@gatech.edu](ostrov@gatech.edu)  \nAadirupa Saha  \nUIC[aadirupa.saha@gmail.com](aadirupa.saha@gmail.com)  \narXiv :2607 . 10936v 1 [ cs .LG] 12 Jul 2026  \nAbstract  \nWe study the bandit-feedback version of online principal component analysis (Bandit PCA): in each round t = 1 ,..., T, the adversary selects a d × d symmetric gain matrix Gt with spectrum in [0 , 1] and rank at most r; the learner simultaneously selects a unit vector wt ∈ Sd−1 and receives the reward w⊤tGt wt. The learner receives no other feedback, and aims to minimize the regret against the best unit vector in hindsight. This problem was introduced by Kotłowski and Neu (2019),  who gave an algorithm with regret O (d √rT log T) and showed the lower bound of Ω(r pT/log T) . We improve upon both of these bounds and essentially bridge the gap between them, establishing the minimax regret of order r √dT up to polylogarithmic factors in d and T. The upper bound is attained by a novel algorithm, which combines online mirror descent on the spectrahedron of (real) density matrices with a multiscale exploration scheme in which theeigenspaces with different spectral magnitudes are updated at different rates. For the lower bound, we construct an adaptive adversary that refines a hidden large-reward subspace based on the learner’s actions, in such a way that low regret is impossible without estimating the subspace; asa result, lower-bounding the regret reduces to studying the arising subspace estimation problem. Finally, we discuss connections of Bandit PCA with adaptive-measurement quantum tomography.  \nContents  \n1 Introduction 1  \n2 Problem statement 4  \n3 Near-optimal algorithm for Bandit PCA 5  \n3.1 Algorithm construction ................................... 5  \n3.2 Proof sketch of Theorem 1 ................................. 9  \n4 Lower bound for Bandit PCA 11  \n4.1 Adaptive adversary construction ............................. 11  \n4.2 Proof sketch of Theorem 2 ................................. 12  \n5 Conclusion and perspectives 13  \nA Proof of the regret upper bound 15  \nA.1 General properties of gain matrix estimators ....................... 15  \nA.2 Properties of exploration subspaces ............................ 17  \nA.3 OMD regret bound with respect to estimated gain matrices .............. 26  \nA.4 Proof of Theorem 7 ..................................... 30  \nB Computational complexity of Algorithm 1 33  \nC Proof of the regret lower bound 36  \nC.1 Supporting lemmas ..................................... 36  \nC.2 From Bandit PCA to subspace discovery ......................... 39  \nC.3 From subspace discovery to covariance estimation .................... 46  \nC.4 Lower bound for covariance estimation .......................... 48  \nC.5 Completing the proof of Theorem 2 ............................ 54  \nD Concentration inequalities 56  \n1 Introduction  \nOnline learning with bandit feedback stands at the intersection of statistical decision theory and sequential optimization: the learner must act without access to counterfactual outcomes, relying solely on the scalar reward of the chosen action. Class˜ical results for multi-armed and linear  \nbandits [ACBF02 , BCBK12] give regret bounds scaling as O (poly(d)√T) in the number of rounds T , where d is the ambient dimension. In this paper, we study matrix-valued bandit setting, in which actions are unit vectors in Rd, rewards are evaluations of quadratic forms, and the adversary controls a sequence of positive-semidefinite (PSD) gain matrices. Following [KN19] who first introduced and studied this setting, we shall refer to it as Bandit Principal Component Analysis (Bandit PCA) . Simply put, Bandit PCA is the noncommutative generalization of the classic multi-armed bandit (MAB) problem, obtained by replacing the probability simplex by the spectrahedron of unit-t","cbCain2ss1iCVqd2","https://ap.wps.com/l/cbCain2ss1iCVqd2","pdf",1009958,1,62,"English","en",105,"# Introduction\n# Problem statement\n# Near-optimal algorithm for Bandit PCA\n## Algorithm construction\n## Proof sketch of Theorem 1\n# Lower bound for Bandit PCA\n## Adaptive adversary construction\n## Proof sketch of Theorem 2\n# Conclusion and perspectives","[{\"question\":\"What is the Bandit PCA problem studied in the document?\",\"answer\":\"At each round the adversary selects a rank-r gain matrix and the learner chooses a unit vector, receiving only the scalar quadratic reward. The learner minimizes regret against the best fixed direction chosen in hindsight.\"},{\"question\":\"How is the gain matrix constrained and what parameters govern the setting?\",\"answer\":\"The gain matrix is symmetric/PSD with spectrum bounded within [0,1] and rank at most r. The setting is determined by three parameters T, d, and r.\"},{\"question\":\"What regret rate does the paper establish and how is it achieved?\",\"answer\":\"The paper establishes minimax regret of order r√dT up to polylogarithmic factors. The upper bound comes from an algorithm combining online mirror descent on the spectrahedron with multiscale exploration of spectral subspaces.\"}]",1784208390,156,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"bandit-pca-with-minimax-optimal-regret","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/bandit-pca-with-minimax-optimal-regret/86080/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the Bandit PCA problem studied in the document?","Question",{"text":75,"@type":76},"At each round the adversary selects a rank-r gain matrix and the learner chooses a unit vector, receiving only the scalar quadratic reward. The learner minimizes regret against the best fixed direction chosen in hindsight.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the gain matrix constrained and what parameters govern the setting?",{"text":80,"@type":76},"The gain matrix is symmetric/PSD with spectrum bounded within [0,1] and rank at most r. The setting is determined by three parameters T, d, and r.",{"name":82,"@type":73,"acceptedAnswer":83},"What regret rate does the paper establish and how is it achieved?",{"text":84,"@type":76},"The paper establishes minimax regret of order r√dT up to polylogarithmic factors. The upper bound comes from an algorithm combining online mirror descent on the spectrahedron with multiscale exploration of spectral subspaces.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]