[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85534-en":3,"doc-seo-85534-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85534,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance","Bilateral trade with full feedback is studied under contextual settings where valuations have bounded conditional densities but infinite variance. The work extends Bachoc et al. self-bounding from bounded to real-valued valuations, deriving a regret bound driven by squared estimation error using bounded density alone and minimal moment assumptions. With epoch-based truncated-mean estimation, the paper identifies the exact minimax regret rate (up to logarithmic factors), interpolating between the classical nonparametric case at p=2 and the limiting linear regime as p approaches 1+. Additional assumptions ensure unique optimal pricing.","arXiv :2603 .06851v2 [ stat .ML] 13 Jul 2026  \nBilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance  \nHangyi Zhao  \n[hyz0815@stanford. edu](hyz0815@stanford. edu)  \nJuly 14, 2026  \nAbstract  \nWe study contextual bilateral trade under full feedback when, conditionally on the context, trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price π satisfies E[g(m, V, W) − g(π, V, W)] ≤ L|m − π|2 under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based apbrn(it1d,hm2vi)a acanAhdssietoveheuadsm’rsegarmretketetha(Tluw2βuan(1ioe)/nd-(βsipsu+pdβp(1o¨lm)d) ) werixt,haurenndecthwoneestnerostuicsatebiolhaishnsfiaOnmuriteatrepchsu-tilthnsmog Ωchamr( ae)cnlttoefwrioezrre the exact minimax rate for this problem, interpolating between the classical nonparametric rate at p=2 and the trivial linear rate as p → 1+ .  \n1 Introduction  \nBilateral trade—the simplest two-sided market—requires a broker to set prices between a buyer anda seller whose private valuations are unknown. The celebrated impossibility theorem of Myerson and Satterthwaite [13] shows that no incentive-compatible, individually rational, budget-balanced mechanism can achieve full efficiency in a single round. This spurred a rich literature on approximate mechanisms [4] and, more recently, on online bilateral trade, where the broker learns from repeated interactions and regret—the cumulative loss from suboptimal pricing—replaces the single-shot efficiency objective.  \nBachoc, Cesari, and Colomboni [3] initiated the study of contextual online bilateral trade, where a public context vector xt arrives each round and trader valuations depend on an unknown function m(xt) . Under bounded noise densities and finite variance, they established an O (Ld log T) pawaregrasrmsuetetbsofriceqprurieecgrntingetlyabtrtoueaπndteindstaiennadd[a2o]nf (LdyatsTtrm)uobctstouurLna| dlmidigπe|rh2t, twisreotd-bhuiectsinfeeglefrdb-boegaureckndtc; itnohgnetnprrooolnptpeoarrtymae:mtanetheersicetsxpimeteatctitineogdn. However, their algorithms rely on ordinary least squares, which requires finite variance (E[ξ2] \u003C ∞ ) . In many applications—financial markets, insurance, real estate—valuations exhibit heavy tails well-modeled by Student’s t (ν) with ν \u003C 2, where the variance is infinite [11] . This raises a natural question: what regret is achievable when the noise has bounded density but infinite variance?  \nWe answer this question with three contributions. (C1) We extend the self-bounding property from bounded to real-valued valuations (Lemma 3.1), showing that bounded density alone—without any moment condition beyond E[|ξ|] \u003C ∞—suffices to control regret via squared estimation error.(C2) We design epoch-based algorithms using truncated-mean estimation [5] and prove tight  \nrceegasstreaeb,tlrwisathhesermea: t(g(2(po,w)/2p)er)iins tbothheunedpmsaroviammena Aettsripsoc cauaraads’emsaetmneedrthnd(Td[ 11]2βisc(poe)/in(βpHoedddw(s)ma))iofinoxtethhd-enseunossppnopra(tramC3mi)xettrWuricee construction (Proposition 6.1, Remark 6.2), proving that our rates are minimax optimal in T up to logarithmic factors for nondegenerate class parameters (the parametric dimension dependence of the lower bound is left open; Remark 6.2) . The constructed instances also satisfy Assumption 2.4, a mild regularity condition at zero under which the optimal price is unique.  \n1.1 Related work  \nBilateral trade. Online bilateral trade was initiated by Cesa-Bianchi et al. [7] and has since been studied under various feedback models and distributional assumptions; see [3] and references therein. The contextual setting was introduced in [3] (parametric) and [2] (nonparametric), both under finite-variance noise. Our work extends this line to the infinite-variance regime.  \nRobust mean estimation. Estimation un","cbCaicykkQoYXAhx","https://ap.wps.com/l/cbCaicykkQoYXAhx","pdf",1032160,2,1,16,"English","en",105,"# Abstract\n# Introduction\n## Related work\n# Setup and the Structural-Algorithmic Gap","[{\"question\":\"What market setting and feedback model does the paper study?\",\"answer\":\"The paper studies online contextual bilateral trade where each round reveals a context vector and valuations depend on an unknown function, while the broker receives full feedback by observing both traders’ valuations.\"},{\"question\":\"How do the authors model heavy-tailed valuations and what key difficulty arises?\",\"answer\":\"Valuations are assumed to have bounded conditional densities but infinite variance, motivated by heavy-tailed noise such as Student’s t with degrees of freedom ν\\u003c2. The resulting infinite-variance setting makes standard estimators like ordinary least squares fail.\"},{\"question\":\"What estimator and strategy are used to achieve minimax regret?\",\"answer\":\"The method uses epoch-based algorithms combined with truncated-mean estimation. This, together with the extended self-bounding property, yields the exact minimax regret rate up to logarithmic factors.\"}]",1784204268,40,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"bilateral-trade-under-heavy-tailed-valuations-minimax-regret-with-infinite-variance","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/bilateral-trade-under-heavy-tailed-valuations-minimax-regret-with-infinite-variance/85534/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"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 market setting and feedback model does the paper study?","Question",{"text":75,"@type":76},"The paper studies online contextual bilateral trade where each round reveals a context vector and valuations depend on an unknown function, while the broker receives full feedback by observing both traders’ valuations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the authors model heavy-tailed valuations and what key difficulty arises?",{"text":80,"@type":76},"Valuations are assumed to have bounded conditional densities but infinite variance, motivated by heavy-tailed noise such as Student’s t with degrees of freedom ν\u003C2. The resulting infinite-variance setting makes standard estimators like ordinary least squares fail.",{"name":82,"@type":73,"acceptedAnswer":83},"What estimator and strategy are used to achieve minimax regret?",{"text":84,"@type":76},"The method uses epoch-based algorithms combined with truncated-mean estimation. This, together with the extended self-bounding property, yields the exact minimax regret rate up to logarithmic factors.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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":29,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]