[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83692-en":3,"doc-seo-83692-105":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},83692,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Open Problem Is Interaction Necessary for Order Optimal 1-bit Mean Estimation","The study investigates whether interaction/adaptivity is required to achieve the order-optimal minimax sample rate for nonparametric one-dimensional 1-bit mean estimation over finite-moment distribution classes. Adaptive threshold-query protocols attain the optimal rate, and general 1-bit queries can match it using only one adaptive transition (two-stage querying). Known results show fully non-adaptive threshold and interval families are highly suboptimal, while the expressive case of arbitrary non-adaptive quantizers remains open.","arXiv :2607 .02896v 1 [ cs .IT] 3 Jul 2026  \nOpen Problem: Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?  \nIvan Lau IVAN . LAU @U . NUS . EDU  \nNational University of Singapore  \nJonathan Scarlett SCARLETT @COMP. NUS . EDU . SG  \nNational University of Singapore  \nEditors: Steve Hanneke and Tor Lattimore  \nAbstract  \nWe ask whether interaction is necessary for order-optimal 1-bit mean estimation over nonparametric finite-moment classes. Adaptive threshold-query protocols achieve the order-optimal 1-bit minimax rate, and the same rate is attainable with general 1-bit queries using only one adaptive transition (i.e., two stages of querying) . In the non-adaptive setting, threshold and interval queries are known to be highly suboptimal, but the case of arbitrary non-adaptive quantizers remains unresolved. Can such quantizers match the adaptive rate, yielding an optimal one-shot protocol? Or is the known two-stage estimator stage-optimal, with a single adaptive transition being necessary and sufficient?  \nKeywords: Communication constraints; 1-bit estimation; adaptivity; non-interactive protocols  \n1. Motivation  \nMean estimation is one of the most fundamental primitives in statistics, machine learning, and theoretical computer science. In many modern data-acquisition systems, samples are not observed directly by the learner. Instead, they are held by agents, such as low-power sensors, edge devices, or clients in a federated system, that may transmit only a few bits per observation. A fundamental limiting case is 1-bit communication: each sample is compressed to a single binary message before being transmitted to the learner.  \nAt first glance, one bit per sample may appear too restrictive for nonparametric inference. However, recent work shows that this intuition is misleading. With interaction/adaptivity (i.e., 1-bit queries that may be designed based on the previous bits observed), 1-bit mean estimation remains nearly as sample-efficient as unquantized mean estimation: adaptive 1-bit estimators match the unquantized minimax rate up to at most one (unavoidable) logarithmic factor. In fact, simple adaptive threshold queries already suffice (Lau and Scarlett, 2026b), and the resulting rates are order-optimal in the 1-bit model. However, the 1-bit constraint itself is not the whole story, and a more refined question is: Is interaction/adaptivity truly necessary to achieve this order-optimal rate?  \nWe study this question in a nonparametric setting. The learner receives one bit per sample and must estimate the mean of an unknown distribution whose mean lies in a known bounded interval and whose k-th central moment is bounded. The quantizers may either be chosen adaptively, using previously received bits, or fixed non-adaptively before any messages are observed. The fully non-adaptive model is operationally appealing: devices can be pre-programmed and communicate without feedback or multiple adaptive rounds.  \n© 2026 I. Lau & J. Scarlett.  \nLAU SCARLETT  \nThe current landscape is sharp but incomplete. Fully non-adaptive threshold and interval queries are known to be highly suboptimal (Lau and Scarlett, 2026a), i.e., these query families cannot match the adaptive rate. However, this does not rule out non-adaptive protocols using general 1-bit quantizers, which are potentially much more expressive. If just one adaptive transition is allowed, then the order-optimal rate is known to be achievable using general 1-bit queries (Lau and Scarlett, 2026b) . The unresolved case is therefore precisely the zero-adaptivity, general-query regime:  \nCan fully non-adaptive arbitrary 1-bit quantizers match the adaptive 1-bit minimax rate for nonparametric mean estimation, or is interaction necessary?  \nEither answer would be significant. A positive answer would give an order-optimal non-adaptive protocol for nonparametric 1-bit mean estimation. A negative answer would establish a genuine interaction/adaptivity gap, and certify that","cbCaiv5WW0LhUJq3","https://ap.wps.com/l/cbCaiv5WW0LhUJq3","pdf",230524,4,1,6,"English","en",105,"# Motivation\n# Problem Setup and Open Problem\n## Distributional assumption\n## 1-bit protocols\n## Order-optimal rate","[{\"question\":\"What core question does the paper address about 1-bit mean estimation?\",\"answer\":\"Whether interaction/adaptivity is necessary to achieve the order-optimal minimax rate when only one bit per sample is allowed.\"},{\"question\":\"What is known about adaptive and two-stage 1-bit protocols?\",\"answer\":\"Adaptive threshold-query protocols achieve the order-optimal rate, and the same rate is attainable with general 1-bit queries using just one adaptive transition (two stages).\"},{\"question\":\"Why is the non-adaptive case still unresolved for arbitrary quantizers?\",\"answer\":\"Fully non-adaptive threshold and interval queries are known to be highly suboptimal, but it is unknown whether arbitrary fixed (one-shot) 1-bit quantizers can still match the adaptive minimax rate.\"}]",1784189760,15,{"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},"open-problem-is-interaction-necessary-for-order-optimal-1-bit-mean-estimation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/open-problem-is-interaction-necessary-for-order-optimal-1-bit-mean-estimation/83692/",{"url":52,"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","2026-07-27","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 core question does the paper address about 1-bit mean estimation?","Question",{"text":75,"@type":76},"Whether interaction/adaptivity is necessary to achieve the order-optimal minimax rate when only one bit per sample is allowed.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is known about adaptive and two-stage 1-bit protocols?",{"text":80,"@type":76},"Adaptive threshold-query protocols achieve the order-optimal rate, and the same rate is attainable with general 1-bit queries using just one adaptive transition (two stages).",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the non-adaptive case still unresolved for arbitrary quantizers?",{"text":84,"@type":76},"Fully non-adaptive threshold and interval queries are known to be highly suboptimal, but it is unknown whether arbitrary fixed (one-shot) 1-bit quantizers can still match the adaptive minimax rate.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"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":22,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"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"]