[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82050-en":3,"doc-seo-82050-105":29,"detail-sidebar-cat-0-en-105":90},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},82050,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Adaptive Bayes exactly tracks information over intrinsic time","Bayesian and multiplicative-weights updates reweight experts, models, or actions in response to sequential feedback. The paper derives an exact information-accounting identity: on each round, a learner’s excess loss versus any comparator splits into an immediate uncertainty payment from the exposed feedback and a reduction in the information distance to the benchmark. Summing one-step balances defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Two exact adaptive decompositions of cumulative regret follow, yielding selfbounding behavior in low-noise regimes rather than slack worst-case bounds.","arXiv :2607 .08789v1 [ cs .LG] 26 Jun 2026  \nAdaptive Bayes exactly tracks information over intrinsic time  \nAkshay Balsubramani  \n[akshay@vac.bio](akshay@vac.bio)  \nAbstract  \nBayesian and multiplicative-weights updates reweight experts, models, or actions from sequential feedback. We show that the regret of any such update obeys an exact information-accounting identity. On each round, the learner’s excess loss to any chosen comparator is the sum of an immediate payment for the uncertainty exposed by the round and a reduction in the information distance from the learner’s current weights to the comparator. The cumulative payment defines a pathwise uncertainty clock, the intrinsic time of the realized sequence. Summing one-step balances yields two exact adaptive decompositions of cumulative regret, one for each natural way of composing the update across rounds. Because the decompositions are exact rather than upper bounds, favorable stochastic or low-noise regimes appear as selfbounding properties of the realized intrinsic time, not as slack in worst-case analyses. The same calculus covers Hedge, optimistic and side-information variants, continuous priors, boosting, online convex optimization, contextual bandits, and repeated games: the pathwise account is the same in every case.  \n1 Introduction  \nBayesian and multiplicative updates are the main mechanisms for learning from sequential feedback. In the classical experts problem they reweight experts after each loss vector; in Bayesian model averaging they reweight models after each observation; and in many modern pipelines they reweight candidate actions, hypotheses, or responses after partial evidence. The common primitive is simple: keep a distribution over candidates, observe feedback, and move mass toward candidates that explain that feedback better.  \nThe main point of this paper is that the performance of these updates can be written as an exact ledger of information. On each round, the learner’s excess loss to a chosen benchmark splits into an immediate payment for the uncertainty exposed by the current feedback and a reduction in the information gap to the benchmark. Summing those one-step balances produces two exact adaptive decompositions: one for a prior-retempered update that recomputes the current posterior from the original prior using cumulative scores, and one for a local update that moves only from the current weights using the current score vector. In both cases, information is the common accounting currency. It plays many simultaneous roles: measuring comparator complexity, governing learning-rate effects, and recording how difficult the realized sequence actually was.  \nThis update template applies not only to prediction with expert advice, but also much more broadly. It encompasses Bayesian model averaging, optimistic online learning, bandit algorithms, boosting, online convex optimization over continuous domains, repeated-game play, contextual decision-making, multiscale aggregation, and softmax preference learning. All are instances of a single Bayes-rule update with different choices of scores, side information, and learning rate.  \nOn round t the learner chooses weights pt 2 ∆([K]) over K experts, observes expert losses ℓt 2 RK , and incurs mixture loss hpt , ℓti. For a comparator distribution ρ 2 ∆([K]), the cumulative regret up to time T is  \nT T  \nRℓT(ρ) :=Xhpt , ℓti 􀀀 hρ, LTi LT :=X ℓt  \nt=1 t=1  \nPoint-mass comparators recover the usual regret to a single expert, while diffuse ρ compare to arbitrary mixtures. A comparator is simply the benchmark distribution we want to match in hindsight, and regret is the learner’s extra cumulative loss relative to that benchmark. Throughout the paper the loss sequence may be arbitrary, so every statement is pathwise, over the realized trajectory of the learner.  \nThe classic Hedge/exponential-weights algorithm [27] is the special case ct = ℓt. More generally, one may first incorporate side information through po","cbCaidij65fqj7Pe","https://ap.wps.com/l/cbCaidij65fqj7Pe","pdf",3754990,1,108,"English","en",105,"# Introduction\n## Regret and comparator setup\n## Prior and update templates\n## Relation to game-theoretic concentration","[{\"question\":\"What exact relationship does the paper establish for regret in Bayesian or multiplicative-weights updates?\",\"answer\":\"It shows that the regret of such updates satisfies an exact information-accounting identity. Each round’s excess loss decomposes into an immediate uncertainty payment and a reduction in information distance to the comparator.\"},{\"question\":\"What does the paper mean by “intrinsic time”?\",\"answer\":\"The cumulative uncertainty payment over the realized sequence defines a pathwise uncertainty clock. This clock is called the intrinsic time of the realized trajectory.\"},{\"question\":\"Which learning algorithms does the framework cover beyond prediction with expert advice?\",\"answer\":\"The calculus applies to Bayesian model averaging, optimistic online learning, contextual bandits, boosting, online convex optimization, repeated games, contextual decision-making, multiscale aggregation, and softmax preference learning.\"}]",1784177823,272,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"adaptive-bayes-exactly-tracks-information-over-intrinsic-time","",{"@graph":35,"@context":84},[36,53,67],{"@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/adaptive-bayes-exactly-tracks-information-over-intrinsic-time/82050/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What exact relationship does the paper establish for regret in Bayesian or multiplicative-weights updates?","Question",{"text":74,"@type":75},"It shows that the regret of such updates satisfies an exact information-accounting identity. Each round’s excess loss decomposes into an immediate uncertainty payment and a reduction in information distance to the comparator.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What does the paper mean by “intrinsic time”?",{"text":79,"@type":75},"The cumulative uncertainty payment over the realized sequence defines a pathwise uncertainty clock. This clock is called the intrinsic time of the realized trajectory.",{"name":81,"@type":72,"acceptedAnswer":82},"Which learning algorithms does the framework cover beyond prediction with expert advice?",{"text":83,"@type":75},"The calculus applies to Bayesian model averaging, optimistic online learning, contextual bandits, boosting, online convex optimization, repeated games, contextual decision-making, multiscale aggregation, and softmax preference learning.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]