[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85462-en":3,"doc-seo-85462-105":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},85462,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Online Conformal Inference with Retrospective Adjustment for Faster Adaptation to Distribution Shift","Conformal prediction constructs distribution-free prediction sets with guaranteed coverage under the exchangeability assumption, but online settings often violate it when data distributions evolve. Prior online methods typically adapt slowly because they update only forward-looking predictions while leaving historical ones unchanged. This paper introduces an online conformal inference approach using retrospective adjustment: regression-based efficient leave-one-out updates retroactively revise past predictions as new data arrive. Experiments on synthetic and real-world data show coverage near nominal and interval-width reduction up to about 30%, improving statistical efficiency and speeding adaptation.","arXiv :2511 .04275v2 [ stat .ML] 13 Jul 2026  \nOnline Conformal Inference with Retrospective Adjustment for Faster Adaptation to Distribution Shift  \nJungbin Jun and Ilsang Ohn∗  \nDepartment of Statistics, Inha University  \nJuly 14, 2026  \nAbstract  \nConformal prediction has emerged as a powerful framework for constructing distribution-free prediction sets with guaranteed coverage assuming only the exchangeability assumption. However, this assumption is often violated in online environments where data distributions evolve over time. Several recent approaches have been proposed to address this limitation, but, typically, they slowly adapt to distribution shifts because they update predictions only in a forward manner, that is, they generate a prediction for a newly observed data point while previously computed predictions are not updated. In this paper, we propose a novel online conformal inference method with retrospective adjustment, which is designed to achieve faster adaptation to distributional shifts. Our method leverages regression approaches with efficient leave-one-out update formulas to retroactively adjust past predictions when new data arrive, thereby aligning the entire set of predictions with the mostrecent data distribution. Through extensive numerical studies performed on both synthetic and realworld data sets, we show that the proposed approach achieves coverage close to the nominal level while reducing predictive interval width by up to approximately 30% compared to existing online conformal prediction methods, demonstrating improved statistical efficiency alongside faster adaptation.  \n1 Introduction  \nIn this paper, we consider a task of quantifying the uncertainty around prediction in an online learning setup, where our aim is, for each time t = 1 , 2 , . . . , to construct a prediction set for the target output Yt+1 ∈ R associated with a feature vector Xt+1 ∈ Rd by using the information of the previously observed data Dt := { (Xi , Yi)}i=1 , ...,t. Specifically, for a specified target miscoverage level α ∈ (0 , 1), we wish to construct a set-valued statistic ˆCt+1 : Rd 7→ 2R depending on Dt and Xt+1, which guarantees P (Yt+1 ∈ tr+ed1i)c≥ 1tionsαan. TbehesusetbstˆCantt+1ial isinreferredmoderntomaascha 1 −ine leαarpredictioning applnicsaett foionsr, Yitt+is[1](1.es)[.es](1.es)Asesnttihae risl tokpofrovincideorvraelctid  \nand well-calibrated prediction sets to enable more robust and reliable decision-making.  \nConformal inference has gained its popularity for the construction of prediction sets, particularly due to its generality and broad applicability. Assuming only the exchangeability of the data, this offers a versatile framework that converts the outputs from any black-box prediction algorithm into a valid prediction set (Vovk et al., 2005; Shafer and Vovk, 2008; Lei et al., 2018) . However, applying conformal inference methods in an online learning setup presents significant challenges, as the exchangeability assumption on the data often fails in practice. Non-stationary time series data serve as illustrative examples that are frequently observed in both natural phenomena and economic contexts (Raza et al. , 2015; Liu et al., 2018) . Distribution shift is also common in modern data analysis (Zhang et al. , 2025, 2026; Wu et al., 2021), for instance, a credit scoring model trained on data from an older population may  \n∗ Corresponding author. Email: [ilsang.ohn@inha.ac.kr](ilsang.ohn@inha.ac.kr)  \nperform poorly when applied to a younger demographic, due to shifts in underlying data distribution (Hand and Henley, 1997) . When exchangeability no longer holds, standard conformal inference methods may fail to achieve the nominal coverage level (Barber et al., 2023; Gibbs and Cand`es, 2021) . To tackle this problem, a number of approaches have been proposed to extend conformal prediction to non-exchangeable and/or distribution-shifted data sets. Representative directions include covariate shift (Tibshirani e","cbCaiuxlU8z0FVKA","https://ap.wps.com/l/cbCaiuxlU8z0FVKA","pdf",974330,1,22,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Challenges of online conformal prediction under distribution shift\n## Retrospective adjustment approach","[{\"question\":\"What limitation of the exchangeability assumption motivates this work?\",\"answer\":\"In online environments, the data distribution changes over time, so exchangeability can fail in practice. This causes standard conformal methods to miss the nominal coverage level.\"},{\"question\":\"How does the proposed method achieve faster adaptation to distribution shift?\",\"answer\":\"It retrospectively adjusts past predictions when new data arrive, using regression estimators with efficient leave-one-out update formulas. This updates earlier outputs so residuals align with the latest distribution.\"},{\"question\":\"What performance improvements are reported in the paper’s experiments?\",\"answer\":\"Numerical studies on synthetic and real-world datasets show coverage close to the nominal level. The method also reduces predictive interval width by up to approximately 30% versus existing online conformal prediction methods.\"}]",1784203718,55,{"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},"online-conformal-inference-with-retrospective-adjustment-for-faster-adaptation-to-distribution-shift","",{"@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/online-conformal-inference-with-retrospective-adjustment-for-faster-adaptation-to-distribution-shift/85462/",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","2026-07-17","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 limitation of the exchangeability assumption motivates this work?","Question",{"text":75,"@type":76},"In online environments, the data distribution changes over time, so exchangeability can fail in practice. This causes standard conformal methods to miss the nominal coverage level.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method achieve faster adaptation to distribution shift?",{"text":80,"@type":76},"It retrospectively adjusts past predictions when new data arrive, using regression estimators with efficient leave-one-out update formulas. This updates earlier outputs so residuals align with the latest distribution.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance improvements are reported in the paper’s experiments?",{"text":84,"@type":76},"Numerical studies on synthetic and real-world datasets show coverage close to the nominal level. The method also reduces predictive interval width by up to approximately 30% versus existing online conformal prediction methods.","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"]