{"id":"https://openalex.org/W2087789467","doi":"https://doi.org/10.1109/tit.2014.2332531","title":"Online Learning as Stochastic Approximation of Regularization Paths: Optimality and Almost-Sure Convergence","display_name":"Online Learning as Stochastic Approximation of Regularization Paths: Optimality and Almost-Sure Convergence","publication_year":2014,"publication_date":"2014-06-24","ids":{"openalex":"https://openalex.org/W2087789467","doi":"https://doi.org/10.1109/tit.2014.2332531","mag":"2087789467"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2014.2332531","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2332531","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1103.5538","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054758400","display_name":"Pierre Tarr\u00e8s","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Pierre Tarres","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Oxford, U.K","Math. Institute, University of Oxford, Oxford, UK"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Oxford, U.K","institution_ids":["https://openalex.org/I40120149"]},{"raw_affiliation_string":"Math. Institute, University of Oxford, Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5051299066","display_name":"Yuan Yao","orcid":"https://orcid.org/0000-0001-5814-1162"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan Yao","raw_affiliation_strings":["School of Mathematical Sciences, LMAM-LMEQF-LMPR, Peking University, Beijing, China","LMAM-LMEQF-LMPR, Peking Univ., Beijing, China"],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, LMAM-LMEQF-LMPR, Peking University, Beijing, China","institution_ids":["https://openalex.org/I20231570"]},{"raw_affiliation_string":"LMAM-LMEQF-LMPR, Peking Univ., Beijing, China","institution_ids":["https://openalex.org/I20231570"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5054758400"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":null,"apc_paid":null,"fwci":6.7577,"has_fulltext":false,"cited_by_count":69,"citation_normalized_percentile":{"value":0.97342345,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"60","issue":"9","first_page":"5716","last_page":"5735"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11233","display_name":"Advanced Adaptive Filtering Techniques","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11233","display_name":"Advanced Adaptive Filtering Techniques","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7699100971221924},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"Reproducing kernel Hilbert space","score":0.647873044013977},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5969603061676025},{"id":"https://openalex.org/keywords/minimax","display_name":"Minimax","score":0.583779513835907},{"id":"https://openalex.org/keywords/stochastic-approximation","display_name":"Stochastic approximation","score":0.5783519148826599},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.5641988515853882},{"id":"https://openalex.org/keywords/martingale","display_name":"Martingale (probability theory)","score":0.5344998240470886},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.515910267829895},{"id":"https://openalex.org/keywords/sample-size-determination","display_name":"Sample size determination","score":0.45914244651794434},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4074747562408447},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.36690229177474976},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.24965280294418335},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17460092902183533},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16248926520347595},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10503745079040527},{"id":"https://openalex.org/keywords/asynchronous-communication","display_name":"Asynchronous communication","score":0.1007583737373352}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7699100971221924},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.647873044013977},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5969603061676025},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.583779513835907},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.5783519148826599},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.5641988515853882},{"id":"https://openalex.org/C48406656","wikidata":"https://www.wikidata.org/wiki/Q534112","display_name":"Martingale (probability theory)","level":2,"score":0.5344998240470886},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.515910267829895},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.45914244651794434},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4074747562408447},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.36690229177474976},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.24965280294418335},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17460092902183533},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16248926520347595},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10503745079040527},{"id":"https://openalex.org/C151319957","wikidata":"https://www.wikidata.org/wiki/Q752739","display_name":"Asynchronous communication","level":2,"score":0.1007583737373352},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tit.2014.2332531","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2332531","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1103.5538","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1103.5538","pdf_url":"https://arxiv.org/pdf/1103.5538","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:HAL:hal-01016561v1","is_oa":false,"landing_page_url":"https://hal.science/hal-01016561","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Transactions on Information Theory, 2014, ⟨10.1109/TIT.2014.2332531⟩","raw_type":"Journal articles"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-80431","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-80431","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1103.5538","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1103.5538","pdf_url":"https://arxiv.org/pdf/1103.5538","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[{"score":0.4399999976158142,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G307296080","display_name":null,"funder_award_id":"6137004","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5583891269","display_name":null,"funder_award_id":"2011CB809105","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G5761754475","display_name":null,"funder_award_id":"2012CB825501","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G6320015136","display_name":null,"funder_award_id":"61071157","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W649943522","https://openalex.org/W1484867920","https://openalex.org/W1491706803","https://openalex.org/W1499021337","https://openalex.org/W1510073064","https://openalex.org/W1565176583","https://openalex.org/W1570089119","https://openalex.org/W1602844557","https://openalex.org/W1970781863","https://openalex.org/W1986280275","https://openalex.org/W1994616650","https://openalex.org/W1996652862","https://openalex.org/W2000021956","https://openalex.org/W2009797711","https://openalex.org/W2012501405","https://openalex.org/W2022829242","https://openalex.org/W2037335644","https://openalex.org/W2047278710","https://openalex.org/W2047311948","https://openalex.org/W2052398345","https://openalex.org/W2057682568","https://openalex.org/W2059571389","https://openalex.org/W2063978378","https://openalex.org/W2073750241","https://openalex.org/W2096840748","https://openalex.org/W2108280195","https://openalex.org/W2110652811","https://openalex.org/W2118285149","https://openalex.org/W2135046866","https://openalex.org/W2142563104","https://openalex.org/W2146766088","https://openalex.org/W2150621701","https://openalex.org/W2156779765","https://openalex.org/W2165867509","https://openalex.org/W2461858179","https://openalex.org/W3105120000","https://openalex.org/W4230058301","https://openalex.org/W4231022371","https://openalex.org/W4255272544","https://openalex.org/W4298876635","https://openalex.org/W6662091678","https://openalex.org/W6683245038"],"related_works":["https://openalex.org/W2016058626","https://openalex.org/W2474724840","https://openalex.org/W56241044","https://openalex.org/W1907128005","https://openalex.org/W2514909387","https://openalex.org/W2120339885","https://openalex.org/W4300374091","https://openalex.org/W1490902169","https://openalex.org/W2032229253","https://openalex.org/W4246247437"],"abstract_inverted_index":{"In":[0,91],"this":[1,89],"paper,":[2],"an":[3,162],"online":[4],"learning":[5,119],"algorithm":[6],"is":[7,32,74],"proposed":[8],"as":[9,150],"sequential":[10],"stochastic":[11],"approximation":[12,126],"of":[13,44,51,139],"a":[14,48,112,151],"regularization":[15,132],"path":[16],"converging":[17],"to":[18,34,116],"the":[19,36,52,63,77,82,117,121,125,131,136,142,146,166,169],"regression":[20,64],"function":[21],"in":[22,76,88,104,124],"reproducing":[23],"kernel":[24],"Hilbert":[25,105],"spaces":[26],"(RKHSs).":[27],"We":[28],"show":[29],"that":[30,79],"it":[31,80],"possible":[33],"produce":[35],"best":[37],"known":[38],"strong":[39],"(RKHS":[40],"norm)":[41],"convergence":[42,72],"rate":[43,73],"batch":[45,118],"learning,":[46],"through":[47],"careful":[49],"choice":[50],"gain":[53],"or":[54],"step":[55],"size":[56],"sequences,":[57],"depending":[58],"on":[59,62],"regularity":[60],"assumptions":[61],"function.":[65],"The":[66,156],"corresponding":[67],"weak":[68],"(mean":[69],"square":[70],"distance)":[71],"optimal":[75,163],"sense":[78],"reaches":[81],"minimax":[83],"and":[84,127,141,168],"individual":[85],"lower":[86],"rates":[87,138,157],"paper.":[90],"both":[92],"cases,":[93],"we":[94,110],"deduce":[95],"almost":[96],"sure":[97],"convergence,":[98,140],"using":[99],"Bernstein-type":[100],"inequalities":[101],"for":[102],"martingales":[103],"spaces.":[106],"To":[107],"achieve":[108],"this,":[109],"develop":[111],"bias-variance":[113],"decomposition":[114],"similar":[115],"setting;":[120],"bias":[122,167],"consists":[123],"drift":[128],"errors":[129],"along":[130],"path,":[133],"which":[134],"display":[135],"same":[137],"variance":[143],"arises":[144],"from":[145],"sample":[147],"error":[148],"analyzed":[149],"(reverse)":[152],"martingale":[153],"difference":[154],"sequence.":[155],"above":[158],"are":[159],"obtained":[160],"by":[161],"tradeoff":[164],"between":[165],"variance.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":7},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2016-06-24T00:00:00"}