{"id":"https://openalex.org/W2963518130","doi":"https://doi.org/10.1073/pnas.1903070116","title":"Reconciling modern machine-learning practice and the classical bias\u2013variance trade-off","display_name":"Reconciling modern machine-learning practice and the classical bias\u2013variance trade-off","publication_year":2019,"publication_date":"2019-07-24","ids":{"openalex":"https://openalex.org/W2963518130","doi":"https://doi.org/10.1073/pnas.1903070116","mag":"2963518130","pmid":"https://pubmed.ncbi.nlm.nih.gov/31341078","pmcid":"https://www.ncbi.nlm.nih.gov/pmc/articles/6689936"},"language":"en","primary_location":{"id":"doi:10.1073/pnas.1903070116","doi":"https://doi.org/10.1073/pnas.1903070116","is_oa":false,"landing_page_url":"https://doi.org/10.1073/pnas.1903070116","pdf_url":null,"source":{"id":"https://openalex.org/S125754415","display_name":"Proceedings of the National Academy of Sciences","issn_l":"0027-8424","issn":["0027-8424","1091-6490"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-a","doyens","erih-plus","jufo-3","ki-jl-3","medline","norway-2","poland-200","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320052","host_organization_name":"National Academy of Sciences","host_organization_lineage":["https://openalex.org/P4310320052"],"host_organization_lineage_names":["National Academy of Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the National Academy of Sciences","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["arxiv","crossref","datacite","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6689936","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102796459","display_name":"Mikhail Belkin","orcid":"https://orcid.org/0009-0005-8633-3926","observed_orcids":["https://orcid.org/0009-0005-8633-3926"]},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mikhail Belkin","raw_affiliation_strings":["Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;","Department of Statistics, The Ohio State University, Columbus, OH 43210;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;","institution_ids":["https://openalex.org/I52357470"]},{"raw_affiliation_string":"Department of Statistics, The Ohio State University, Columbus, OH 43210;","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061246300","display_name":"Daniel Hsu","orcid":"https://orcid.org/0000-0002-3495-7113","observed_orcids":["https://orcid.org/0000-0002-3495-7113"]},"institutions":[{"id":"https://openalex.org/I78577930","display_name":"Columbia University","ror":"https://ror.org/00hj8s172","country_code":"US","type":"education","lineage":["https://openalex.org/I78577930"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Daniel Hsu","raw_affiliation_strings":["Computer Science Department and Data Science Institute, Columbia University, New York, NY 10027"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Science Department and Data Science Institute, Columbia University, New York, NY 10027","institution_ids":["https://openalex.org/I78577930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032794241","display_name":"Siyuan Ma","orcid":"https://orcid.org/0000-0002-9207-1062","observed_orcids":["https://orcid.org/0000-0002-9207-1062","https://orcid.org/0009-0002-6673-5002","https://orcid.org/0009-0004-6569-6093"]},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Siyuan Ma","raw_affiliation_strings":["Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"last","author":{"id":null,"display_name":"Soumik Mandal","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Soumik Mandal","raw_affiliation_strings":["Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210;","institution_ids":["https://openalex.org/I52357470"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":80.9921,"has_fulltext":true,"cited_by_count":1956,"citation_normalized_percentile":{"value":0.9994844,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"116","issue":"32","first_page":"15849","last_page":"15854"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.3984000086784363,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.3984000086784363,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.28929999470710754,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12535","display_name":"Machine Learning and Data Classification","score":0.08389999717473984,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/double-descent","display_name":"double descent","score":0.8220000267028809},{"id":"https://openalex.org/keywords/bias-variance-trade-off","display_name":"bias-variance trade-off","score":0.9980000257492065},{"id":"https://openalex.org/keywords/overfitting","display_name":"overfitting","score":0.8029999732971191},{"id":"https://openalex.org/keywords/underfitting","display_name":"underfitting","score":0.7820000052452087},{"id":"https://openalex.org/keywords/neural-network","display_name":"artificial neural networks","score":0.6880000233650208},{"id":"https://openalex.org/keywords/interpolation","display_name":"interpolation","score":0.7379999756813049},{"id":"https://openalex.org/keywords/model-capacity","display_name":"model capacity","score":0.949999988079071},{"id":"https://openalex.org/keywords/generalization","display_name":"generalization","score":0.7129999995231628},{"id":"https://openalex.org/keywords/performance-curves","display_name":"performance curves","score":0.4189999997615814},{"id":"https://openalex.org/keywords/overparameterized-models","display_name":"overparameterized models","score":0.4830000102519989}],"concepts":[{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.5892517566680908},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.4546126127243042},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.44328737258911133},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.409856379032135},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.36701834201812744},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.35424476861953735},{"id":"https://openalex.org/C121955636","wikidata":"https://www.wikidata.org/wiki/Q4116214","display_name":"Accounting","level":1,"score":0.07283151149749756}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1073/pnas.1903070116","doi":"https://doi.org/10.1073/pnas.1903070116","is_oa":false,"landing_page_url":"https://doi.org/10.1073/pnas.1903070116","pdf_url":null,"source":{"id":"https://openalex.org/S125754415","display_name":"Proceedings of the National Academy of Sciences","issn_l":"0027-8424","issn":["0027-8424","1091-6490"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-a","doyens","erih-plus","jufo-3","ki-jl-3","medline","norway-2","poland-200","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320052","host_organization_name":"National Academy of Sciences","host_organization_lineage":["https://openalex.org/P4310320052"],"host_organization_lineage_names":["National Academy of Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the National Academy of Sciences","raw_type":"journal-article"},{"id":"pmh:oai:pubmedcentral.nih.gov:6689936","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6689936","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proc Natl Acad Sci U S A","raw_type":"Text"},{"id":"pmid:31341078","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/31341078","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the National Academy of Sciences of the United States of America","raw_type":"Journal Article"},{"id":"pmh:oai:arXiv.org:1812.11118","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1812.11118","pdf_url":"https://arxiv.org/pdf/1812.11118","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"mag:2907127169","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/1812.11118v1","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"arXiv (Cornell University)","raw_type":null},{"id":"mag:2972810859","doi":null,"is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1812.11118","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1812.11118","doi":"https://doi.org/10.48550/arxiv.1812.11118","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1812.11118","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:pubmedcentral.nih.gov:6689936","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6689936","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proc Natl Acad Sci U S A","raw_type":"Text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G1302751805","display_name":"RI: Small: Learning discrete structure from continuous spaces","funder_award_id":"1815697","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3974885098","display_name":"TRIPODS: From Foundations to Practice of Data Science and Back","funder_award_id":"1740833","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G63815674","display_name":null,"funder_award_id":"FG-2016-6295","funder_id":"https://openalex.org/F4320306151","funder_display_name":"Alfred P. Sloan Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306151","display_name":"Alfred P. Sloan Foundation","ror":"https://ror.org/052csg198"},{"id":"https://openalex.org/F4320309480","display_name":"Nvidia","ror":"https://ror.org/03jdj4y14"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963518130.pdf","grobid_xml":"https://content.openalex.org/works/W2963518130.grobid-xml"},"referenced_works_count":46,"referenced_works":["https://openalex.org/W1480376833","https://openalex.org/W1486164486","https://openalex.org/W1493526108","https://openalex.org/W1502922572","https://openalex.org/W1533861849","https://openalex.org/W1571975558","https://openalex.org/W1678356000","https://openalex.org/W1975846642","https://openalex.org/W1996908140","https://openalex.org/W2070902649","https://openalex.org/W2076118331","https://openalex.org/W2087347434","https://openalex.org/W2088883866","https://openalex.org/W2099579348","https://openalex.org/W2112796928","https://openalex.org/W2117539524","https://openalex.org/W2144902422","https://openalex.org/W2156909104","https://openalex.org/W2157581667","https://openalex.org/W2166193626","https://openalex.org/W2167608136","https://openalex.org/W2250539671","https://openalex.org/W2335728318","https://openalex.org/W2419597278","https://openalex.org/W2542136887","https://openalex.org/W2566079294","https://openalex.org/W2612690371","https://openalex.org/W2618069152","https://openalex.org/W2739160792","https://openalex.org/W2763894180","https://openalex.org/W2787348029","https://openalex.org/W2789048837","https://openalex.org/W2810365967","https://openalex.org/W2896549049","https://openalex.org/W2897025488","https://openalex.org/W2899476926","https://openalex.org/W2911964244","https://openalex.org/W2950220847","https://openalex.org/W2962712496","https://openalex.org/W2963013450","https://openalex.org/W2963742538","https://openalex.org/W2964034630","https://openalex.org/W2964161291","https://openalex.org/W2990138404","https://openalex.org/W3118608800","https://openalex.org/W4212774754"],"related_works":["https://openalex.org/W2907127169","https://openalex.org/W4309590236","https://openalex.org/W6910535334","https://openalex.org/W4392933226","https://openalex.org/W3023272737","https://openalex.org/W3196464222","https://openalex.org/W4382567306","https://openalex.org/W3008906732","https://openalex.org/W7204754271","https://openalex.org/W7108665889"],"abstract_inverted_index":{"Breakthroughs":[0],"in":[1,45,67,78,171],"machine":[2,125,229],"learning":[3,126],"are":[4,88],"rapidly":[5],"changing":[6],"science":[7],"and":[8,59,69,104,127,140,180,191,193,206,218,225],"society,":[9],"yet":[10,105],"our":[11],"fundamental":[12],"understanding":[13,139],"of":[14,23,27,42,124,168,182,189,209,215,228],"this":[15,133],"technology":[16],"has":[17,117,219],"lagged":[18],"far":[19],"behind.":[20],"Indeed,":[21],"one":[22],"the":[24,28,30,39,95,121,137,141,153,166,178,204,207,213,223,226],"central":[25],"tenets":[26],"field,":[29],"bias-variance":[31,50,156],"trade-off,":[32],"appears":[33],"to":[34,63,72,90,130],"be":[35,101],"at":[36],"odds":[37],"with":[38],"observed":[40],"behavior":[41],"methods":[43],"used":[44],"modern":[46,79,142],"machine-learning":[47,210],"practice.":[48],"The":[49],"trade-off":[51,157],"implies":[52],"that":[53],"a":[54,145,186,196],"model":[55,163],"should":[56],"balance":[57],"underfitting":[58],"overfitting:":[60],"Rich":[61],"enough":[62,71],"express":[64],"underlying":[65],"structure":[66,208],"data":[68],"simple":[70],"avoid":[73],"fitting":[74],"spurious":[75],"patterns.":[76],"However,":[77],"practice,":[80],"very":[81],"rich":[82],"models":[83,99,190,211],"such":[84,98],"as":[85],"neural":[86],"networks":[87],"trained":[89],"exactly":[91],"fit":[92],"(i.e.,":[93],"interpolate)":[94],"data.":[96,113],"Classically,":[97],"would":[100],"considered":[102],"overfitted,":[103],"they":[106],"often":[107],"obtain":[108],"high":[109],"accuracy":[110],"on":[111],"test":[112],"This":[114,149,201],"apparent":[115],"contradiction":[116],"raised":[118],"questions":[119],"about":[120],"mathematical":[122],"foundations":[123],"their":[128],"relevance":[129],"practitioners.":[131],"In":[132],"paper,":[134],"we":[135,194],"reconcile":[136],"classical":[138,216],"practice":[143,227],"within":[144],"unified":[146],"performance":[147,205],"curve.":[148],"\"double-descent\"":[150],"curve":[151,158],"subsumes":[152],"textbook":[154],"U-shaped":[155],"by":[159],"showing":[160],"how":[161],"increasing":[162],"capacity":[164],"beyond":[165],"point":[167],"interpolation":[169],"results":[170],"improved":[172],"performance.":[173],"We":[174],"provide":[175],"evidence":[176],"for":[177,185,198,221],"existence":[179],"ubiquity":[181],"double":[183],"descent":[184],"wide":[187],"spectrum":[188],"datasets,":[192],"posit":[195],"mechanism":[197],"its":[199],"emergence.":[200],"connection":[202],"between":[203],"delineates":[212],"limits":[214],"analyses":[217],"implications":[220],"both":[222],"theory":[224],"learning.":[230]},"counts_by_year":[{"year":2026,"cited_by_count":308},{"year":2025,"cited_by_count":320},{"year":2024,"cited_by_count":258},{"year":2023,"cited_by_count":257},{"year":2022,"cited_by_count":264},{"year":2021,"cited_by_count":261},{"year":2020,"cited_by_count":241},{"year":2019,"cited_by_count":39},{"year":2018,"cited_by_count":8}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2019-07-30T00:00:00"}