{"id":"https://openalex.org/W4396677240","doi":"https://doi.org/10.1137/23m1594388","title":"Rigorous Dynamical Mean-Field Theory for Stochastic Gradient Descent Methods","display_name":"Rigorous Dynamical Mean-Field Theory for Stochastic Gradient Descent Methods","publication_year":2024,"publication_date":"2024-05-06","ids":{"openalex":"https://openalex.org/W4396677240","doi":"https://doi.org/10.1137/23m1594388"},"language":"en","primary_location":{"id":"doi:10.1137/23m1594388","is_oa":true,"landing_page_url":"https://doi.org/10.1137/23m1594388","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1137/23m1594388","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076357441","display_name":"C\u00e9dric Gerbelot","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"C\u00e9dric Gerbelot","raw_affiliation_strings":["Laboratoire de Physique de l\u2019Ecole Normale Sup\u00e9rieure, Universit\u00e9 PSL, Paris, France, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA"],"affiliations":[{"raw_affiliation_string":"Laboratoire de Physique de l\u2019Ecole Normale Sup\u00e9rieure, Universit\u00e9 PSL, Paris, France, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012 USA","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5044679630","display_name":"Emanuele Troiani","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Emanuele Troiani","raw_affiliation_strings":["Statistical Physics of Computation Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France"],"affiliations":[{"raw_affiliation_string":"Statistical Physics of Computation Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5054525547","display_name":"Francesca Mignacco","orcid":null},"institutions":[{"id":"https://openalex.org/I277688954","display_name":"Universit\u00e9 Paris-Saclay","ror":"https://ror.org/03xjwb503","country_code":"FR","type":"education","lineage":["https://openalex.org/I277688954"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I2738703131","display_name":"Commissariat \u00e0 l'\u00c9nergie Atomique et aux \u00c9nergies Alternatives","ror":"https://ror.org/00jjx8s55","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131"]},{"id":"https://openalex.org/I4210163633","display_name":"Institut de Physique Th\u00e9orique","ror":"https://ror.org/058rvd314","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I2738703131","https://openalex.org/I2738703131","https://openalex.org/I4210098836","https://openalex.org/I4210113668","https://openalex.org/I4210163633"]},{"id":"https://openalex.org/I4210128565","display_name":"CEA Paris-Saclay","ror":"https://ror.org/03n15ch10","country_code":"FR","type":"government","lineage":["https://openalex.org/I2738703131","https://openalex.org/I277688954","https://openalex.org/I4210128565"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Francesca Mignacco","raw_affiliation_strings":["Universit\u00e9 Paris-Saclay, CNRS, CEA, Institut de physique th\u00e9orique, 91191, Gif-sur-Yvette, France"],"affiliations":[{"raw_affiliation_string":"Universit\u00e9 Paris-Saclay, CNRS, CEA, Institut de physique th\u00e9orique, 91191, Gif-sur-Yvette, France","institution_ids":["https://openalex.org/I4210163633","https://openalex.org/I277688954","https://openalex.org/I1294671590","https://openalex.org/I4210128565","https://openalex.org/I2738703131"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068236230","display_name":"Florent Krz\u0105ka\u0142a","orcid":"https://orcid.org/0000-0003-2313-2578"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Florent Krzakala","raw_affiliation_strings":["Information, Learning and Physics Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France"],"affiliations":[{"raw_affiliation_string":"Information, Learning and Physics Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089268172","display_name":"Lenka Zdeborov\u00e1","orcid":"https://orcid.org/0000-0002-8377-3978"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lenka Zdeborov\u00e1","raw_affiliation_strings":["Statistical Physics of Computation Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France"],"affiliations":[{"raw_affiliation_string":"Statistical Physics of Computation Lab, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), Lausanne, France","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5076357441"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.732,"has_fulltext":true,"cited_by_count":11,"citation_normalized_percentile":{"value":0.9377287,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"6","issue":"2","first_page":"400","last_page":"427"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9993000030517578,"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":0.9993000030517578,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.6895261406898499},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.6054120063781738},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.5997788906097412},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.5420300960540771},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.5361843705177307},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5301024913787842},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5277676582336426},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.4872118830680847},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.4776521921157837},{"id":"https://openalex.org/keywords/acceleration","display_name":"Acceleration","score":0.4560243785381317},{"id":"https://openalex.org/keywords/dynamical-systems-theory","display_name":"Dynamical systems theory","score":0.4220716655254364},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.39577096700668335},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3304895758628845},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3214951753616333},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22406291961669922},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1491318643093109},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.0825757384300232},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.07415905594825745}],"concepts":[{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.6895261406898499},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.6054120063781738},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5997788906097412},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5420300960540771},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5361843705177307},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5301024913787842},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5277676582336426},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.4872118830680847},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.4776521921157837},{"id":"https://openalex.org/C117896860","wikidata":"https://www.wikidata.org/wiki/Q11376","display_name":"Acceleration","level":2,"score":0.4560243785381317},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.4220716655254364},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.39577096700668335},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3304895758628845},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3214951753616333},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22406291961669922},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1491318643093109},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0825757384300232},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.07415905594825745},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/23m1594388","is_oa":true,"landing_page_url":"https://doi.org/10.1137/23m1594388","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},{"id":"pmh:oai:infoscience.epfl.ch:20.500.14299/240445","is_oa":true,"landing_page_url":"https://infoscience.epfl.ch/handle/20.500.14299/240445","pdf_url":"https://infoscience.epfl.ch/bitstreams/95e09338-c165-4de3-bf38-c8fe1b9d9e3b/download","source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"other"}],"best_oa_location":{"id":"doi:10.1137/23m1594388","is_oa":true,"landing_page_url":"https://doi.org/10.1137/23m1594388","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2365339206","display_name":null,"funder_award_id":"200390","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G3406941144","display_name":null,"funder_award_id":"714608-SMiLe","funder_id":"https://openalex.org/F4320332999","funder_display_name":"Horizon 2020 Framework Programme"},{"id":"https://openalex.org/G3694363619","display_name":"Statistical Mechanics of Learning","funder_award_id":"714608","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G472782874","display_name":"Die Sorgfaltspflicht des Bankiers bei der Verm\u00f6- gensverwaltung nach amerikanischem Recht unter Be- r\u00fccksichtigung der Ausgestaltung in der Schweiz.","funder_award_id":"20039","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G4836148144","display_name":null,"funder_award_id":"20002","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G5790682227","display_name":null,"funder_award_id":"714608-SMiLe","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G7567215475","display_name":null,"funder_award_id":"200021_200390","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G8318064016","display_name":null,"funder_award_id":"Horizon","funder_id":"https://openalex.org/F4320320300","funder_display_name":"European Commission"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G993971353","display_name":null,"funder_award_id":"200021","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"},{"id":"https://openalex.org/F4320332999","display_name":"Horizon 2020 Framework Programme","ror":"https://ror.org/00k4n6c32"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W173781251","https://openalex.org/W1415278202","https://openalex.org/W1498436455","https://openalex.org/W1977011475","https://openalex.org/W1979344292","https://openalex.org/W1988150410","https://openalex.org/W1988720110","https://openalex.org/W2001170284","https://openalex.org/W2004602215","https://openalex.org/W2008348370","https://openalex.org/W2026896061","https://openalex.org/W2064512219","https://openalex.org/W2066424095","https://openalex.org/W2067038358","https://openalex.org/W2087327526","https://openalex.org/W2610525766","https://openalex.org/W2610971674","https://openalex.org/W2766016303","https://openalex.org/W2914082785","https://openalex.org/W2963064298","https://openalex.org/W2963072996","https://openalex.org/W2963886912","https://openalex.org/W3007724531","https://openalex.org/W3098848552","https://openalex.org/W3104075626","https://openalex.org/W3104358493","https://openalex.org/W3105008261","https://openalex.org/W3105463807","https://openalex.org/W3167874995","https://openalex.org/W3169803339","https://openalex.org/W3214314869","https://openalex.org/W4212774754","https://openalex.org/W4220685512","https://openalex.org/W4293168170","https://openalex.org/W4360879278"],"related_works":["https://openalex.org/W4287028882","https://openalex.org/W4206903459","https://openalex.org/W2754816816","https://openalex.org/W2891466145","https://openalex.org/W4366280654","https://openalex.org/W4309201680","https://openalex.org/W3160167280","https://openalex.org/W4387963920","https://openalex.org/W4231621013","https://openalex.org/W4362706668"],"abstract_inverted_index":{"We":[0],"prove":[1],"closed-form":[2],"equations":[3,49,60,110],"for":[4,111],"the":[5,54,67,88,109],"exact":[6],"high-dimensional":[7],"asymptotics":[8],"of":[9,12,56,81,108],"a":[10],"family":[11],"first-order":[13],"gradient-based":[14],"methods,":[15],"learning":[16,120],"an":[17,78],"estimator":[18],"(e.g.,":[19],"M-estimator,":[20],"shallow":[21],"neural":[22],"network)":[23],"from":[24,53,61],"observations":[25],"on":[26],"Gaussian":[27],"data":[28],"with":[29,99,113],"empirical":[30],"risk":[31],"minimization.":[32],"This":[33],"includes":[34],"widely":[35],"used":[36],"algorithms":[37],"such":[38],"as":[39],"stochastic":[40],"gradient":[41,69],"descent":[42],"(SGD)":[43],"or":[44],"Nesterov":[45],"acceleration.":[46],"The":[47],"obtained":[48],"match":[50],"those":[51],"resulting":[52],"discretization":[55],"dynamical":[57],"mean-field":[58],"theory":[59],"statistical":[62],"physics":[63],"when":[64],"applied":[65],"to":[66,76,92],"corresponding":[68],"flow.":[70],"Our":[71],"proof":[72],"method":[73],"allows":[74],"us":[75],"give":[77],"explicit":[79],"description":[80],"how":[82],"memory":[83],"kernels":[84],"build":[85],"up":[86],"in":[87],"effective":[89],"dynamics":[90],"and":[91,118],"include":[93],"nonseparable":[94],"update":[95],"functions,":[96],"allowing":[97],"datasets":[98],"nonidentity":[100],"covariance":[101],"matrices.":[102],"Finally,":[103],"we":[104],"provide":[105],"numerical":[106],"implementations":[107],"SGD":[112],"generic":[114],"extensive":[115],"batch":[116],"size":[117],"constant":[119],"rates.":[121]},"counts_by_year":[{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":4}],"updated_date":"2026-04-13T07:58:08.660418","created_date":"2025-10-10T00:00:00"}
