{"id":"https://openalex.org/W3016883099","doi":"https://doi.org/10.1007/s11222-021-10016-8","title":"Analysis of stochastic gradient descent in continuous time","display_name":"Analysis of stochastic gradient descent in continuous time","publication_year":2021,"publication_date":"2021-05-09","ids":{"openalex":"https://openalex.org/W3016883099","doi":"https://doi.org/10.1007/s11222-021-10016-8","mag":"3016883099"},"language":"en","primary_location":{"id":"doi:10.1007/s11222-021-10016-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-021-10016-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-021-10016-8.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11222-021-10016-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026110503","display_name":"Jonas Latz","orcid":"https://orcid.org/0000-0002-4600-0247"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Jonas Latz","raw_affiliation_strings":["Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK"],"raw_orcid":"https://orcid.org/0000-0002-4600-0247","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5026110503"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":{"value":2690,"currency":"USD","value_usd":2690},"apc_paid":{"value":2690,"currency":"USD","value_usd":2690},"fwci":2.6053,"has_fulltext":true,"cited_by_count":33,"citation_normalized_percentile":{"value":0.9097644,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":95,"max":99},"biblio":{"volume":"31","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9998000264167786,"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":0.9998000264167786,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T10136","display_name":"Statistical Methods and Inference","score":0.9954000115394592,"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":0.6989095211029053},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6344842910766602},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6335437893867493},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.4884118437767029},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.4882541000843048},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4832080602645874},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.47961506247520447},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.4496751129627228},{"id":"https://openalex.org/keywords/markov-process","display_name":"Markov process","score":0.4183151423931122},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3980652391910553},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30833321809768677},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.23271912336349487},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08856305480003357}],"concepts":[{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.6989095211029053},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6344842910766602},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6335437893867493},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.4884118437767029},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.4882541000843048},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4832080602645874},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.47961506247520447},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.4496751129627228},{"id":"https://openalex.org/C159886148","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov process","level":2,"score":0.4183151423931122},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3980652391910553},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30833321809768677},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.23271912336349487},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08856305480003357},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s11222-021-10016-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-021-10016-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-021-10016-8.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2004.07177","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2004.07177","pdf_url":"https://arxiv.org/pdf/2004.07177","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,"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":"pmh:oai:www.repository.cam.ac.uk:1810/325629","is_oa":true,"landing_page_url":"https://www.repository.cam.ac.uk/handle/1810/325629","pdf_url":"https://www.repository.cam.ac.uk/handle/1810/325629","source":{"id":"https://openalex.org/S4306401777","display_name":"Apollo (University of Cambridge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I241749","host_organization_name":"University of Cambridge","host_organization_lineage":["https://openalex.org/I241749"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"pmh:oai:www.repository.cam.ac.uk:1810/322164","is_oa":true,"landing_page_url":"https://www.repository.cam.ac.uk/handle/1810/322164","pdf_url":null,"source":{"id":"https://openalex.org/S4306401777","display_name":"Apollo (University of Cambridge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I241749","host_organization_name":"University of Cambridge","host_organization_lineage":["https://openalex.org/I241749"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"doi:10.17863/cam.69623","is_oa":true,"landing_page_url":"https://doi.org/10.17863/cam.69623","pdf_url":null,"source":{"id":"https://openalex.org/S7407050737","display_name":"Apollo","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.17863/cam.73086","is_oa":true,"landing_page_url":"https://doi.org/10.17863/cam.73086","pdf_url":null,"source":{"id":"https://openalex.org/S7407050737","display_name":"Apollo","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"JournalArticle"}],"best_oa_location":{"id":"doi:10.1007/s11222-021-10016-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-021-10016-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-021-10016-8.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2978676218","display_name":null,"funder_award_id":"EP/S026045/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G763414266","display_name":"PET++: Improving Localisation, Diagnosis and Quantification in Clinical and Medical PET Imaging with Randomised Optimisation","funder_award_id":"EP/S026045/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3016883099.pdf","grobid_xml":"https://content.openalex.org/works/W3016883099.grobid-xml"},"referenced_works_count":105,"referenced_works":["https://openalex.org/W48288425","https://openalex.org/W83275800","https://openalex.org/W91439453","https://openalex.org/W581347801","https://openalex.org/W634746429","https://openalex.org/W1491706803","https://openalex.org/W1546689893","https://openalex.org/W1568229137","https://openalex.org/W1791038712","https://openalex.org/W1821151678","https://openalex.org/W1899249567","https://openalex.org/W1906471536","https://openalex.org/W1989303055","https://openalex.org/W1992208280","https://openalex.org/W1992546716","https://openalex.org/W1994616650","https://openalex.org/W2005308357","https://openalex.org/W2043487490","https://openalex.org/W2049435734","https://openalex.org/W2072356249","https://openalex.org/W2073750241","https://openalex.org/W2079925354","https://openalex.org/W2082404215","https://openalex.org/W2092911205","https://openalex.org/W2092922447","https://openalex.org/W2107438106","https://openalex.org/W2108566619","https://openalex.org/W2116614622","https://openalex.org/W2143706916","https://openalex.org/W2146502635","https://openalex.org/W2149600993","https://openalex.org/W2155418451","https://openalex.org/W2163715525","https://openalex.org/W2167433878","https://openalex.org/W2171032649","https://openalex.org/W2171957785","https://openalex.org/W2259695933","https://openalex.org/W2288292100","https://openalex.org/W2320795822","https://openalex.org/W2473295534","https://openalex.org/W2488096466","https://openalex.org/W2550039210","https://openalex.org/W2557207432","https://openalex.org/W2625685405","https://openalex.org/W2738599156","https://openalex.org/W2780299545","https://openalex.org/W2786773456","https://openalex.org/W2790118436","https://openalex.org/W2796146910","https://openalex.org/W2798710209","https://openalex.org/W2802739963","https://openalex.org/W2884512178","https://openalex.org/W2885366791","https://openalex.org/W2891433050","https://openalex.org/W2898866127","https://openalex.org/W2912205739","https://openalex.org/W2949400887","https://openalex.org/W2949432630","https://openalex.org/W2951563002","https://openalex.org/W2951601728","https://openalex.org/W2951817256","https://openalex.org/W2952215077","https://openalex.org/W2962798807","https://openalex.org/W2962915600","https://openalex.org/W2963156201","https://openalex.org/W2963266525","https://openalex.org/W2963398636","https://openalex.org/W2963403654","https://openalex.org/W2963550745","https://openalex.org/W2963613486","https://openalex.org/W2964072432","https://openalex.org/W2964089856","https://openalex.org/W2964286272","https://openalex.org/W2982286654","https://openalex.org/W2995058356","https://openalex.org/W3010526759","https://openalex.org/W3029645440","https://openalex.org/W3033360065","https://openalex.org/W3043072433","https://openalex.org/W3100431062","https://openalex.org/W3104122450","https://openalex.org/W3104193409","https://openalex.org/W3104370808","https://openalex.org/W3105711964","https://openalex.org/W3106453976","https://openalex.org/W3125747381","https://openalex.org/W3163923442","https://openalex.org/W4236501602","https://openalex.org/W4242615509","https://openalex.org/W4247188587","https://openalex.org/W4247619688","https://openalex.org/W4249927918","https://openalex.org/W4255683503","https://openalex.org/W4256522009","https://openalex.org/W4256685026","https://openalex.org/W4289147292","https://openalex.org/W4289753520","https://openalex.org/W4299436927","https://openalex.org/W4299533062","https://openalex.org/W4301909677","https://openalex.org/W6600529748","https://openalex.org/W6610515285","https://openalex.org/W6632807815","https://openalex.org/W6633236887","https://openalex.org/W6633794997"],"related_works":["https://openalex.org/W4287028882","https://openalex.org/W3192094705","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"],"abstract_inverted_index":{"Abstract":[0],"Stochastic":[1],"gradient":[2,11,27,35,39,62,65,104,114,136,208],"descent":[3,12,66],"is":[4,41,46,142],"an":[5],"optimisation":[6],"method":[7],"that":[8,45,108],"combines":[9],"classical":[10],"with":[13,48,116,138],"random":[14,77],"subsampling":[15],"within":[16],"the":[17,25,64,68,71,76,102,113,119,124,134,146,152,155,164,174,179,184,188,195,206],"target":[18,121,166,190],"functional.":[19],"In":[20,171],"this":[21,81,172],"work,":[22],"we":[23,97,150],"introduce":[24],"stochastic":[26,34,38,103,135,207],"process":[28,40,52,69,137,175,209],"as":[29,123],"a":[30,42,49,55,200],"continuous-time":[31,50],"representation":[32],"of":[33,80,101,187,194,202],"descent.":[36],"The":[37,59],"dynamical":[43,60],"system":[44],"coupled":[47],"Markov":[51],"living":[53],"on":[54,70],"finite":[56,72],"state":[57,73],"space.":[58],"system\u2014a":[61],"flow\u2014represents":[63],"part,":[67],"space":[74],"represents":[75],"subsampling.":[78],"Processes":[79],"type":[82],"are,":[83],"for":[84,205],"instance,":[85],"used":[86],"to":[87,112,118,159,178],"model":[88],"clonal":[89],"populations":[90],"in":[91,145,183],"fluctuating":[92],"environments.":[93],"After":[94],"introducing":[95],"it,":[96],"study":[98,151],"theoretical":[99],"properties":[100],"process:":[105],"We":[106,129,197],"show":[107],"it":[109],"converges":[110,176],"weakly":[111,177],"flow":[115],"respect":[117],"full":[120,189],"function,":[122],"learning":[125,140,156],"rate":[126,141,157],"approaches":[127],"zero.":[128],"give":[130],"conditions":[131],"under":[132],"which":[133],"constant":[139],"exponentially":[143],"ergodic":[144],"Wasserstein":[147],"sense.":[148],"Then":[149],"case,":[153,173],"where":[154],"goes":[158],"zero":[160],"sufficiently":[161],"slowly":[162],"and":[163,210],"single":[165],"functions":[167],"are":[168],"strongly":[169],"convex.":[170],"point":[180],"mass":[181],"concentrated":[182],"global":[185],"minimum":[186],"function;":[191],"indicating":[192],"consistency":[193],"method.":[196],"conclude":[198],"after":[199],"discussion":[201],"discretisation":[203],"strategies":[204],"numerical":[211],"experiments.":[212]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}
