{"id":"https://openalex.org/W2940467104","doi":"https://doi.org/10.1137/19m128908x","title":"Making the Last Iterate of SGD Information Theoretically Optimal","display_name":"Making the Last Iterate of SGD Information Theoretically Optimal","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W2940467104","doi":"https://doi.org/10.1137/19m128908x","mag":"2940467104"},"language":"en","primary_location":{"id":"doi:10.1137/19m128908x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m128908x","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"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 Optimization","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1904.12443","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101633144","display_name":"Prateek Jain","orcid":"https://orcid.org/0000-0002-9233-5160"},"institutions":[{"id":"https://openalex.org/I4210164937","display_name":"Microsoft Research (United Kingdom)","ror":"https://ror.org/05k87vq12","country_code":"GB","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210164937"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Prateek Jain","raw_affiliation_strings":["(Microsoft)"],"affiliations":[{"raw_affiliation_string":"(Microsoft)","institution_ids":["https://openalex.org/I4210164937"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Dheeraj M. Nagaraj","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dheeraj M. Nagaraj","raw_affiliation_strings":["Massachusetts Institute Of Technology#TAB#"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute Of Technology#TAB#","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5007012613","display_name":"Praneeth Netrapalli","orcid":"https://orcid.org/0000-0003-3863-6162"},"institutions":[{"id":"https://openalex.org/I4210164937","display_name":"Microsoft Research (United Kingdom)","ror":"https://ror.org/05k87vq12","country_code":"GB","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210164937"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Praneeth Netrapalli","raw_affiliation_strings":["(Microsoft)"],"affiliations":[{"raw_affiliation_string":"(Microsoft)","institution_ids":["https://openalex.org/I4210164937"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101633144"],"corresponding_institution_ids":["https://openalex.org/I4210164937"],"apc_list":null,"apc_paid":null,"fwci":1.2597,"has_fulltext":true,"cited_by_count":18,"citation_normalized_percentile":{"value":0.83028065,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"31","issue":"2","first_page":"1108","last_page":"1130"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9998999834060669,"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":0.9998999834060669,"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/T12101","display_name":"Advanced Bandit Algorithms Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"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/subgradient-method","display_name":"Subgradient method","score":0.7782998085021973},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7116116285324097},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.7078851461410522},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6967594623565674},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.6112371683120728},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5750632882118225},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5621105432510376},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.520530104637146},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.43536296486854553},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.41623786091804504},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36763226985931396},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3280297815799713},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3119111657142639},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16656851768493652},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08722609281539917},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07121235132217407}],"concepts":[{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.7782998085021973},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7116116285324097},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.7078851461410522},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6967594623565674},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6112371683120728},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5750632882118225},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5621105432510376},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.520530104637146},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.43536296486854553},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.41623786091804504},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36763226985931396},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3280297815799713},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3119111657142639},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16656851768493652},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08722609281539917},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07121235132217407},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/19m128908x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m128908x","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"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 Optimization","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1904.12443","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.12443","pdf_url":"https://arxiv.org/pdf/1904.12443","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":"mag:2940467104","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1904.12443v2","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1904.12443","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1904.12443","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:3153737196","is_oa":false,"landing_page_url":"http://proceedings.mlr.press/v99/jain19a/jain19a.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306418075","display_name":"Conference on Learning Theory","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":"conference"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Conference on Learning Theory","raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1904.12443","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1904.12443","pdf_url":"https://arxiv.org/pdf/1904.12443","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":[],"awards":[{"id":"https://openalex.org/G4504108201","display_name":null,"funder_award_id":"N00014-17-1","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G4751570384","display_name":null,"funder_award_id":"4-17-1-","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G5137248134","display_name":null,"funder_award_id":"N00014-17-1-2147","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G5357900919","display_name":null,"funder_award_id":"N00014-17-1-21","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G6914117704","display_name":null,"funder_award_id":"N00014-17-1-214","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G8876996369","display_name":null,"funder_award_id":"N00014","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"}],"funders":[{"id":"https://openalex.org/F4320309369","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2940467104.pdf","grobid_xml":"https://content.openalex.org/works/W2940467104.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1505731132","https://openalex.org/W1516903196","https://openalex.org/W1994616650","https://openalex.org/W2086161653","https://openalex.org/W2091825929","https://openalex.org/W2109339818","https://openalex.org/W2112269233","https://openalex.org/W2125993116","https://openalex.org/W2138682935","https://openalex.org/W2148825261","https://openalex.org/W2151695970","https://openalex.org/W2154682027","https://openalex.org/W2398337421","https://openalex.org/W2769856846","https://openalex.org/W2903893215","https://openalex.org/W2919115771","https://openalex.org/W2963326510","https://openalex.org/W2963433607","https://openalex.org/W4206742934"],"related_works":["https://openalex.org/W2981302852","https://openalex.org/W2990219166","https://openalex.org/W2896572968","https://openalex.org/W3100770245","https://openalex.org/W1516903196","https://openalex.org/W2811201469","https://openalex.org/W2963834323","https://openalex.org/W3046380121","https://openalex.org/W3127328563","https://openalex.org/W3080474418","https://openalex.org/W2950397435","https://openalex.org/W3214289948","https://openalex.org/W3101271792","https://openalex.org/W3034241250","https://openalex.org/W2964271537","https://openalex.org/W1866529865","https://openalex.org/W2971899460","https://openalex.org/W3198258820","https://openalex.org/W2791562219","https://openalex.org/W2914239853"],"abstract_inverted_index":{"Stochastic":[0],"gradient":[1],"descent":[2,117],"(SGD)":[3],"is":[4,79,94,147],"one":[5,182],"of":[6,21,29,42,60,82,101,144,163,166,184,194,207],"the":[7,39,47,80,125,138,161,191,200,205,230,237,243],"most":[8,48],"widely":[9],"used":[10],"algorithms":[11],"for":[12,23,57,96,105,115],"large":[13],"scale":[14],"optimization":[15],"problems.":[16],"While":[17],"classical":[18],"theoretical":[19],"analysis":[20],"SGD":[22,43,61,167],"convex":[24,107,110,123],"problems":[25],"studies":[26],"(suffix)":[27],"\\emph{averages}":[28],"iterates":[30],"and":[31,103,108],"obtains":[32],"information":[33,68,156],"theoretically":[34,157],"optimal":[35,158],"bounds":[36,71,159],"on":[37,160],"suboptimality,":[38],"\\emph{last":[40,164],"point}":[41,165],"is,":[44],"by":[45,72,175],"far,":[46],"preferred":[49],"choice":[50],"in":[51,87],"practice.":[52],"The":[53,141],"best":[54,126],"known":[55,127],"results":[56,224],"last":[58,192],"point":[59,193],"\\cite{shamir2013stochastic}":[62],"however,":[63],"are":[64],"suboptimal":[65],"compared":[66,241],"to":[67,121,132,148,187,242],"theoretic":[69],"lower":[70],"a":[73,177],"$\\log":[74,91],"T$":[75,92],"factor,":[76],"where":[77],"$T$":[78],"number":[81],"iterations.":[83],"\\cite{harvey2018tight}":[84],"shows":[85],"that":[86,154,180,190,215,229],"fact,":[88],"this":[89,145,174],"additional":[90],"factor":[93],"tight":[95],"standard":[97,244],"step":[98,151,185,232,245],"size":[99,152,233,246],"sequences":[100,129,153],"$\\OTheta{\\frac{1}{\\sqrt{t}}}$":[102],"$\\OTheta{\\frac{1}{t}}$":[104],"non-strongly":[106],"strongly":[109],"settings,":[111],"respectively.":[112],"Similarly,":[113],"even":[114],"subgradient":[116],"(GD)":[118],"when":[119],"applied":[120],"non-smooth,":[122],"functions,":[124],"step-size":[128],"still":[130],"lead":[131],"$O(\\log":[133],"T)$-suboptimal":[134],"convergence":[135],"rates":[136],"(on":[137],"final":[139,238],"iterate).":[140],"main":[142],"contribution":[143],"work":[146],"design":[149],"new":[150,231],"enjoy":[155],"suboptimality":[162,202],"as":[168,170,204],"well":[169],"GD.":[171],"We":[172,212,221],"achieve":[173],"designing":[176],"modification":[178],"scheme,":[179],"converts":[181],"sequence":[183,198,234],"sizes":[186],"another":[188],"so":[189],"SGD/GD":[195,208],"with":[196,209,219],"modified":[197],"has":[199],"same":[201],"guarantees":[203],"average":[206],"original":[210],"sequence.":[211],"also":[213],"show":[214],"our":[216,223],"result":[217],"holds":[218],"high-probability.":[220],"validate":[222],"through":[225],"simulations":[226],"which":[227],"demonstrate":[228],"indeed":[235],"improves":[236],"iterate":[239],"significantly":[240],"sequences.":[247]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":3}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}