{"id":"https://openalex.org/W2772283936","doi":"https://doi.org/10.1561/2200000058","title":"Non-convex Optimization for Machine Learning","display_name":"Non-convex Optimization for Machine Learning","publication_year":2017,"publication_date":"2017-12-04","ids":{"openalex":"https://openalex.org/W2772283936","doi":"https://doi.org/10.1561/2200000058","mag":"2772283936"},"language":"en","primary_location":{"id":"doi:10.1561/2200000058","is_oa":false,"landing_page_url":"https://doi.org/10.1561/2200000058","pdf_url":null,"source":{"id":"https://openalex.org/S4210188176","display_name":"Foundations and Trends\u00ae in Machine Learning","issn_l":"1935-8237","issn":["1935-8237","1935-8245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318575","host_organization_name":"Now Publishers","host_organization_lineage":["https://openalex.org/P4310318575"],"host_organization_lineage_names":["Now Publishers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations and Trends\u00ae in Machine Learning","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1712.07897","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/I4210124949","display_name":"Microsoft Research (India)","ror":"https://ror.org/02w7f3w92","country_code":"IN","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210124949"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Prateek Jain","raw_affiliation_strings":["Microsoft Research India","Microsoft Research, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Microsoft Research India","institution_ids":["https://openalex.org/I4210124949"]},{"raw_affiliation_string":"Microsoft Research, India","institution_ids":["https://openalex.org/I4210124949"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5081725635","display_name":"Purushottam Kar","orcid":"https://orcid.org/0000-0003-2096-5267"},"institutions":[{"id":"https://openalex.org/I94234084","display_name":"Indian Institute of Technology Kanpur","ror":"https://ror.org/05pjsgx75","country_code":"IN","type":"education","lineage":["https://openalex.org/I94234084"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Purushottam Kar","raw_affiliation_strings":["IIT Kanpur ,","IIT Kanpur, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"IIT Kanpur ,","institution_ids":["https://openalex.org/I94234084"]},{"raw_affiliation_string":"IIT Kanpur, India","institution_ids":["https://openalex.org/I94234084"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101633144"],"corresponding_institution_ids":["https://openalex.org/I4210124949"],"apc_list":null,"apc_paid":null,"fwci":16.0315,"has_fulltext":true,"cited_by_count":346,"citation_normalized_percentile":{"value":0.99218955,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"10","issue":"3-4","first_page":"142","last_page":"336"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12535","display_name":"Machine Learning and Data Classification","score":0.5935999751091003,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.5935999751091003,"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/T12702","display_name":"Brain Tumor Detection and Classification","score":0.5672000050544739,"subfield":{"id":"https://openalex.org/subfields/2808","display_name":"Neurology"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.5285000205039978,"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/computer-science","display_name":"Computer science","score":0.6277998685836792},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5925203561782837},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.5532273650169373},{"id":"https://openalex.org/keywords/inference","display_name":"Inference","score":0.4825103282928467},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4711792767047882},{"id":"https://openalex.org/keywords/heuristics","display_name":"Heuristics","score":0.44905662536621094},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4296186566352844},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.41549721360206604},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.3432806730270386},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.267630398273468},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.26437246799468994},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2370293140411377},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.20088863372802734}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6277998685836792},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5925203561782837},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.5532273650169373},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.4825103282928467},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4711792767047882},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.44905662536621094},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4296186566352844},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.41549721360206604},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.3432806730270386},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.267630398273468},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.26437246799468994},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2370293140411377},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.20088863372802734},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1561/2200000058","is_oa":false,"landing_page_url":"https://doi.org/10.1561/2200000058","pdf_url":null,"source":{"id":"https://openalex.org/S4210188176","display_name":"Foundations and Trends\u00ae in Machine Learning","issn_l":"1935-8237","issn":["1935-8237","1935-8245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318575","host_organization_name":"Now Publishers","host_organization_lineage":["https://openalex.org/P4310318575"],"host_organization_lineage_names":["Now Publishers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations and Trends\u00ae in Machine Learning","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1712.07897","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1712.07897","pdf_url":"https://arxiv.org/pdf/1712.07897","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":null,"raw_type":"text"},{"id":"pmh:oai:aleph.bib-bvb.de:BVB01-031602327","is_oa":false,"landing_page_url":"https://www.nowpublishers.com/article/Details/MAL-058","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"software, multimedia"},{"id":"pmh:oai:cds.cern.ch:2762158","is_oa":false,"landing_page_url":"http://cds.cern.ch/record/2762158","pdf_url":null,"source":{"id":"https://openalex.org/S4306402195","display_name":"CERN Document Server (European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":""}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1712.07897","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1712.07897","pdf_url":"https://arxiv.org/pdf/1712.07897","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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2772283936.pdf","grobid_xml":"https://content.openalex.org/works/W2772283936.grobid-xml"},"referenced_works_count":119,"referenced_works":["https://openalex.org/W54422097","https://openalex.org/W149129625","https://openalex.org/W616376197","https://openalex.org/W947105725","https://openalex.org/W1484131664","https://openalex.org/W1484412996","https://openalex.org/W1511796008","https://openalex.org/W1511814458","https://openalex.org/W1569098853","https://openalex.org/W1697075315","https://openalex.org/W1712752320","https://openalex.org/W1899249567","https://openalex.org/W1933990309","https://openalex.org/W1969515697","https://openalex.org/W1974524469","https://openalex.org/W1977520307","https://openalex.org/W1986814016","https://openalex.org/W2000157792","https://openalex.org/W2001801912","https://openalex.org/W2002182974","https://openalex.org/W2004544971","https://openalex.org/W2009941369","https://openalex.org/W2013850411","https://openalex.org/W2015418199","https://openalex.org/W2020390700","https://openalex.org/W2028781966","https://openalex.org/W2030449718","https://openalex.org/W2053742104","https://openalex.org/W2054141820","https://openalex.org/W2054443446","https://openalex.org/W2056145308","https://openalex.org/W2056979797","https://openalex.org/W2057690238","https://openalex.org/W2059958128","https://openalex.org/W2080108722","https://openalex.org/W2093417350","https://openalex.org/W2095984592","https://openalex.org/W2100875869","https://openalex.org/W2103972604","https://openalex.org/W2105724942","https://openalex.org/W2106005123","https://openalex.org/W2106334384","https://openalex.org/W2107151623","https://openalex.org/W2110451175","https://openalex.org/W2113118796","https://openalex.org/W2118425803","https://openalex.org/W2118550318","https://openalex.org/W2118585731","https://openalex.org/W2118838680","https://openalex.org/W2119385818","https://openalex.org/W2127271355","https://openalex.org/W2129131372","https://openalex.org/W2129185311","https://openalex.org/W2129812935","https://openalex.org/W2134332047","https://openalex.org/W2134618502","https://openalex.org/W2138473013","https://openalex.org/W2138967244","https://openalex.org/W2150593711","https://openalex.org/W2152701363","https://openalex.org/W2158923808","https://openalex.org/W2162312215","https://openalex.org/W2162708633","https://openalex.org/W2164452299","https://openalex.org/W2168893847","https://openalex.org/W2169501582","https://openalex.org/W2184753682","https://openalex.org/W2185602187","https://openalex.org/W2189137528","https://openalex.org/W2221507943","https://openalex.org/W2263105607","https://openalex.org/W2276652794","https://openalex.org/W2289096394","https://openalex.org/W2289917018","https://openalex.org/W2343583226","https://openalex.org/W2399116993","https://openalex.org/W2407904812","https://openalex.org/W2474090883","https://openalex.org/W2498631646","https://openalex.org/W2564702731","https://openalex.org/W2566240941","https://openalex.org/W2566608758","https://openalex.org/W2592651140","https://openalex.org/W2609037894","https://openalex.org/W2611328865","https://openalex.org/W2618398196","https://openalex.org/W2625063094","https://openalex.org/W2751645773","https://openalex.org/W2755553805","https://openalex.org/W2772283936","https://openalex.org/W2950190315","https://openalex.org/W2951539721","https://openalex.org/W2952160444","https://openalex.org/W2952318479","https://openalex.org/W2952594493","https://openalex.org/W2962737134","https://openalex.org/W2963206527","https://openalex.org/W2963254349","https://openalex.org/W2963586744","https://openalex.org/W2963684688","https://openalex.org/W2963697946","https://openalex.org/W2963874210","https://openalex.org/W2963972677","https://openalex.org/W2964156132","https://openalex.org/W2964326464","https://openalex.org/W3010434925","https://openalex.org/W3098045837","https://openalex.org/W3102206315","https://openalex.org/W3105703423","https://openalex.org/W3141595720","https://openalex.org/W4230471307","https://openalex.org/W4238983424","https://openalex.org/W4243493583","https://openalex.org/W4254315244","https://openalex.org/W4293775970","https://openalex.org/W4293867496","https://openalex.org/W4298263336","https://openalex.org/W4300485810","https://openalex.org/W4300634415"],"related_works":["https://openalex.org/W2756132392","https://openalex.org/W3035814349","https://openalex.org/W4285101096","https://openalex.org/W4320477335","https://openalex.org/W2518949622","https://openalex.org/W4382725876","https://openalex.org/W2084892497","https://openalex.org/W2115614142","https://openalex.org/W1561889708","https://openalex.org/W2275184629"],"abstract_inverted_index":{"A":[0,101],"vast":[1],"majority":[2],"of":[3,54,162,191,202,213,225,237,271],"machine":[4],"learning":[5,22,78],"algorithms":[6,55],"train":[7,63],"their":[8,192],"models":[9,65,69,242],"and":[10,23,70,115,132,158,180,194,240,293],"perform":[11],"inference":[12],"by":[13],"solving":[14],"optimization":[15,83,124,149,262],"problems.":[16,125,303],"In":[17],"order":[18],"to":[19,45,75,89,99,104,108,112,119,147,232,248,275,296],"capture":[20],"the":[21,40,77,90,121,142,160,165,222,230,234,255,278,288,291],"prediction":[24],"problems":[25,96,111],"accurately,":[26],"structural":[27],"constraints":[28],"such":[29,66,95],"as":[30,67,80,167,264,266,284,286],"sparsity":[31],"or":[32,38,61],"low":[33],"rank":[34],"are":[35,97,185],"frequently":[36,169],"imposed":[37],"else":[39],"objective":[41],"itself":[42],"is":[43,51,274],"designed":[44],"be":[46,130],"a":[47,81,200,207],"non-convex":[48,82,110,148,302],"function.":[49],"This":[50,197],"especially":[52],"true":[53],"that":[56,62,205,218,243],"operate":[57],"in":[58,155,189,210,281],"high-dimensional":[59],"spaces":[60],"non-linear":[64],"tensor":[68],"deep":[71],"networks.":[72],"The":[73,251,269],"freedom":[74],"express":[76],"problem":[79,84],"gives":[85],"immense":[86],"modeling":[87],"power":[88],"algorithm":[91],"designer,":[92],"but":[93],"often":[94,186],"NP-hard":[98],"solve.":[100],"popular":[102,174],"workaround":[103],"this":[105,127,272,282],"has":[106],"been":[107],"relax":[109],"convex":[113],"ones":[114],"use":[116],"traditional":[117],"methods":[118,161,227],"solve":[120],"(convex)":[122],"relaxed":[123],"However":[126],"approach":[128],"may":[129],"lossy":[131],"nevertheless":[133],"presents":[134,199],"significant":[135],"challenges":[136],"for":[137,164,301],"large":[138],"scale":[139],"optimization.":[140],"On":[141],"other":[143,195],"hand,":[144],"direct":[145],"approaches":[146],"have":[150],"met":[151],"with":[152,290],"resounding":[153],"success":[154],"several":[156,258],"domains":[157],"remain":[159],"choice":[163],"practitioner,":[166],"they":[168],"outperform":[170],"relaxation-based":[171],"techniques":[172,247,294],"-":[173],"heuristics":[175],"include":[176],"projected":[177],"gradient":[178],"descent":[179],"alternating":[181],"minimization.":[182],"However,":[183],"these":[184,214,226,245,298],"poorly":[187],"understood":[188],"terms":[190],"convergence":[193],"properties.":[196],"monograph":[198,252,273],"selection":[201],"recent":[203],"advances":[204],"bridge":[206],"long-standing":[208],"gap":[209],"our":[211],"understanding":[212],"heuristics.":[215],"We":[216],"hope":[217],"an":[219],"insight":[220],"into":[221],"inner":[223],"workings":[224],"will":[228,253],"allow":[229,244],"reader":[231,256,289],"appreciate":[233],"unique":[235],"marriage":[236],"task":[238],"structure":[239],"generative":[241],"heuristic":[246],"(provably)":[249],"succeed.":[250],"lead":[254],"through":[257],"widely":[259],"used":[260],"nonconvex":[261],"techniques,":[263],"well":[265,285],"applications":[267],"thereof.":[268],"goal":[270],"both,":[276],"introduce":[277],"rich":[279],"literature":[280],"area,":[283],"equip":[287],"tools":[292],"needed":[295],"analyze":[297],"simple":[299],"procedures":[300]},"counts_by_year":[{"year":2026,"cited_by_count":17},{"year":2025,"cited_by_count":30},{"year":2024,"cited_by_count":54},{"year":2023,"cited_by_count":53},{"year":2022,"cited_by_count":45},{"year":2021,"cited_by_count":70},{"year":2020,"cited_by_count":35},{"year":2019,"cited_by_count":29},{"year":2018,"cited_by_count":12},{"year":2017,"cited_by_count":1}],"updated_date":"2026-05-31T08:46:17.908082","created_date":"2025-10-10T00:00:00"}