{"id":"https://openalex.org/W3087176855","doi":"https://doi.org/10.1145/3406325.3451125","title":"The complexity of constrained min-max optimization","display_name":"The complexity of constrained min-max optimization","publication_year":2021,"publication_date":"2021-06-15","ids":{"openalex":"https://openalex.org/W3087176855","doi":"https://doi.org/10.1145/3406325.3451125","mag":"3087176855"},"language":"en","primary_location":{"id":"doi:10.1145/3406325.3451125","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3406325.3451125","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3406325.3451125","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","raw_type":"proceedings-article"},"type":"conference-paper","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://dl.acm.org/doi/pdf/10.1145/3406325.3451125","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034822411","display_name":"Constantinos Daskalakis","orcid":"https://orcid.org/0000-0002-5451-0490"},"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":"Constantinos Daskalakis","raw_affiliation_strings":["Massachusetts Institute of Technology, USA","Massachusetts Institute Of Technology#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute Of Technology#TAB#","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020247743","display_name":"Stratis Skoulakis","orcid":"https://orcid.org/0000-0003-3419-1776"},"institutions":[{"id":"https://openalex.org/I152815399","display_name":"Singapore University of Technology and Design","ror":"https://ror.org/05j6fvn87","country_code":"SG","type":"education","lineage":["https://openalex.org/I152815399"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Stratis Skoulakis","raw_affiliation_strings":["Singapore University of Technology and Design, Singapore","Singapore University of Technology and Design"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Singapore University of Technology and Design, Singapore","institution_ids":["https://openalex.org/I152815399"]},{"raw_affiliation_string":"Singapore University of Technology and Design","institution_ids":["https://openalex.org/I152815399"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083461583","display_name":"Manolis Zampetakis","orcid":"https://orcid.org/0009-0005-4967-5927"},"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"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Manolis Zampetakis","raw_affiliation_strings":["University of California at Berkeley, USA","Massachusetts Institute Of Technology#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of California at Berkeley, USA","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Massachusetts Institute Of Technology#TAB#","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.0558,"has_fulltext":true,"cited_by_count":18,"citation_normalized_percentile":{"value":0.89074258,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"1466","last_page":"1478"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9995999932289124,"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":0.9995999932289124,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T12101","display_name":"Advanced Bandit Algorithms Research","score":0.9993000030517578,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6800273656845093},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.6602970957756042},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5999881029129028},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5630619525909424},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.5590455532073975},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5052159428596497},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.49117156863212585},{"id":"https://openalex.org/keywords/point","display_name":"Point (geometry)","score":0.46440768241882324},{"id":"https://openalex.org/keywords/oracle","display_name":"Oracle","score":0.4508426785469055},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4443742632865906},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.42930787801742554},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34805649518966675},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1980176568031311},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1264301836490631},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10346981883049011}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6800273656845093},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.6602970957756042},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5999881029129028},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5630619525909424},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.5590455532073975},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5052159428596497},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.49117156863212585},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.46440768241882324},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.4508426785469055},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4443742632865906},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.42930787801742554},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34805649518966675},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1980176568031311},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1264301836490631},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10346981883049011},{"id":"https://openalex.org/C115903868","wikidata":"https://www.wikidata.org/wiki/Q80993","display_name":"Software engineering","level":1,"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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1145/3406325.3451125","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3406325.3451125","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3406325.3451125","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:2009.09623","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2009.09623","pdf_url":"https://arxiv.org/pdf/2009.09623","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":"mag:3087176855","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2009.09623","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,"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":"pmh:oai:dspace.mit.edu:1721.1/143467","is_oa":true,"landing_page_url":"https://hdl.handle.net/1721.1/143467","pdf_url":null,"source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc","license_id":"https://openalex.org/licenses/cc-by-nc","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ACM","raw_type":"Article"},{"id":"doi:10.48550/arxiv.2009.09623","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2009.09623","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,"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":"Preprint"}],"best_oa_location":{"id":"doi:10.1145/3406325.3451125","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3406325.3451125","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3406325.3451125","source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","raw_type":"proceedings-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G125460123","display_name":null,"funder_award_id":"DE-AC05\u201376RL01830","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G2506945562","display_name":null,"funder_award_id":"76RL01830","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G2510423872","display_name":null,"funder_award_id":"AC05-76RL01830","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G2988778103","display_name":null,"funder_award_id":"DE-AC05?","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G3066980937","display_name":null,"funder_award_id":"HR00111990021","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G3567753379","display_name":"AF: Medium: Collaborative Research: Theoretical Foundations of Deep  Generative Models and High-Dimensional Distributions","funder_award_id":"1901292","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5270222991","display_name":"BIGDATA: F: Testing High Dimensional Distributions without the Curse of Dimensionality","funder_award_id":"1741137","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5653134238","display_name":null,"funder_award_id":"DEAC05\u201376RL01830","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7560550554","display_name":null,"funder_award_id":"76RL01830","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8679789787","display_name":"AF: SMALL: Frontiers in Algorithmic Game Theory","funder_award_id":"1617730","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320309327","display_name":"Google","ror":"https://ror.org/00njsd438"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3087176855.pdf","grobid_xml":"https://content.openalex.org/works/W3087176855.grobid-xml"},"referenced_works_count":88,"referenced_works":["https://openalex.org/W607505555","https://openalex.org/W1505731132","https://openalex.org/W1519983590","https://openalex.org/W1522301498","https://openalex.org/W1543439990","https://openalex.org/W1804423330","https://openalex.org/W1972221945","https://openalex.org/W1980813953","https://openalex.org/W1982813377","https://openalex.org/W1991624978","https://openalex.org/W1991713925","https://openalex.org/W2002373723","https://openalex.org/W2017781790","https://openalex.org/W2019883273","https://openalex.org/W2023815950","https://openalex.org/W2049934117","https://openalex.org/W2057913812","https://openalex.org/W2062459871","https://openalex.org/W2063416709","https://openalex.org/W2067482046","https://openalex.org/W2075567596","https://openalex.org/W2077723394","https://openalex.org/W2099471712","https://openalex.org/W2103981148","https://openalex.org/W2104375208","https://openalex.org/W2109693504","https://openalex.org/W2120177315","https://openalex.org/W2126211987","https://openalex.org/W2145297839","https://openalex.org/W2146502635","https://openalex.org/W2166717942","https://openalex.org/W2169359757","https://openalex.org/W2186160814","https://openalex.org/W2430369560","https://openalex.org/W2513180554","https://openalex.org/W2577946330","https://openalex.org/W2591713661","https://openalex.org/W2606445249","https://openalex.org/W2609037894","https://openalex.org/W2739748921","https://openalex.org/W2785523195","https://openalex.org/W2791323206","https://openalex.org/W2792263949","https://openalex.org/W2809314237","https://openalex.org/W2893314716","https://openalex.org/W2895628298","https://openalex.org/W2914904265","https://openalex.org/W2915116423","https://openalex.org/W2919464796","https://openalex.org/W2947761153","https://openalex.org/W2947996939","https://openalex.org/W2948013958","https://openalex.org/W2948127421","https://openalex.org/W2948411390","https://openalex.org/W2950041755","https://openalex.org/W2950929549","https://openalex.org/W2953246223","https://openalex.org/W2962723467","https://openalex.org/W2962981216","https://openalex.org/W2963086090","https://openalex.org/W2963092340","https://openalex.org/W2963147599","https://openalex.org/W2963647223","https://openalex.org/W2963674652","https://openalex.org/W2964067523","https://openalex.org/W2964070557","https://openalex.org/W2964253222","https://openalex.org/W2964334726","https://openalex.org/W2965299328","https://openalex.org/W2970416783","https://openalex.org/W2970552792","https://openalex.org/W2972812036","https://openalex.org/W2995151593","https://openalex.org/W2995291884","https://openalex.org/W3003619226","https://openalex.org/W3004694802","https://openalex.org/W3007720024","https://openalex.org/W3008554935","https://openalex.org/W3014065175","https://openalex.org/W3016604587","https://openalex.org/W3035764496","https://openalex.org/W3036302304","https://openalex.org/W3113835508","https://openalex.org/W3164209919","https://openalex.org/W3168185438","https://openalex.org/W4205841652","https://openalex.org/W4292022450","https://openalex.org/W6815275497"],"related_works":["https://openalex.org/W3168185438","https://openalex.org/W3035665052","https://openalex.org/W2963981733","https://openalex.org/W2099471712","https://openalex.org/W3034862928","https://openalex.org/W2970416783","https://openalex.org/W3171002556","https://openalex.org/W3007729382","https://openalex.org/W2995128076","https://openalex.org/W2964334726","https://openalex.org/W2964249351","https://openalex.org/W2964070557","https://openalex.org/W2962981216","https://openalex.org/W2962723467","https://openalex.org/W2895628298","https://openalex.org/W2619503996","https://openalex.org/W3167725861","https://openalex.org/W2557444915","https://openalex.org/W2956358468","https://openalex.org/W3134655137"],"abstract_inverted_index":{"Despite":[0],"its":[1,169],"important":[2,131],"applications":[3],"in":[4,67,74,143,214,238,249,280],"Machine":[5],"Learning,":[6],"min-max":[7,30,77,86,128,201],"optimization":[8,78,148,278],"of":[9,40,54,61,64,88,107,113,133,159,208,218,234],"objective":[10],"functions":[11],"that":[12,73,185,188,211],"are":[13,19,152,228],"nonconvex-nonconcave":[14,81],"remains":[15],"elusive.":[16],"Not":[17],"only":[18],"there":[20],"no":[21],"known":[22,180],"first-order":[23,65,191],"methods":[24,66],"converging":[25],"to":[26,94,124,137,156,194,204,241,269],"even":[27],"approximate":[28,33,84,110,126,246],"local":[29,85,127,200,247],"equilibria":[31],"(a.k.a.":[32],"saddle":[34],"points),":[35],"but":[36,96],"the":[37,55,62,114,144,157,230,250,267,281],"computational":[38,56],"complexity":[39,57],"identifying":[41],"them":[42],"is":[43,92,101,105,121,136,178,212,266],"also":[44],"poorly":[45],"understood.":[46],"In":[47],"this":[48,68],"paper,":[49],"we":[50,71,151],"provide":[51],"a":[52,99,179,206],"characterization":[53],"as":[58,60],"well":[59],"limitations":[63],"problem.":[69],"Specifically,":[70],"show":[72,184,270],"linearly":[75],"constrained":[76],"problems":[79,279],"with":[80,256],"objectives":[82],"an":[83,109,139,198,271],"equilibrium":[87,202],"large":[89],"enough":[90],"approximation":[91],"guaranteed":[93],"exist,":[95],"computing":[97,108,125],"such":[98,190],"point":[100,112],"PPAD-complete.":[102],"The":[103],"same":[104,251],"true":[106],"fixed":[111],"(Projected)":[115],"Gradient":[116,258],"Descent/Ascent":[117],"update":[118],"dynamics,":[119],"which":[120],"computationally":[122],"equivalent":[123],"equilibria.":[129],"An":[130],"byproduct":[132],"our":[134],"proof":[135],"establish":[138],"unconditional":[140],"hardness":[141],"result":[142,265],"Nemirovsky-Yudin":[145],"1983":[146],"oracle":[147,154,192,209,282],"model,":[149],"where":[150,173,224,244],"given":[153],"access":[155,193],"values":[158],"some":[160],"function":[161],"f":[162,195],":":[163],"P":[164,174],"\u2192":[165],"[\u22121,":[166],"1]":[167],"and":[168,196,226,232],"gradient":[170],"\u2207":[171],"f,":[172],"\u2286":[175],"[0,":[176],"1]d":[177],"convex":[181],"polytope.":[182],"We":[183],"any":[186],"algorithm":[187],"uses":[189],"finds":[197],"\u03b5-approximate":[199],"needs":[203],"make":[205],"number":[207],"queries":[210],"exponential":[213,272],"at":[215],"least":[216],"one":[217],"1/\u03b5,":[219],"L,":[220],"G,":[221],"or":[222],"d,":[223],"L":[225],"G":[227],"respectively":[229],"smoothness":[231],"Lipschitzness":[233],"f.":[235],"This":[236],"comes":[237],"sharp":[239],"contrast":[240],"minimization":[242],"problems,":[243],"finding":[245],"minima":[248],"setting":[252],"can":[253],"be":[254],"done":[255],"Projected":[257],"Descent":[259],"using":[260],"O(L/\u03b5)":[261],"many":[262],"queries.":[263],"Our":[264],"first":[268],"separation":[273],"between":[274],"these":[275],"two":[276],"fundamental":[277],"model.":[283]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":4}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
