{"id":"https://openalex.org/W2905371339","doi":"https://doi.org/10.1109/tit.2019.2893916","title":"Data-Dependent Generalization Bounds for Multi-Class Classification","display_name":"Data-Dependent Generalization Bounds for Multi-Class Classification","publication_year":2019,"publication_date":"2019-01-22","ids":{"openalex":"https://openalex.org/W2905371339","doi":"https://doi.org/10.1109/tit.2019.2893916","mag":"2905371339"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2019.2893916","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2019.2893916","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046468616","display_name":"Yunwen Lei","orcid":"https://orcid.org/0000-0002-5383-467X"},"institutions":[{"id":"https://openalex.org/I3045169105","display_name":"Southern University of Science and Technology","ror":"https://ror.org/049tv2d57","country_code":"CN","type":"education","lineage":["https://openalex.org/I3045169105"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yunwen Lei","raw_affiliation_strings":["Shenzhen Key Laboratory of Computational Intelligence, Southern University of Science and Technology, Shenzhen, China"],"raw_orcid":"https://orcid.org/0000-0002-5383-467X","affiliations":[{"raw_affiliation_string":"Shenzhen Key Laboratory of Computational Intelligence, Southern University of Science and Technology, Shenzhen, China","institution_ids":["https://openalex.org/I3045169105"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086993787","display_name":"\u00dcr\u00fcn Do\u01e7an","orcid":null},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Urun Dogan","raw_affiliation_strings":["Microsoft, Sunnyvale, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Microsoft, Sunnyvale, CA, USA","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006802589","display_name":"Ding\u2010Xuan Zhou","orcid":"https://orcid.org/0000-0003-0224-9216"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Ding-Xuan Zhou","raw_affiliation_strings":["Department of Mathematics, City University of Hong Kong, Hong Kong"],"raw_orcid":"https://orcid.org/0000-0003-0224-9216","affiliations":[{"raw_affiliation_string":"Department of Mathematics, City University of Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5091841504","display_name":"Marius Kloft","orcid":"https://orcid.org/0000-0001-6829-3725"},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Marius Kloft","raw_affiliation_strings":["Department of Computer Science, University of Southern California, Los Angeles, CA, USA"],"raw_orcid":"https://orcid.org/0000-0001-6829-3725","affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Southern California, Los Angeles, CA, USA","institution_ids":["https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.5297,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.94083618,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"65","issue":"5","first_page":"2995","last_page":"3021"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11307","display_name":"Domain Adaptation and Few-Shot Learning","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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","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/T12676","display_name":"Machine Learning and ELM","score":0.9987999796867371,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9983000159263611,"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/dependency","display_name":"Dependency (UML)","score":0.6387763023376465},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5467175841331482},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5267678499221802},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5197162628173828},{"id":"https://openalex.org/keywords/empirical-risk-minimization","display_name":"Empirical risk minimization","score":0.49891066551208496},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4513395130634308},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.43264830112457275},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.42074185609817505},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4203597903251648},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4092060327529907},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4022444486618042},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.35173237323760986}],"concepts":[{"id":"https://openalex.org/C19768560","wikidata":"https://www.wikidata.org/wiki/Q320727","display_name":"Dependency (UML)","level":2,"score":0.6387763023376465},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5467175841331482},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5267678499221802},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5197162628173828},{"id":"https://openalex.org/C107321475","wikidata":"https://www.wikidata.org/wiki/Q5374254","display_name":"Empirical risk minimization","level":2,"score":0.49891066551208496},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4513395130634308},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.43264830112457275},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.42074185609817505},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4203597903251648},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4092060327529907},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4022444486618042},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.35173237323760986},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tit.2019.2893916","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2019.2893916","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/4df1eab1-8a82-40a2-8811-05797dd53557","is_oa":false,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/4df1eab1-8a82-40a2-8811-05797dd53557","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Lei, Y, Dogan, U, Zhou, D-X & Kloft, M 2019, 'Data-dependent Generalization Bounds for Multi-class Classification', IEEE Transactions on Information Theory, vol. 65, no. 5, pp. 2995-3021.","raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1332273124","display_name":null,"funder_award_id":"KL 2698/2-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1855494640","display_name":null,"funder_award_id":"11461161006","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2607551084","display_name":null,"funder_award_id":"ZDSYS201703031748284","funder_id":"https://openalex.org/F4320326705","funder_display_name":"Science, Technology and Innovation Commission of Shenzhen Municipality"},{"id":"https://openalex.org/G3814908347","display_name":null,"funder_award_id":"KQTD2016112514355531","funder_id":"https://openalex.org/F4320335790","funder_display_name":"Shenzhen Peacock Plan"},{"id":"https://openalex.org/G612290940","display_name":null,"funder_award_id":"61806091","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7290307066","display_name":null,"funder_award_id":"GRK1589/2","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8805597337","display_name":null,"funder_award_id":"2017YFC0804003","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320326705","display_name":"Science, Technology and Innovation Commission of Shenzhen Municipality","ror":null},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null},{"id":"https://openalex.org/F4320335790","display_name":"Shenzhen Peacock Plan","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":119,"referenced_works":["https://openalex.org/W905619","https://openalex.org/W104302408","https://openalex.org/W131187193","https://openalex.org/W175058090","https://openalex.org/W583140124","https://openalex.org/W602904462","https://openalex.org/W756166754","https://openalex.org/W1480538416","https://openalex.org/W1493526108","https://openalex.org/W1544839292","https://openalex.org/W1550206324","https://openalex.org/W1569616332","https://openalex.org/W1608733719","https://openalex.org/W1663973292","https://openalex.org/W1698155719","https://openalex.org/W1734803745","https://openalex.org/W1895356522","https://openalex.org/W1940008012","https://openalex.org/W1968436459","https://openalex.org/W1982032418","https://openalex.org/W1984876737","https://openalex.org/W1992736075","https://openalex.org/W1994642189","https://openalex.org/W2000655290","https://openalex.org/W2000908731","https://openalex.org/W2001619934","https://openalex.org/W2021440127","https://openalex.org/W2023163512","https://openalex.org/W2029538739","https://openalex.org/W2029922822","https://openalex.org/W2056625421","https://openalex.org/W2068074736","https://openalex.org/W2069754508","https://openalex.org/W2087258353","https://openalex.org/W2096520669","https://openalex.org/W2097455965","https://openalex.org/W2105463715","https://openalex.org/W2108598243","https://openalex.org/W2112274905","https://openalex.org/W2112528242","https://openalex.org/W2112531253","https://openalex.org/W2112796928","https://openalex.org/W2115046017","https://openalex.org/W2116444583","https://openalex.org/W2136158847","https://openalex.org/W2136885855","https://openalex.org/W2140204814","https://openalex.org/W2140956554","https://openalex.org/W2145862222","https://openalex.org/W2148126650","https://openalex.org/W2148603752","https://openalex.org/W2150102617","https://openalex.org/W2151211319","https://openalex.org/W2153635508","https://openalex.org/W2155144632","https://openalex.org/W2157791002","https://openalex.org/W2161824996","https://openalex.org/W2168645005","https://openalex.org/W2172000360","https://openalex.org/W2180530054","https://openalex.org/W2183087644","https://openalex.org/W2198114369","https://openalex.org/W2282078507","https://openalex.org/W2362855512","https://openalex.org/W2403763698","https://openalex.org/W2501948295","https://openalex.org/W2515754823","https://openalex.org/W2517520041","https://openalex.org/W2540093921","https://openalex.org/W2553870915","https://openalex.org/W2558151686","https://openalex.org/W2579923771","https://openalex.org/W2580921317","https://openalex.org/W2962708723","https://openalex.org/W2963203807","https://openalex.org/W2963880114","https://openalex.org/W2963899139","https://openalex.org/W2964249944","https://openalex.org/W2989661724","https://openalex.org/W2999771887","https://openalex.org/W3000669664","https://openalex.org/W3004533406","https://openalex.org/W3013820469","https://openalex.org/W3120740533","https://openalex.org/W4212863985","https://openalex.org/W4213201446","https://openalex.org/W4238284510","https://openalex.org/W4253712714","https://openalex.org/W4285719527","https://openalex.org/W4293583073","https://openalex.org/W4301014524","https://openalex.org/W4391712018","https://openalex.org/W6604275778","https://openalex.org/W6605290811","https://openalex.org/W6617064350","https://openalex.org/W6628642759","https://openalex.org/W6629455646","https://openalex.org/W6632865047","https://openalex.org/W6634146089","https://openalex.org/W6639568146","https://openalex.org/W6640485552","https://openalex.org/W6674516588","https://openalex.org/W6675743382","https://openalex.org/W6675999653","https://openalex.org/W6680020180","https://openalex.org/W6680729576","https://openalex.org/W6680741563","https://openalex.org/W6681875376","https://openalex.org/W6682939927","https://openalex.org/W6682953061","https://openalex.org/W6684462589","https://openalex.org/W6685655662","https://openalex.org/W6685974025","https://openalex.org/W6705120047","https://openalex.org/W6725805838","https://openalex.org/W6730061259","https://openalex.org/W6732517885","https://openalex.org/W6770207831","https://openalex.org/W7046297837"],"related_works":["https://openalex.org/W1978042415","https://openalex.org/W63071447","https://openalex.org/W1971753667","https://openalex.org/W2163330118","https://openalex.org/W2052275678","https://openalex.org/W3166409534","https://openalex.org/W4229773585","https://openalex.org/W2334791747","https://openalex.org/W1997271576","https://openalex.org/W2348187822"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,121,167],"study":[4],"data-dependent":[5,97,170],"generalization":[6],"error":[7,98],"bounds":[8,33,99],"that":[9],"exhibit":[10],"a":[11,26,106,112,140,169,173,178],"mild":[12],"dependency":[13,142],"on":[14,111,143],"the":[15,47,72,76,82,103,130,144,149,152,157,182,193],"number":[16,28,145],"of":[17,29,75,102,105,146,151,162,181,195],"classes,":[18],"making":[19],"them":[20],"suitable":[21],"for":[22,36,57,119,156],"multi-class":[23,38,58,135,158],"learning":[24,136],"with":[25,79,172],"large":[27],"label":[30],"classes.":[31],"The":[32,186],"generally":[34],"hold":[35],"empirical":[37,62],"risk":[39],"minimization":[40],"algorithms":[41],"using":[42],"an":[43],"arbitrary":[44],"norm":[45],"as":[46],"regularizer.":[48],"Key":[49],"to":[50,81,132,191],"our":[51,196],"analysis":[52],"is":[53,177],"new":[54],"structural":[55],"results":[56,131,188],"Gaussian":[59],"complexities":[60,104],"and":[61,88,125,138,164],"\u2113":[63,83,89],"<sub":[64,84,90],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[65,85,91],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\u221e</sub>":[66,92],"-norm":[67],"covering":[68],"numbers,":[69],"which":[70,120,176],"exploit":[71],"Lipschitz":[73],"continuity":[74],"loss":[77],"function":[78,108],"respect":[80],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[86],"-":[87],"-norm,":[93],"respectively.":[94],"We":[95,128],"establish":[96],"in":[100],"terms":[101],"linear":[107],"class":[109],"defined":[110],"finite":[113],"set":[114],"induced":[115],"by":[116],"training":[117],"examples,":[118],"show":[122,139],"tight":[123],"lower":[124],"upper":[126],"bounds.":[127],"apply":[129],"several":[133],"prominent":[134],"machines":[137],"tighter":[141],"classes":[147],"than":[148],"state":[150],"art.":[153],"For":[154],"instance,":[155],"support":[159],"vector":[160],"machine":[161],"Crammer":[163],"Singer":[165],"(2002),":[166],"obtain":[168],"bound":[171],"logarithmic":[174],"dependency,":[175],"significant":[179],"improvement":[180],"previous":[183],"square-root":[184],"dependency.":[185],"experimental":[187],"are":[189],"reported":[190],"verify":[192],"effectiveness":[194],"theoretical":[197],"findings.":[198]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":5}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}