{"id":"https://openalex.org/W7212284229","doi":"https://doi.org/10.48550/arxiv.2609.10886","title":"Relatively Smart II: Tractable or Semi-Supervised Instance-Optimal Learning","display_name":"Relatively Smart II: Tractable or Semi-Supervised Instance-Optimal Learning","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212284229","doi":"https://doi.org/10.48550/arxiv.2609.10886"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.10886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10886","pdf_url":"https://arxiv.org/pdf/2609.10886","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.10886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001275905","display_name":"Shaddin Dughmi","orcid":"https://orcid.org/0000-0002-2784-1868","observed_orcids":["https://orcid.org/0000-0002-2784-1868"]},"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":"Dughmi, Shaddin","raw_affiliation_strings":["University of Southern California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I1174212"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089510952","display_name":"Alireza F. Pour","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Pour, Alireza F.","raw_affiliation_strings":["University of Waterloo"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Waterloo","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9990000128746033,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9990000128746033,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.0003000000142492354,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.00019999999494757503,"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/semi-supervised-learning","display_name":"semi-supervised learning","score":0.8690000176429749},{"id":"https://openalex.org/keywords/sample-complexity","display_name":"sample complexity","score":0.6000000238418579},{"id":"https://openalex.org/keywords/empirical-risk-minimization","display_name":"empirical risk minimization","score":0.8709999918937683},{"id":"https://openalex.org/keywords/transductive-learning","display_name":"transductive learning","score":0.6830000281333923},{"id":"https://openalex.org/keywords/intractability","display_name":"computational intractability","score":0.6499999761581421},{"id":"https://openalex.org/keywords/distribution-free-learning","display_name":"distribution-free learning","score":0.5460000038146973},{"id":"https://openalex.org/keywords/binary-classification","display_name":"binary classification","score":0.6869999766349792},{"id":"https://openalex.org/keywords/unlabeled-data","display_name":"unlabeled data","score":0.4880000054836273}],"concepts":[{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.8023999929428101},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5493000149726868},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.5432999730110168},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5303999781608582},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4821000099182129},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.4772000014781952},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4722999930381775},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.4544000029563904},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.41589999198913574},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4047999978065491},{"id":"https://openalex.org/C203313322","wikidata":"https://www.wikidata.org/wiki/Q5158394","display_name":"Concept class","level":2,"score":0.40130001306533813},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3935000002384186},{"id":"https://openalex.org/C94284585","wikidata":"https://www.wikidata.org/wiki/Q228184","display_name":"Random oracle","level":4,"score":0.38179999589920044},{"id":"https://openalex.org/C117765406","wikidata":"https://www.wikidata.org/wiki/Q5362437","display_name":"Generalization error","level":3,"score":0.3772999942302704},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.36469998955726624},{"id":"https://openalex.org/C66905080","wikidata":"https://www.wikidata.org/wiki/Q17005494","display_name":"Binary classification","level":3,"score":0.3643999993801117},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3643999993801117},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.3359000086784363},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32019999623298645},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.2919999957084656},{"id":"https://openalex.org/C103088060","wikidata":"https://www.wikidata.org/wiki/Q1062839","display_name":"Error detection and correction","level":2,"score":0.28619998693466187},{"id":"https://openalex.org/C176248197","wikidata":"https://www.wikidata.org/wiki/Q458526","display_name":"Probably approximately correct learning","level":4,"score":0.28610000014305115},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.28380000591278076},{"id":"https://openalex.org/C167085575","wikidata":"https://www.wikidata.org/wiki/Q6803654","display_name":"Mean squared prediction error","level":2,"score":0.2822999954223633},{"id":"https://openalex.org/C2778445095","wikidata":"https://www.wikidata.org/wiki/Q18354077","display_name":"Sample complexity","level":2,"score":0.26660001277923584},{"id":"https://openalex.org/C58973888","wikidata":"https://www.wikidata.org/wiki/Q1041418","display_name":"Semi-supervised learning","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.2639999985694885},{"id":"https://openalex.org/C2982736386","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Statistical learning","level":2,"score":0.2590000033378601},{"id":"https://openalex.org/C136389625","wikidata":"https://www.wikidata.org/wiki/Q334384","display_name":"Supervised learning","level":3,"score":0.2535000145435333}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.10886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10886","pdf_url":"https://arxiv.org/pdf/2609.10886","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,"listed_in":[],"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":"doi:10.48550/arxiv.2609.10886","doi":"https://doi.org/10.48550/arxiv.2609.10886","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.10886","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.10886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10886","pdf_url":"https://arxiv.org/pdf/2609.10886","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,"listed_in":[],"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":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G7289584270","display_name":"AF:Small:A Combinatorial Lens on Learning","funder_award_id":"2432219","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/F4320309475","display_name":"Northwestern University","ror":"https://ror.org/000e0be47"},{"id":"https://openalex.org/F4320309626","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212284229.pdf","grobid_xml":"https://content.openalex.org/works/W7212284229.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2045888327","https://openalex.org/W2081177492","https://openalex.org/W7164540382"],"related_works":["https://openalex.org/W7133309093","https://openalex.org/W7101081865","https://openalex.org/W7095431935","https://openalex.org/W2120409953","https://openalex.org/W2909563803","https://openalex.org/W7133183159","https://openalex.org/W7100432044","https://openalex.org/W7077865444","https://openalex.org/W7098638210","https://openalex.org/W2154455818"],"abstract_inverted_index":{"We":[0,106,156],"continue":[1],"the":[2,37,77,103,122,147,207,225,255],"study":[3],"of":[4,129,135,149,189,198,200,273],"relatively":[5,43,53,97,161,238],"smart":[6,44,54,98,162,239],"learning,":[7],"introduced":[8],"by":[9,22],"Dughmi":[10],"and":[11,50,176,206,222,260],"Pour":[12],"(2026),":[13],"which":[14],"asks":[15],"a":[16,46,109,117,126,133,168,186,192,201,218],"supervised":[17],"learner":[18,41,55,68,184,240],"to":[19,82,145,191],"compete,":[20],"marginal":[21,256],"marginal,":[23],"with":[24,45,113,153,167,241],"every":[25],"distribution-fixed":[26,268],"error":[27,112,120,148],"guarantee":[28],"soundly":[29],"certifiable":[30,111],"from":[31],"unlabeled":[32,83,173],"data.":[33,84],"They":[34,72],"showed":[35],"that":[36,51,89,108,141,159,270],"One-Inclusion":[38],"Graph":[39],"(OIG)":[40],"is":[42,164,228,257],"quadratic":[47,169],"sample-complexity":[48],"blowup,":[49],"no":[52,177],"can":[56,79],"do":[57],"better,":[58],"leaving":[59],"open":[60,75],"whether":[61,76],"ERM":[62,234],"or":[63,66],"another":[64],"natural":[65,187],"tractable":[67],"achieves":[69],"comparable":[70],"guarantees.":[71],"also":[73,262],"left":[74],"blowup":[78,170,178,245],"be":[80,272],"restricted":[81],"Our":[85],"firs":[86],"results":[87],"shows":[88],"ERM---and":[90],"in":[91,102,172,179,220],"fact":[92],"any":[93,236],"proper":[94,150],"consistent":[95,151],"learner---is":[96],"for":[99,267],"binary":[100],"classification":[101],"distribution-free":[104],"setting.":[105],"show":[107,158],"small":[110,119],"$m$":[114],"samples":[115],"implies":[116],"similarly":[118],"on":[121,140],"uniform":[123],"distribution":[124],"over":[125],"random":[127],"sample":[128,174,181],"size":[130,136],"$O(m^2)$,":[131],"yielding":[132],"cover":[134],"at":[137,217],"most":[138],"$2^{m+1}$":[139],"sample.":[142],"This":[143,251],"suffices":[144],"control":[146],"learners":[152],"$O(m^2)$":[154],"samples.":[155],"then":[157],"semi-supervised":[160,237],"learning":[163,269],"information-theoretically":[165],"possible":[166],"only":[171,230],"complexity":[175],"labeled":[180],"complexity.":[182],"The":[183],"uses":[185],"generalization":[188],"OIG":[190],"leave-most-out":[193],"transductive":[194],"problem,":[195],"where":[196],"labels":[197,209],"part":[199],"finite":[202],"pool":[203],"are":[204,210],"revealed":[205],"remaining":[208],"predicted.":[211],"Finally,":[212],"this":[213],"label":[214],"efficiency":[215],"comes":[216],"cost":[219],"simplicity":[221],"tractability.":[223],"If":[224],"hypothesis":[226],"class":[227],"accessed":[229],"through":[231],"an":[232,264],"agnostic":[233],"oracle,":[235],"substantially":[242],"sub-quadratic":[243],"labeled-sample":[244],"requires":[246],"super-polynomially":[247],"many":[248],"oracle":[249],"calls.":[250],"holds":[252],"even":[253],"when":[254],"given":[258],"explicitly,":[259],"thus":[261],"yields":[263],"intractability":[265],"result":[266],"may":[271],"independent":[274],"interest.":[275]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-12T00:00:00"}