{"id":"https://openalex.org/W7155186209","doi":"https://doi.org/10.48550/arxiv.2604.19560","title":"Separating Geometry from Probability in the Analysis of Generalization","display_name":"Separating Geometry from Probability in the Analysis of Generalization","publication_year":2026,"publication_date":"2026-04-21","ids":{"openalex":"https://openalex.org/W7155186209","doi":"https://doi.org/10.48550/arxiv.2604.19560"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.19560","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19560","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":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.19560","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044844161","display_name":"Maxim Raginsky","orcid":"https://orcid.org/0000-0002-5586-9219"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Raginsky, Maxim","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5012870568","display_name":"Benjamin Recht","orcid":"https://orcid.org/0000-0002-0293-593X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Recht, Benjamin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.28760001063346863,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.28760001063346863,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.17430000007152557,"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/T12261","display_name":"Statistical Mechanics and Entropy","score":0.06319999694824219,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.8759999871253967},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.6662999987602234},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.6348999738693237},{"id":"https://openalex.org/keywords/generalization-error","display_name":"Generalization error","score":0.4593999981880188},{"id":"https://openalex.org/keywords/sample","display_name":"Sample (material)","score":0.45190000534057617},{"id":"https://openalex.org/keywords/sensitivity","display_name":"Sensitivity (control systems)","score":0.4332999885082245},{"id":"https://openalex.org/keywords/statistical-model","display_name":"Statistical model","score":0.3903000056743622}],"concepts":[{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.8759999871253967},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.6662999987602234},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.6348999738693237},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.48249998688697815},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4684000015258789},{"id":"https://openalex.org/C117765406","wikidata":"https://www.wikidata.org/wiki/Q5362437","display_name":"Generalization error","level":3,"score":0.4593999981880188},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.45190000534057617},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.4332999885082245},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4242999851703644},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38760000467300415},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.37599998712539673},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.3244999945163727},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2759000062942505},{"id":"https://openalex.org/C2776537626","wikidata":"https://www.wikidata.org/wiki/Q4047883","display_name":"Long-term prediction","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2596000134944916},{"id":"https://openalex.org/C3018824978","wikidata":"https://www.wikidata.org/wiki/Q2894891","display_name":"Error analysis","level":2,"score":0.25189998745918274},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.25049999356269836}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.19560","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19560","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":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.19560","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.19560","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":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0,46],"goal":[1],"of":[2,56,69,74,106,111,114,116,140],"machine":[3],"learning":[4],"is":[5,179],"to":[6,26,119,160,171],"find":[7],"models":[8],"that":[9,15,77,143,153],"minimize":[10],"prediction":[11],"error":[12,151,177],"on":[13,42,182],"data":[14,82,158],"has":[16],"not":[17],"yet":[18],"been":[19],"seen.":[20],"Its":[21],"operational":[22],"paradigm":[23],"assumes":[24,76],"access":[25],"a":[27,32,38],"dataset":[28],"$S$":[29,50],"and":[30,80,136,146],"articulates":[31],"scheme":[33],"for":[34],"evaluating":[35],"how":[36,155],"well":[37],"given":[39],"model":[40],"performs":[41],"an":[43,87,103,150],"arbitrary":[44],"sample.":[45],"sample":[47],"can":[48,165],"be":[49,95,167],"(in":[51,64],"which":[52,65],"case":[53,66],"we":[54,67],"speak":[55,68],"``in-sample''":[57],"performance)":[58],"or":[59,184],"some":[60],"entirely":[61],"new":[62],"$S'$":[63],"``out-of-sample''":[70],"performance).":[71],"Traditional":[72],"analysis":[73,113],"generalization":[75,107,128],"both":[78],"in-":[79],"out-of-sample":[81,147,157],"are":[83,130,159],"i.i.d.\\":[84],"draws":[85],"from":[86],"infinite":[88],"population.":[89],"However,":[90],"these":[91],"probabilistic":[92],"assumptions":[93,164],"cannot":[94],"verified":[96],"even":[97],"in":[98,121],"principle.":[99],"This":[100],"paper":[101],"presents":[102],"alternative":[104],"view":[105],"through":[108,149],"the":[109,122,138,173],"lens":[110],"sensitivity":[112],"solutions":[115],"optimization":[117],"problems":[118],"perturbations":[120],"problem":[123],"data.":[124,162],"Under":[125],"this":[126,176],"framework,":[127],"bounds":[129],"obtained":[131],"by":[132],"purely":[133],"deterministic":[134],"means":[135],"take":[137],"form":[139],"variational":[141],"principles":[142],"relate":[144],"in-sample":[145,161],"evaluations":[148],"term":[152,178],"quantifies":[154],"close":[156],"Statistical":[163],"then":[166],"used":[168],"\\textit{ex":[169],"post}":[170],"characterize":[172],"situations":[174],"when":[175],"small":[180],"(either":[181],"average":[183],"with":[185],"high":[186],"probability).":[187]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-23T00:00:00"}