{"id":"https://openalex.org/W7128811410","doi":"https://doi.org/10.48550/arxiv.2602.11557","title":"The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient","display_name":"The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient","publication_year":2026,"publication_date":"2026-02-12","ids":{"openalex":"https://openalex.org/W7128811410","doi":"https://doi.org/10.48550/arxiv.2602.11557"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.11557","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125985338","display_name":"Jichu Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Jichu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086117235","display_name":"Xuan Tang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tang, Xuan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5085848346","display_name":"Difan Zou","orcid":"https://orcid.org/0000-0002-6544-2593"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zou, Difan","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":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.18989345,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9207000136375427,"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":0.9207000136375427,"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/T10100","display_name":"Metaheuristic Optimization Algorithms Research","score":0.01759999990463257,"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/T10764","display_name":"Privacy-Preserving Technologies in Data","score":0.015799999237060547,"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/convergence","display_name":"Convergence (economics)","score":0.7858999967575073},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.7294999957084656},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.715499997138977},{"id":"https://openalex.org/keywords/method-of-steepest-descent","display_name":"Method of steepest descent","score":0.6043000221252441},{"id":"https://openalex.org/keywords/variance-reduction","display_name":"Variance reduction","score":0.5874999761581421},{"id":"https://openalex.org/keywords/variance","display_name":"Variance (accounting)","score":0.4832000136375427},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4318000078201294},{"id":"https://openalex.org/keywords/descent","display_name":"Descent (aeronautics)","score":0.3917999863624573}],"concepts":[{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7858999967575073},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.7294999957084656},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.715499997138977},{"id":"https://openalex.org/C158847443","wikidata":"https://www.wikidata.org/wiki/Q1997812","display_name":"Method of steepest descent","level":2,"score":0.6043000221252441},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5911999940872192},{"id":"https://openalex.org/C62644790","wikidata":"https://www.wikidata.org/wiki/Q3454689","display_name":"Variance reduction","level":3,"score":0.5874999761581421},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.4832000136375427},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44269999861717224},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4318000078201294},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.40790000557899475},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.3917999863624573},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.38449999690055847},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.358599990606308},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.34130001068115234},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.3181000053882599},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.30730000138282776},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.30480000376701355},{"id":"https://openalex.org/C774472","wikidata":"https://www.wikidata.org/wiki/Q6760393","display_name":"Margin (machine learning)","level":2,"score":0.3025999963283539},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28060001134872437},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.2766000032424927}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.11557","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.11557","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.11557","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":"pmh:doi:10.48550/arxiv.2602.11557","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[{"score":0.4155465066432953,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"A":[0],"variety":[1],"of":[2,16,31,161,185,189],"widely":[3],"used":[4],"optimization":[5,172],"methods":[6],"like":[7],"SignSGD":[8],"and":[9,44,52,58,68,142,177],"Muon":[10],"can":[11,71,115],"be":[12,73],"interpreted":[13],"as":[14],"instances":[15],"steepest":[17,34,138,192],"descent":[18,35,139,193],"under":[19,55],"different":[20,149],"norm-induced":[21],"geometries.":[22],"In":[23,78],"this":[24],"work,":[25],"we":[26,110,133],"study":[27],"the":[28,48,117,136,179,186],"implicit":[29,120],"bias":[30,121,150],"mini-batch":[32],"stochastic":[33,163,171,190],"in":[36],"multi-class":[37],"classification,":[38],"characterizing":[39],"how":[40],"batch":[41,124],"size,":[42,125],"momentum,":[43,65,141],"variance":[45,113],"reduction":[46,114],"shape":[47],"limiting":[49],"max-margin":[50,87],"behavior":[51,188],"convergence":[53,67,83,130,145],"rates":[54,103],"general":[56],"entry-wise":[57],"Schatten-$p$":[59],"norms.":[60],"We":[61],"show":[62],"that,":[63],"without":[64,140],"worst-case":[66],"successful":[69],"classification":[70],"only":[72],"guaranteed":[74],"with":[75,174],"full-batch":[76,119,175],"gradient.":[77],"contrast,":[79],"momentum":[80],"enables":[81],"small-batch":[82],"to":[84,146],"an":[85],"approximate":[86],"solution":[88],"through":[89],"a":[90,128,147,152,158],"batch-momentum":[91],"trade-off,":[92],"though":[93],"it":[94],"slows":[95],"convergence.":[96],"This":[97],"approach":[98],"provides":[99],"fully":[100],"explicit,":[101],"dimension-free":[102],"that":[104,112],"improve":[105],"upon":[106],"prior":[107],"results.":[108],"Moreover,":[109],"prove":[111],"recover":[116],"exact":[118],"for":[122,181],"any":[123],"albeit":[126],"at":[127],"slower":[129],"rate.":[131],"Finally,":[132],"further":[134],"investigate":[135],"batch-size-one":[137],"reveal":[143],"its":[144],"fundamentally":[148],"via":[151],"concrete":[153],"data":[154],"example,":[155],"which":[156],"reveals":[157],"key":[159],"limitation":[160],"purely":[162],"updates.":[164],"Overall,":[165],"our":[166],"unified":[167],"analysis":[168],"clarifies":[169],"when":[170],"aligns":[173],"behavior,":[176],"paves":[178],"way":[180],"perform":[182],"deeper":[183],"explorations":[184],"training":[187],"gradient":[191],"algorithms.":[194]},"counts_by_year":[],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2026-02-14T00:00:00"}