{"id":"https://openalex.org/W7214701992","doi":"https://doi.org/10.48550/arxiv.2609.33152","title":"Convergence of Practical Muon","display_name":"Convergence of Practical Muon","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214701992","doi":"https://doi.org/10.48550/arxiv.2609.33152"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33152","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33152","pdf_url":"https://arxiv.org/pdf/2609.33152","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/2609.33152","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153498213","display_name":"Haonan Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Haonan","raw_affiliation_strings":["Tongji University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tongji University","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5153412044","display_name":"Yu Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wu, Yu","raw_affiliation_strings":["Tongji University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tongji University","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5021636395","display_name":"Minghui Liwang","orcid":"https://orcid.org/0000-0002-1289-0130","observed_orcids":["https://orcid.org/0000-0002-1289-0130"]},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liwang, Minghui","raw_affiliation_strings":["Shanghai Research Institute for Intelligent Autonomous Systems","Tongji University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Research Institute for Intelligent Autonomous Systems","institution_ids":[]},{"raw_affiliation_string":"Tongji University","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016417704","display_name":"Xinlei Yi","orcid":"https://orcid.org/0000-0003-4299-0471","observed_orcids":["https://orcid.org/0000-0003-4299-0471","https://orcid.org/0000-0002-3530-5395"]},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yi, Xinlei","raw_affiliation_strings":["Shanghai Research Institute for Intelligent Autonomous Systems","Tongji University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Research Institute for Intelligent Autonomous Systems","institution_ids":[]},{"raw_affiliation_string":"Tongji University","institution_ids":["https://openalex.org/I116953780"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153474030","display_name":"Yiguang Hong","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I116953780","display_name":"Tongji University","ror":"https://ror.org/03rc6as71","country_code":"CN","type":"education","lineage":["https://openalex.org/I116953780"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hong, Yiguang","raw_affiliation_strings":["Shanghai Research Institute for Intelligent Autonomous Systems","Tongji University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Research Institute for Intelligent Autonomous Systems","institution_ids":[]},{"raw_affiliation_string":"Tongji University","institution_ids":["https://openalex.org/I116953780"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I116953780"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":1.0,"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":1.0,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"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":0.0,"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/muon-optimizer","display_name":"Muon optimizer","score":0.9139999747276306},{"id":"https://openalex.org/keywords/adamw","display_name":"AdamW","score":0.7910000085830688},{"id":"https://openalex.org/keywords/convergence-analysis","display_name":"convergence analysis","score":0.6430000066757202},{"id":"https://openalex.org/keywords/newton-schulz-iteration","display_name":"Newton-Schulz iteration","score":0.8029999732971191},{"id":"https://openalex.org/keywords/convergence-rate","display_name":"convergence rate","score":0.7229999899864197},{"id":"https://openalex.org/keywords/large-scale-neural-network-training","display_name":"large-scale neural network training","score":0.7210000157356262}],"concepts":[{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.7093999981880188},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6625999808311462},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5532000064849854},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5442000031471252},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5074999928474426},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5009999871253967},{"id":"https://openalex.org/C205334942","wikidata":"https://www.wikidata.org/wiki/Q3151","display_name":"Muon","level":2,"score":0.48410001397132874},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44290000200271606},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4043000042438507},{"id":"https://openalex.org/C538505008","wikidata":"https://www.wikidata.org/wiki/Q3979448","display_name":"Convergence tests","level":4,"score":0.4000000059604645},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.38029998540878296},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.3630000054836273},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3456000089645386},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.3027999997138977},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2994000017642975},{"id":"https://openalex.org/C146324458","wikidata":"https://www.wikidata.org/wiki/Q1202673","display_name":"Uniform norm","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.27900001406669617},{"id":"https://openalex.org/C28761237","wikidata":"https://www.wikidata.org/wiki/Q7805321","display_name":"Time horizon","level":2,"score":0.26829999685287476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33152","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33152","pdf_url":"https://arxiv.org/pdf/2609.33152","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.2609.33152","doi":"https://doi.org/10.48550/arxiv.2609.33152","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33152","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.33152","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33152","pdf_url":"https://arxiv.org/pdf/2609.33152","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/G1176662209","display_name":null,"funder_award_id":"62503365","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2133048905","display_name":null,"funder_award_id":"62573319","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2575069074","display_name":null,"funder_award_id":"62088101","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6263580406","display_name":null,"funder_award_id":"62271424","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214701992.pdf","grobid_xml":"https://content.openalex.org/works/W7214701992.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W4414846785","https://openalex.org/W4414939355","https://openalex.org/W4415938525","https://openalex.org/W4416046424","https://openalex.org/W4417156118","https://openalex.org/W4417357118","https://openalex.org/W4417511325","https://openalex.org/W7077044503","https://openalex.org/W7125282725","https://openalex.org/W7127380112","https://openalex.org/W7130402983","https://openalex.org/W7158880064","https://openalex.org/W7160835758","https://openalex.org/W7160952933","https://openalex.org/W7161784592","https://openalex.org/W7162141601","https://openalex.org/W7163542449","https://openalex.org/W7165385035","https://openalex.org/W7165625859","https://openalex.org/W7171550366","https://openalex.org/W7196924427"],"related_works":["https://openalex.org/W4417156118","https://openalex.org/W4415250134","https://openalex.org/W7131890839","https://openalex.org/W7214607794","https://openalex.org/W4416617024","https://openalex.org/W7162566957","https://openalex.org/W7130733643","https://openalex.org/W7161139062","https://openalex.org/W7215116573","https://openalex.org/W7160960074"],"abstract_inverted_index":{"Muon":[0,73,115],"is":[1,91,154,160],"emerging":[2],"as":[3,23,74,89],"a":[4,82,148],"promising":[5],"alternative":[6],"to":[7,105],"AdamW":[8],"for":[9,48,61,66,113],"large-scale":[10],"neural":[11],"network":[12],"training,":[13],"yet":[14],"theoretical":[15,168],"understanding":[16],"of":[17,77,128,133,140,150],"its":[18],"practical":[19,34,62,72,114],"implementation":[20],"remains":[21,98],"incomplete,":[22],"existing":[24],"analyses":[25],"often":[26],"simplify":[27],"or":[28],"omit":[29],"two":[30],"key":[31],"components:":[32],"(i)":[33],"Newton--Schulz":[35],"iterations":[36],"with":[37,81,121],"empirically":[38],"tuned":[39],"polynomial":[40],"coefficients":[41],"$(3.4445,-4.7750,2.0315)$;":[42],"and":[43,58,158],"(ii)":[44],"decoupled":[45],"weight":[46],"decay":[47],"regularization.":[49],"In":[50],"this":[51],"paper,":[52],"we":[53,70],"provide":[54],"an":[55,122],"optimization":[56,76,96],"interpretation":[57],"establish":[59],"convergence":[60,111,124,145,169],"Muon,":[63],"jointly":[64],"accounting":[65],"both":[67],"components.":[68],"Specifically,":[69],"interpret":[71],"right-preconditioned":[75],"the":[78,95,99,109,117,129,134,137,141,155,161,167],"original":[79,100],"loss":[80],"dynamic":[83],"weighted":[84],"$\\ell_2$":[85],"regularizer":[86],"that":[87,94],"vanishes":[88],"stationarity":[90],"approached,":[92],"so":[93],"target":[97],"objective.":[101],"We":[102],"then":[103],"establish,":[104],"our":[106],"best":[107,142],"knowledge,":[108],"first":[110],"guarantee":[112],"in":[116,126],"stochastic":[118],"nonconvex":[119],"setting,":[120],"$\\mathcal{O}(T^{-1/4})$":[123],"rate":[125,146],"terms":[127],"expected":[130],"Frobenius":[131],"norm":[132],"gradient,":[135],"improving":[136],"dimension":[138],"dependence":[139],"known":[143],"AdamW's":[144],"by":[147],"factor":[149],"$\\sqrt{d}$,":[151],"where":[152],"$T$":[153],"iteration":[156],"horizon":[157],"$d$":[159],"parameter":[162],"dimension.":[163],"Experiments":[164],"further":[165],"support":[166],"results.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}
