{"id":"https://openalex.org/W7160835758","doi":"https://doi.org/10.48550/arxiv.2605.06884","title":"Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition","display_name":"Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition","publication_year":2026,"publication_date":"2026-05-07","ids":{"openalex":"https://openalex.org/W7160835758","doi":"https://doi.org/10.48550/arxiv.2605.06884"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2605.06884","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.06884","pdf_url":"https://arxiv.org/pdf/2605.06884","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/2605.06884","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5135889439","display_name":"Sayantan Choudhury","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Choudhury, Sayantan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135857847","display_name":"Xiaoran Cheng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I130769515","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231","country_code":"US","type":"education","lineage":["https://openalex.org/I130769515"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Cheng, Xiaoran","raw_affiliation_strings":["Penn State University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Penn State University","institution_ids":["https://openalex.org/I130769515"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138691307","display_name":"Martin Tak\u00e1\u010d","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tak\u00e1\u010d, Martin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056179845","display_name":"Sen Na","orcid":"https://orcid.org/0000-0002-7977-5276","observed_orcids":["https://orcid.org/0000-0002-7977-5276"]},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Na, Sen","raw_affiliation_strings":["Georgia Institute of Technology"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Georgia Institute of Technology","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016868056","display_name":"Mladen Kolar","orcid":"https://orcid.org/0000-0001-7353-3404","observed_orcids":["https://orcid.org/0000-0001-7353-3404"]},"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":"Kolar, Mladen","raw_affiliation_strings":["University of Southern California"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":10,"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":0.9961000084877014,"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":0.9961000084877014,"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":0.0010000000474974513,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.0007999999797903001,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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":0.9350000023841858},{"id":"https://openalex.org/keywords/nesterov-momentum","display_name":"Nesterov momentum","score":0.9900000095367432},{"id":"https://openalex.org/keywords/polar-decomposition","display_name":"polar decomposition","score":0.8100000023841858},{"id":"https://openalex.org/keywords/heavy-tailed-noise","display_name":"heavy-tailed noise","score":0.9409999847412109},{"id":"https://openalex.org/keywords/convergence-analysis","display_name":"convergence analysis","score":0.4970000088214874},{"id":"https://openalex.org/keywords/newton-schulz-iteration","display_name":"Newton-Schulz iteration","score":0.8289999961853027},{"id":"https://openalex.org/keywords/iteration-complexity","display_name":"iteration complexity","score":0.49900001287460327},{"id":"https://openalex.org/keywords/sample-complexity","display_name":"sample complexity","score":0.8920000195503235},{"id":"https://openalex.org/keywords/neural-network-training","display_name":"neural network training","score":0.5799999833106995},{"id":"https://openalex.org/keywords/first-order-methods","display_name":"first-order methods","score":0.4950000047683716}],"concepts":[{"id":"https://openalex.org/C29705727","wikidata":"https://www.wikidata.org/wiki/Q294562","display_name":"Polar","level":2,"score":0.6351000070571899},{"id":"https://openalex.org/C130956294","wikidata":"https://www.wikidata.org/wiki/Q2101158","display_name":"Polar decomposition","level":3,"score":0.629800021648407},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5846999883651733},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.517300009727478},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.47909998893737793},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.476500004529953},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.45879998803138733},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.40149998664855957},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39820000529289246},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.38909998536109924},{"id":"https://openalex.org/C205334942","wikidata":"https://www.wikidata.org/wiki/Q3151","display_name":"Muon","level":2,"score":0.3824000060558319},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.3813000023365021},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36890000104904175},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33980000019073486},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.334199994802475},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.32120001316070557},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.3199999928474426},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.31529998779296875},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.30720001459121704},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.29249998927116394},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.28600001335144043},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.27469998598098755},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2605.06884","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.06884","pdf_url":"https://arxiv.org/pdf/2605.06884","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.2605.06884","doi":"https://doi.org/10.48550/arxiv.2605.06884","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.06884","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:2605.06884","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2605.06884","pdf_url":"https://arxiv.org/pdf/2605.06884","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7160835758.pdf","grobid_xml":"https://content.openalex.org/works/W7160835758.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W2048305372","https://openalex.org/W4322718191","https://openalex.org/W4392575839","https://openalex.org/W4403580163","https://openalex.org/W4407212433","https://openalex.org/W4414846785","https://openalex.org/W4415089271","https://openalex.org/W4415250134","https://openalex.org/W4416613161","https://openalex.org/W4416617024","https://openalex.org/W4417156118","https://openalex.org/W4417298428","https://openalex.org/W6966842224"],"related_works":["https://openalex.org/W4415145305","https://openalex.org/W4415250134","https://openalex.org/W7215116573","https://openalex.org/W7131890839","https://openalex.org/W7125960423","https://openalex.org/W7196924427","https://openalex.org/W7128817298","https://openalex.org/W7161681801","https://openalex.org/W7223289972","https://openalex.org/W4417156118"],"abstract_inverted_index":{"Most":[0],"first-order":[1],"optimizers":[2],"treat":[3],"matrix-valued":[4],"parameters":[5],"as":[6,104],"vectors,":[7],"ignoring":[8],"the":[9,25,48,113,141,145,187,190],"intrinsic":[10],"geometry":[11],"of":[12,28,130,160,189],"hidden-layer":[13],"weights":[14],"in":[15,80],"neural":[16],"networks.":[17],"Muon":[18,72],"addresses":[19],"this":[20,64],"mismatch":[21],"by":[22,66],"updating":[23],"along":[24],"polar":[26,49,78,96,120,167],"factor":[27,50],"a":[29,68,91,164],"momentum":[30,75],"matrix,":[31],"but":[32],"its":[33],"theoretical":[34],"understanding":[35],"has":[36],"lagged":[37],"behind":[38],"practice.":[39],"In":[40],"particular,":[41],"practical":[42,100],"implementations":[43],"incorporate":[44],"Nesterov":[45,74],"momentum,":[46],"compute":[47],"only":[51],"approximately,":[52],"and":[53,76,106,127,193],"operate":[54],"with":[55,73,148,180],"stochastic":[56],"gradients":[57],"that":[58,98,154,169],"may":[59],"be":[60],"heavy-tailed.":[61],"We":[62,162],"close":[63],"gap":[65],"developing":[67],"convergence":[69],"theory":[70],"for":[71,94,134],"inexact":[77,95,192],"decomposition":[79,97,168],"non-convex":[81],"matrix":[82],"optimization":[83,114],"under":[84],"heavy-tailed":[85],"noise.":[86],"Our":[87],"analysis":[88],"builds":[89],"on":[90,119],"unified":[92],"framework":[93],"captures":[99],"iterative":[101],"approximations":[102],"such":[103],"Newton-Schulz":[105],"quantifies":[107],"how":[108],"their":[109],"errors":[110],"propagate":[111],"through":[112],"dynamics.":[115],"Under":[116],"uniform":[117],"bounds":[118],"errors,":[121],"we":[122,150],"establish":[123],"an":[124,136],"optimal":[125],"iteration":[126],"sample":[128],"complexity":[129],"$O":[131],"\\left(\\varepsilon^{\\frac{-(3\u03b1-2)}{(\u03b1-1)}}":[132],"\\right)$":[133],"finding":[135],"$\\varepsilon$-stationary":[137],"point,":[138],"where":[139],"$\u03b1\\in(1,2]$~denotes":[140],"heavy-tail":[142],"index.":[143],"For":[144],"inexact-polar":[146],"setting":[147],"$\u03c3_1=0$,":[149],"also":[151],"provide":[152],"guarantees":[153],"do":[155],"not":[156],"require":[157],"prior":[158],"knowledge":[159],"$\u03b1$.":[161],"analyze":[163],"randomized":[165],"low-rank":[166],"is":[170],"substantially":[171],"more":[172],"efficient":[173],"than":[174],"full-space":[175],"methods":[176],"while":[177],"remaining":[178],"compatible":[179],"our":[181],"theory.":[182],"Numerical":[183],"experiments":[184],"further":[185],"demonstrate":[186],"effectiveness":[188],"proposed":[191],"randomized~variants.":[194]},"counts_by_year":[{"year":2026,"cited_by_count":10}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-05-12T00:00:00"}