{"id":"https://openalex.org/W7160626267","doi":"https://doi.org/10.48550/arxiv.2605.06615","title":"When and Why SignSGD Outperforms SGD: A Theoretical Study Based on $\\ell_1$-norm Lower Bounds","display_name":"When and Why SignSGD Outperforms SGD: A Theoretical Study Based on $\\ell_1$-norm Lower Bounds","publication_year":2026,"publication_date":"2026-05-07","ids":{"openalex":"https://openalex.org/W7160626267","doi":"https://doi.org/10.48550/arxiv.2605.06615"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.06615","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.06615","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":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.06615","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125541846","display_name":"Hongyi Tao","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Tao, Hongyi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109689425","display_name":"Dingzhi Yu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yu, Dingzhi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5135687931","display_name":"Lijun Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Lijun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9531999826431274,"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":0.9531999826431274,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.006399999838322401,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.004699999932199717,"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/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7123000025749207},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.6769999861717224},{"id":"https://openalex.org/keywords/minimax","display_name":"Minimax","score":0.6563000082969666},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.46219998598098755},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4253999888896942},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.423799991607666},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.4212999939918518},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.40779998898506165}],"concepts":[{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7123000025749207},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.6769999861717224},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.6563000082969666},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5245000123977661},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5217000246047974},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.46219998598098755},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4361000061035156},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4253999888896942},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.423799991607666},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.4212999939918518},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.40779998898506165},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.3718999922275543},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3662000000476837},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.36579999327659607},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.36079999804496765},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.3440000116825104},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.3271999955177307},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.32359999418258667},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32109999656677246},{"id":"https://openalex.org/C2776650193","wikidata":"https://www.wikidata.org/wiki/Q264661","display_name":"Obstacle","level":2,"score":0.31839999556541443},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.31709998846054077},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.302700012922287},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.2797999978065491},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.2727000117301941},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2712000012397766}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.06615","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.06615","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":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.06615","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.06615","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":"article"},"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":{"Sign-based":[0],"optimization":[1],"algorithms,":[2],"such":[3],"as":[4],"SignSGD":[5,119,128,148,214],"and":[6,33,49,91,115,120],"Muon,":[7],"have":[8],"garnered":[9],"significant":[10],"attention":[11],"for":[12,60,73,118,182],"their":[13],"remarkable":[14],"performance":[15],"in":[16,76,126],"training":[17],"large":[18],"foundation":[19],"models.":[20],"Despite":[21],"this":[22,81,107,170,195],"empirical":[23],"success,":[24],"we":[25,83,111,133,168,201],"still":[26],"lack":[27],"a":[28,92,154,224],"theoretical":[29,204,211],"understanding":[30],"of":[31,103,138,144,156,213,223],"when":[32],"why":[34],"these":[35],"sign-based":[36,74,85],"methods":[37,75],"outperform":[38],"vanilla":[39],"SGD.":[40,131],"The":[41],"core":[42],"obstacle":[43],"is":[44,54,163],"that":[45,147,187,209],"under":[46,158],"standard":[47,77],"smoothness":[48],"finite":[50],"variance":[51],"conditions,":[52],"SGD":[53],"known":[55],"to":[56,172,192,206],"be":[57],"minimax":[58],"optimal":[59,179,196],"finding":[61],"stationary":[62],"points":[63],"measured":[64],"by":[65,153],"$\\ell_2$-norms,":[66],"thereby":[67],"fundamentally":[68],"precluding":[69],"any":[70],"complexity":[71,152],"gains":[72],"settings.":[78],"To":[79],"overcome":[80],"barrier,":[82],"analyze":[84],"optimizers":[86],"leveraging":[87],"$\\ell_1$-norm":[88],"stationarity,":[89],"$\\ell_\\infty$-smoothness,":[90],"separable":[93],"noise":[94],"model,":[95],"which":[96,127],"can":[97],"better":[98],"capture":[99],"the":[100,123,135,141,151,164,173,183,189,210,221],"coordinate-wise":[101],"nature":[102],"signed":[104],"updates.":[105],"Under":[106],"distinct":[108],"problem":[109,124,165],"geometry,":[110],"derive":[112],"matched":[113],"upper":[114],"lower":[116,180],"bounds":[117,205],"explicitly":[121],"characterize":[122],"class":[125],"provably":[129],"dominates":[130],"Specifically,":[132],"compare":[134],"\\emph{upper":[136],"bound":[137,143,181],"SignSGD}":[139],"with":[140,198],"\\emph{lower":[142],"SGD},":[145],"illustrating":[146],"effectively":[149],"reduces":[150],"factor":[155],"$d$":[157,162],"\\emph{sparse":[159],"noise},":[160],"where":[161],"dimension.":[166],"Furthermore,":[167],"elevate":[169],"framework":[171],"matrix":[174],"domain,":[175],"providing":[176],"an":[177],"equivalent":[178],"Muon":[184],"optimizer,":[185],"proving":[186],"extending":[188],"sign":[190],"operator":[191],"matrices":[193],"preserves":[194],"scaling":[197],"dimensionality.":[199],"Finally,":[200],"bridge":[202],"our":[203],"practice,":[207],"demonstrating":[208],"superiority":[212],"accurately":[215],"predicts":[216],"its":[217],"faster":[218],"convergence":[219],"during":[220],"pretraining":[222],"124M":[225],"parameter":[226],"GPT-2":[227],"model.":[228]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-09T00:00:00"}