{"id":"https://openalex.org/W7161726329","doi":"https://doi.org/10.48550/arxiv.2605.18106","title":"Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers","display_name":"Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers","publication_year":2026,"publication_date":"2026-05-18","ids":{"openalex":"https://openalex.org/W7161726329","doi":"https://doi.org/10.48550/arxiv.2605.18106"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.18106","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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.2605.18106","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014387645","display_name":"Tim Tsz-Kit Lau","orcid":"https://orcid.org/0000-0002-5653-7232"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lau, Tim Tsz-Kit","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5136463091","display_name":"Weijie Su","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Su, Weijie","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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.6801000237464905,"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":0.6801000237464905,"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/T11948","display_name":"Machine Learning in Materials Science","score":0.0731000006198883,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.028200000524520874,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/equivariant-map","display_name":"Equivariant map","score":0.8148999810218811},{"id":"https://openalex.org/keywords/permutation","display_name":"Permutation (music)","score":0.5264999866485596},{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.5188000202178955},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.46480000019073486},{"id":"https://openalex.org/keywords/symmetry","display_name":"Symmetry (geometry)","score":0.4390999972820282},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.37439998984336853},{"id":"https://openalex.org/keywords/bottleneck","display_name":"Bottleneck","score":0.3671000003814697},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3395000100135803}],"concepts":[{"id":"https://openalex.org/C171036898","wikidata":"https://www.wikidata.org/wiki/Q256355","display_name":"Equivariant map","level":2,"score":0.8148999810218811},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.5264999866485596},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5212000012397766},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.5188000202178955},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.46480000019073486},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.44359999895095825},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.4390999972820282},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3824000060558319},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.37439998984336853},{"id":"https://openalex.org/C2780513914","wikidata":"https://www.wikidata.org/wiki/Q18210350","display_name":"Bottleneck","level":2,"score":0.3671000003814697},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3395000100135803},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.3337000012397766},{"id":"https://openalex.org/C192576344","wikidata":"https://www.wikidata.org/wiki/Q194706","display_name":"Boltzmann machine","level":3,"score":0.3312000036239624},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.32910001277923584},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.328000009059906},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.30970001220703125},{"id":"https://openalex.org/C43711488","wikidata":"https://www.wikidata.org/wiki/Q7534783","display_name":"Skew","level":2,"score":0.2992999851703644},{"id":"https://openalex.org/C150921843","wikidata":"https://www.wikidata.org/wiki/Q1170431","display_name":"Resampling","level":2,"score":0.29809999465942383},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.28929999470710754},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.27489998936653137},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2680000066757202},{"id":"https://openalex.org/C153083717","wikidata":"https://www.wikidata.org/wiki/Q6535263","display_name":"Leverage (statistics)","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.25699999928474426},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.25690001249313354},{"id":"https://openalex.org/C1893757","wikidata":"https://www.wikidata.org/wiki/Q3653001","display_name":"Inversion (geology)","level":3,"score":0.25110000371932983},{"id":"https://openalex.org/C20892832","wikidata":"https://www.wikidata.org/wiki/Q3271879","display_name":"Circular symmetry","level":2,"score":0.2508000135421753}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.18106","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.18106","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18106","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"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],"striking":[1],"geometric":[2],"disparity":[3,51],"has":[4],"long":[5],"persisted":[6],"in":[7,168,224,229],"the":[8,41,45,60,68,73,244],"practice":[9],"of":[10,44,130],"deep":[11],"learning.":[12],"While":[13],"modern":[14],"neural":[15],"network":[16],"architectures":[17],"naturally":[18],"exhibit":[19],"rich":[20],"symmetry":[21,69,183],"and":[22,30,102,115,135,142,158,194,205,228,239],"equivariance":[23,42,180],"properties,":[24],"popular":[25],"optimizers":[26,121],"such":[27],"as":[28,94],"Adam":[29],"its":[31,182],"variants":[32],"operate":[33],"inherently":[34],"coordinate-wise,":[35],"rendering":[36],"them":[37],"unable":[38],"to":[39,113],"respect":[40],"structures":[43],"parameter":[46,123,173],"space.":[47],"We":[48,185],"address":[49],"this":[50,78,187],"by":[52,96,108],"introducing":[53],"a":[54,83],"symmetry-compatible":[55,120,212],"principle":[56,188],"for":[57,90,122],"optimizer":[58,166],"design:":[59],"gradient":[61,104],"update":[62,178,213],"rule":[63],"should":[64],"be":[65],"equivariant":[66,88],"under":[67],"group":[70],"acting":[71],"on":[72,86,192],"corresponding":[74,245],"weight":[75],"block.":[76],"Following":[77],"principle,":[79],"we":[80,118],"first":[81],"provide":[82],"unified":[84],"perspective":[85],"bi-orthogonally":[87],"updates":[89],"general":[91,131],"matrix":[92,132],"layers,":[93],"employed":[95],"stochastic":[97],"spectral":[98],"descent,":[99],"Muon,":[100],"Scion,":[101],"polar":[103],"methods.":[105],"More":[106],"importantly,":[107],"moving":[109],"from":[110,128],"orthogonal":[111],"groups":[112],"permutation":[114],"shared-shift":[116],"symmetries,":[117],"derive":[119],"blocks":[124],"whose":[125,179],"symmetries":[126],"differ":[127],"those":[129],"layers:":[133],"embedding":[134],"LM":[136],"head":[137],"matrices,":[138],"SwiGLU":[139],"MLP":[140],"projections,":[141],"MoE":[143,196,226],"router":[144,237],"matrices.":[145],"These":[146],"constructions":[147],"include":[148],"one-sided":[149],"spectral,":[150],"row-norm,":[151,157],"hybrid":[152],"row-norm/spectral,":[153],"row-aware,":[154],"column-aware,":[155],"centered":[156],"left-spectral":[159],"updates.":[160,247],"They":[161],"yield":[162],"an":[163,177],"end-to-end":[164],"layerwise":[165],"stack":[167],"which":[169],"each":[170],"major":[171],"matrix-valued":[172],"class":[174],"is":[175],"assigned":[176],"matches":[181],"group.":[184],"corroborate":[186],"through":[189],"pre-training":[190],"experiments":[191],"dense":[193],"sparse":[195,225],"language":[197],"models,":[198,227],"including":[199],"Qwen3-0.6B-style,":[200],"Gemma":[201],"3":[202],"1B-style,":[203],"OLMoE-1B-7B-style,":[204],"downsized":[206],"gpt-oss":[207],"architectures.":[208],"Across":[209],"these":[210],"experiments,":[211],"rules":[214],"consistently":[215],"improve":[216,236],"final":[217,233],"validation":[218],"loss,":[219],"reduce":[220],"expert":[221],"load":[222],"imbalance":[223],"several":[230],"cases":[231],"control":[232],"vocabulary-logit":[234],"growth,":[235],"stability,":[238],"overall":[240],"training":[241],"stability":[242],"over":[243],"AdamW":[246]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-20T00:00:00"}