{"id":"https://openalex.org/W7220722482","doi":"https://doi.org/10.48550/arxiv.2610.05595","title":"Moment-Accurate Gaussian Mixtures for Constant-Step Stochastic Approximation","display_name":"Moment-Accurate Gaussian Mixtures for Constant-Step Stochastic Approximation","publication_year":2026,"publication_date":"2026-10-04","ids":{"openalex":"https://openalex.org/W7220722482","doi":"https://doi.org/10.48550/arxiv.2610.05595"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.05595","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05595","pdf_url":"https://arxiv.org/pdf/2610.05595","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/2610.05595","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157735190","display_name":"Xiaoli Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Li, Xiaoli","raw_affiliation_strings":["University of Chicago"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Chicago","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157726838","display_name":"Wei Biao Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wu, Wei Biao","raw_affiliation_strings":["University of Chicago"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Chicago","institution_ids":["https://openalex.org/I40347166"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I40347166"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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":0.9452999830245972,"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":0.9452999830245972,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.02019999921321869,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.006300000008195639,"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/gaussian-mixture-model","display_name":"Gaussian mixture model","score":0.7749999761581421},{"id":"https://openalex.org/keywords/moment-matching","display_name":"moment matching","score":0.6899999976158142},{"id":"https://openalex.org/keywords/ornstein-uhlenbeck-process","display_name":"Ornstein-Uhlenbeck process","score":0.7250000238418579},{"id":"https://openalex.org/keywords/invariant-measures","display_name":"invariant measures","score":0.5609999895095825},{"id":"https://openalex.org/keywords/saddle-points","display_name":"saddle points","score":0.5289999842643738},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"stochastic gradient descent","score":0.7089999914169312},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"weak convergence","score":0.46000000834465027}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8199999928474426},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.6832000017166138},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.6157000064849854},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.558899998664856},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5214999914169312},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.49619999527931213},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.4740999937057495},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.43140000104904175},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4275999963283539},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.4106999933719635},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40849998593330383},{"id":"https://openalex.org/C51267290","wikidata":"https://www.wikidata.org/wiki/Q5527848","display_name":"Gaussian random field","level":4,"score":0.398499995470047},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.38830000162124634},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.37869998812675476},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.3767000138759613},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.33329999446868896},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.33309999108314514},{"id":"https://openalex.org/C4199805","wikidata":"https://www.wikidata.org/wiki/Q2725903","display_name":"Gaussian noise","level":2,"score":0.322299987077713},{"id":"https://openalex.org/C36299963","wikidata":"https://www.wikidata.org/wiki/Q1369844","display_name":"Observability","level":2,"score":0.32089999318122864},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.319599986076355},{"id":"https://openalex.org/C115667082","wikidata":"https://www.wikidata.org/wiki/Q4200518","display_name":"Invariant measure","level":3,"score":0.3133000135421753},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.29739999771118164},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.2939999997615814},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.2800000011920929},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.2630000114440918},{"id":"https://openalex.org/C171078966","wikidata":"https://www.wikidata.org/wiki/Q111029","display_name":"Root (linguistics)","level":2,"score":0.2606000006198883},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.25870001316070557}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.05595","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05595","pdf_url":"https://arxiv.org/pdf/2610.05595","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.2610.05595","doi":"https://doi.org/10.48550/arxiv.2610.05595","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.05595","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":"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.05595","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05595","pdf_url":"https://arxiv.org/pdf/2610.05595","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/W7220722482.pdf","grobid_xml":"https://content.openalex.org/works/W7220722482.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1557324374","https://openalex.org/W2075167161","https://openalex.org/W2119489903","https://openalex.org/W2164864577","https://openalex.org/W3127649930","https://openalex.org/W4214535770","https://openalex.org/W4233762729","https://openalex.org/W4383860522","https://openalex.org/W4398796931","https://openalex.org/W4412097067","https://openalex.org/W6963997914","https://openalex.org/W7129244992"],"related_works":["https://openalex.org/W7215086941","https://openalex.org/W7129244992","https://openalex.org/W7220739644","https://openalex.org/W7127396070","https://openalex.org/W3214203167","https://openalex.org/W7221764958","https://openalex.org/W4284890751","https://openalex.org/W4416983543","https://openalex.org/W7212171236","https://openalex.org/W7129616961"],"abstract_inverted_index":{"Local":[0],"Gaussian":[1,25],"models":[2],"of":[3,112,132],"constant-step":[4],"learning":[5],"predict":[6],"output":[7],"variability":[8],"and":[9,74,88,98,146],"expected":[10,85],"losses,":[11],"but":[12],"weak":[13,42],"convergence":[14],"alone":[15],"does":[16],"not":[17],"justify":[18],"these":[19],"moment":[20],"predictions.":[21],"We":[22],"establish":[23],"moment-accurate":[24],"mixtures":[26],"by":[27,107],"matching":[28],"stationary":[29],"energy":[30],"with":[31,77],"local":[32],"Ornstein--Uhlenbeck":[33],"limits,":[34],"ruling":[35],"out":[36],"quadratic":[37],"tail":[38],"mass":[39],"invisible":[40],"to":[41],"convergence.":[43],"For":[44,103],"step":[45,142],"size":[46],"$a$,":[47],"the":[48,130],"second-order":[49],"Wasserstein":[50],"error":[51],"is":[52],"$o(\\sqrt":[53],"a)$,":[54],"uniformly":[55],"over":[56],"invariant":[57],"laws,":[58],"using":[59],"each":[60],"law's":[61],"actual":[62],"root":[63,75],"weights.":[64],"The":[65,80],"assumptions":[66],"combine":[67],"confinement,":[68],"descent,":[69],"finitely":[70],"many":[71],"hyperbolic":[72],"equilibria":[73],"continuity":[76],"finite-variance":[78],"innovations.":[79],"result":[81],"yields":[82],"observable":[83,139],"covariances,":[84,95],"objective":[86],"gaps":[87],"first-order":[89],"mean":[90],"shifts,":[91],"while":[92],"allowing":[93],"singular":[94],"compatible":[96],"saddles":[97],"weights":[99],"without":[100],"a":[101,108],"limit.":[102],"additive":[104],"noise":[105],"given":[106],"fixed":[109],"invertible":[110],"transform":[111],"independent":[113],"standardized":[114],"Student":[115],"$t_3$":[116],"coordinates,":[117],"symmetry":[118],"gives":[119],"an":[120],"order-sharp":[121],"$\\sqrt":[122],"a$":[123],"smooth-test":[124],"bound.":[125],"Numerical":[126],"transport":[127],"calculations":[128],"demonstrate":[129],"value":[131],"root-specific":[133],"covariances;":[134],"controlled":[135],"SGD":[136],"studies":[137],"assess":[138],"predictions":[140],"across":[141],"sizes,":[143],"batch":[144],"sizes":[145],"model":[147],"geometries.":[148]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}