{"id":"https://openalex.org/W4392796749","doi":"https://doi.org/10.1109/tnnls.2024.3371025","title":"Learning Rates of Deep Nets for Geometrically Strongly Mixing Sequence","display_name":"Learning Rates of Deep Nets for Geometrically Strongly Mixing Sequence","publication_year":2024,"publication_date":"2024-03-14","ids":{"openalex":"https://openalex.org/W4392796749","doi":"https://doi.org/10.1109/tnnls.2024.3371025","pmid":"https://pubmed.ncbi.nlm.nih.gov/38466602"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2024.3371025","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2024.3371025","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009698197","display_name":"Yueyang Men","orcid":"https://orcid.org/0000-0001-9392-738X"},"institutions":[{"id":"https://openalex.org/I75390827","display_name":"Beijing University of Chemical Technology","ror":"https://ror.org/00df5yc52","country_code":"CN","type":"education","lineage":["https://openalex.org/I75390827"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yueyang Men","raw_affiliation_strings":["College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China"],"raw_orcid":"https://orcid.org/0000-0001-9392-738X","affiliations":[{"raw_affiliation_string":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China","institution_ids":["https://openalex.org/I75390827"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100420733","display_name":"Liang Li","orcid":"https://orcid.org/0000-0003-4723-101X"},"institutions":[{"id":"https://openalex.org/I75390827","display_name":"Beijing University of Chemical Technology","ror":"https://ror.org/00df5yc52","country_code":"CN","type":"education","lineage":["https://openalex.org/I75390827"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liang Li","raw_affiliation_strings":["College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China"],"raw_orcid":"https://orcid.org/0000-0003-4723-101X","affiliations":[{"raw_affiliation_string":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China","institution_ids":["https://openalex.org/I75390827"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052715307","display_name":"Ziqing Hu","orcid":"https://orcid.org/0009-0000-3383-2426"},"institutions":[{"id":"https://openalex.org/I75390827","display_name":"Beijing University of Chemical Technology","ror":"https://ror.org/00df5yc52","country_code":"CN","type":"education","lineage":["https://openalex.org/I75390827"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ziqing Hu","raw_affiliation_strings":["College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China"],"raw_orcid":"https://orcid.org/0009-0000-3383-2426","affiliations":[{"raw_affiliation_string":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China","institution_ids":["https://openalex.org/I75390827"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042212433","display_name":"Yongli Xu","orcid":"https://orcid.org/0000-0002-9625-3410"},"institutions":[{"id":"https://openalex.org/I37796252","display_name":"Beijing University of Technology","ror":"https://ror.org/037b1pp87","country_code":"CN","type":"education","lineage":["https://openalex.org/I37796252"]},{"id":"https://openalex.org/I75390827","display_name":"Beijing University of Chemical Technology","ror":"https://ror.org/00df5yc52","country_code":"CN","type":"education","lineage":["https://openalex.org/I75390827"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yongli Xu","raw_affiliation_strings":["College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China","Faculty of Science, College of Statistics and Data Science, Beijing University of Technology, Beijing, China"],"raw_orcid":"https://orcid.org/0000-0002-9625-3410","affiliations":[{"raw_affiliation_string":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, China","institution_ids":["https://openalex.org/I75390827"]},{"raw_affiliation_string":"Faculty of Science, College of Statistics and Data Science, Beijing University of Technology, Beijing, China","institution_ids":["https://openalex.org/I37796252"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5009698197"],"corresponding_institution_ids":["https://openalex.org/I75390827"],"apc_list":null,"apc_paid":null,"fwci":1.3245,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.82393564,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":96,"max":98},"biblio":{"volume":"36","issue":"3","first_page":"5826","last_page":"5832"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9994000196456909,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9994000196456909,"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/T10136","display_name":"Statistical Methods and Inference","score":0.9991000294685364,"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/T10320","display_name":"Neural Networks and Applications","score":0.9990000128746033,"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/mixing","display_name":"Mixing (physics)","score":0.7507701516151428},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.7028030157089233},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6730460524559021},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.6537717580795288},{"id":"https://openalex.org/keywords/empirical-risk-minimization","display_name":"Empirical risk minimization","score":0.647567093372345},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5964739918708801},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5458531379699707},{"id":"https://openalex.org/keywords/sample","display_name":"Sample (material)","score":0.509009599685669},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.5080280303955078},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4933084547519684},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.4361257255077362},{"id":"https://openalex.org/keywords/deep-neural-networks","display_name":"Deep neural networks","score":0.4297560751438141},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.4163312315940857},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.35717329382896423},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.32458966970443726},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.29014578461647034},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.09654757380485535},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06683158874511719},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06240624189376831}],"concepts":[{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.7507701516151428},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.7028030157089233},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6730460524559021},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6537717580795288},{"id":"https://openalex.org/C107321475","wikidata":"https://www.wikidata.org/wiki/Q5374254","display_name":"Empirical risk minimization","level":2,"score":0.647567093372345},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5964739918708801},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5458531379699707},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.509009599685669},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5080280303955078},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4933084547519684},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.4361257255077362},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.4297560751438141},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.4163312315940857},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.35717329382896423},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.32458966970443726},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.29014578461647034},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.09654757380485535},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06683158874511719},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06240624189376831},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2024.3371025","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2024.3371025","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Neural Networks and Learning Systems","raw_type":"journal-article"},{"id":"pmid:38466602","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/38466602","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE transactions on neural networks and learning systems","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6700000166893005,"display_name":"Climate action","id":"https://metadata.un.org/sdg/13"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":36,"referenced_works":["https://openalex.org/W1482196458","https://openalex.org/W1484867920","https://openalex.org/W1973881932","https://openalex.org/W1996700041","https://openalex.org/W2004567286","https://openalex.org/W2017332415","https://openalex.org/W2032882463","https://openalex.org/W2046464909","https://openalex.org/W2047278710","https://openalex.org/W2051904535","https://openalex.org/W2056871913","https://openalex.org/W2075672181","https://openalex.org/W2094547532","https://openalex.org/W2136922672","https://openalex.org/W2139548827","https://openalex.org/W2147303334","https://openalex.org/W2149614431","https://openalex.org/W2163922914","https://openalex.org/W2171798319","https://openalex.org/W2528305538","https://openalex.org/W2919115771","https://openalex.org/W2962761333","https://openalex.org/W2963146412","https://openalex.org/W2963245453","https://openalex.org/W2963482148","https://openalex.org/W2965767282","https://openalex.org/W2993802668","https://openalex.org/W3014385693","https://openalex.org/W3071135787","https://openalex.org/W3093499264","https://openalex.org/W3130671418","https://openalex.org/W4200241466","https://openalex.org/W4238519965","https://openalex.org/W6685370539","https://openalex.org/W6686712754","https://openalex.org/W6801019524"],"related_works":["https://openalex.org/W3162204513","https://openalex.org/W2371138613","https://openalex.org/W2155195660","https://openalex.org/W2177289819","https://openalex.org/W2060518740","https://openalex.org/W4367626110","https://openalex.org/W2052343155","https://openalex.org/W2262246290","https://openalex.org/W2103875979","https://openalex.org/W2785875001"],"abstract_inverted_index":{"The":[0],"great":[1,31],"success":[2],"of":[3,24,86,94,103,116],"deep":[4,25],"learning":[5,65],"poses":[6],"an":[7],"urgent":[8],"challenge":[9],"to":[10,51,54],"establish":[11,62],"the":[12,22,34,40,43,68,80,92,99,117],"theoretical":[13],"basis":[14],"for":[15,67],"its":[16],"working":[17],"mechanism.":[18],"Recently,":[19],"research":[20],"on":[21,39,73,107],"convergence":[23,101],"neural":[26],"networks":[27],"(DNNs)":[28],"has":[29],"made":[30],"progress.":[32],"However,":[33],"existing":[35],"studies":[36],"are":[37,45],"based":[38,106],"assumption":[41],"that":[42],"samples":[44],"independent,":[46],"which":[47],"is":[48,82,98,112],"too":[49],"strong":[50],"be":[52],"applied":[53],"many":[55],"real-world":[56],"scenarios.":[57],"In":[58],"this":[59,97],"brief,":[60],"we":[61],"a":[63,113],"fast":[64],"rate":[66],"empirical":[69],"risk":[70],"minimization":[71],"(ERM)":[72],"DNN":[74,104],"regression":[75],"with":[76],"dependent":[77],"samples,":[78],"and":[79],"dependence":[81],"expressed":[83],"in":[84],"terms":[85],"geometrically":[87],"strongly":[88],"mixing":[89,108],"sequence.":[90],"To":[91],"best":[93],"our":[95],"knowledge,":[96],"first":[100],"result":[102,111],"methods":[105],"sequences.":[109],"This":[110],"natural":[114],"generalization":[115],"independent":[118],"sample":[119],"case.":[120]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}