{"id":"https://openalex.org/W3089805682","doi":"https://doi.org/10.1109/tnnls.2020.3025157","title":"Nonergodic Complexity of Proximal Inertial Gradient Descents","display_name":"Nonergodic Complexity of Proximal Inertial Gradient Descents","publication_year":2020,"publication_date":"2020-09-30","ids":{"openalex":"https://openalex.org/W3089805682","doi":"https://doi.org/10.1109/tnnls.2020.3025157","mag":"3089805682","pmid":"https://pubmed.ncbi.nlm.nih.gov/32997636"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2020.3025157","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2020.3025157","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/A5037318331","display_name":"Tao Sun","orcid":"https://orcid.org/0000-0001-5024-1900"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Tao Sun","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha, China"],"raw_orcid":"https://orcid.org/0000-0001-5024-1900","affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, China","institution_ids":["https://openalex.org/I170215575"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5043045076","display_name":"Linbo Qiao","orcid":"https://orcid.org/0000-0002-8285-2738"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Linbo Qiao","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha, China"],"raw_orcid":"https://orcid.org/0000-0002-8285-2738","affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, China","institution_ids":["https://openalex.org/I170215575"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100440903","display_name":"Dongsheng Li","orcid":"https://orcid.org/0000-0001-9743-2034"},"institutions":[{"id":"https://openalex.org/I170215575","display_name":"National University of Defense Technology","ror":"https://ror.org/05d2yfz11","country_code":"CN","type":"education","lineage":["https://openalex.org/I170215575"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dongsheng Li","raw_affiliation_strings":["College of Computer, National University of Defense Technology, Changsha, China"],"raw_orcid":"https://orcid.org/0000-0001-9743-2034","affiliations":[{"raw_affiliation_string":"College of Computer, National University of Defense Technology, Changsha, China","institution_ids":["https://openalex.org/I170215575"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I170215575"],"apc_list":null,"apc_paid":null,"fwci":0.8749,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.69583919,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"32","issue":"10","first_page":"4613","last_page":"4626"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9998000264167786,"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/T12879","display_name":"Distributed Sensor Networks and Detection Algorithms","score":0.9890999794006348,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/sublinear-function","display_name":"Sublinear function","score":0.8098032474517822},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.646512508392334},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.5931060314178467},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.591823935508728},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5913547873497009},{"id":"https://openalex.org/keywords/inertial-frame-of-reference","display_name":"Inertial frame of reference","score":0.5281527042388916},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5091435313224792},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.46993619203567505},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4600368142127991},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.44831541180610657},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.4229428768157959},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.41465556621551514},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.37206172943115234},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3171600103378296},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.29500994086265564},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1888917088508606},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.12850388884544373}],"concepts":[{"id":"https://openalex.org/C117160843","wikidata":"https://www.wikidata.org/wiki/Q338652","display_name":"Sublinear function","level":2,"score":0.8098032474517822},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.646512508392334},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.5931060314178467},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.591823935508728},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5913547873497009},{"id":"https://openalex.org/C173386949","wikidata":"https://www.wikidata.org/wiki/Q192735","display_name":"Inertial frame of reference","level":2,"score":0.5281527042388916},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5091435313224792},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.46993619203567505},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4600368142127991},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.44831541180610657},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4229428768157959},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.41465556621551514},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.37206172943115234},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3171600103378296},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.29500994086265564},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1888917088508606},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.12850388884544373},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.0},{"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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2020.3025157","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2020.3025157","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:32997636","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/32997636","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":[],"awards":[{"id":"https://openalex.org/G285475101","display_name":null,"funder_award_id":"61932001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3627478391","display_name":null,"funder_award_id":"61906200","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4488484741","display_name":null,"funder_award_id":"2018YFB0204300","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":41,"referenced_works":["https://openalex.org/W6908809","https://openalex.org/W1686810756","https://openalex.org/W1984481261","https://openalex.org/W1988720110","https://openalex.org/W1994616650","https://openalex.org/W2000462146","https://openalex.org/W2000769684","https://openalex.org/W2006262045","https://openalex.org/W2009501510","https://openalex.org/W2013918710","https://openalex.org/W2027982384","https://openalex.org/W2063357812","https://openalex.org/W2087714402","https://openalex.org/W2088630746","https://openalex.org/W2100556411","https://openalex.org/W2101095383","https://openalex.org/W2118550318","https://openalex.org/W2124450277","https://openalex.org/W2137731592","https://openalex.org/W2146502635","https://openalex.org/W2194775991","https://openalex.org/W2221366174","https://openalex.org/W2302721240","https://openalex.org/W2460144244","https://openalex.org/W2549127690","https://openalex.org/W2610880400","https://openalex.org/W2767036780","https://openalex.org/W2780752111","https://openalex.org/W2785523195","https://openalex.org/W2903382683","https://openalex.org/W2913535645","https://openalex.org/W2962811981","https://openalex.org/W2962834223","https://openalex.org/W2963248893","https://openalex.org/W2963430672","https://openalex.org/W2963674932","https://openalex.org/W2963965485","https://openalex.org/W2964092527","https://openalex.org/W2964102336","https://openalex.org/W2964121744","https://openalex.org/W3118608800"],"related_works":["https://openalex.org/W4206903459","https://openalex.org/W2754816816","https://openalex.org/W4366280654","https://openalex.org/W3160167280","https://openalex.org/W4231621013","https://openalex.org/W4362706668","https://openalex.org/W3008318776","https://openalex.org/W1977633006","https://openalex.org/W1971945429","https://openalex.org/W2041416246"],"abstract_inverted_index":{"The":[0,46],"proximal":[1],"inertial":[2,79],"gradient":[3],"descent":[4],"(PIGD)":[5],"is":[6,50,62,97],"efficient":[7],"for":[8,14,52],"the":[9,25,38,42,59,65,74,82,85,89,94,107,116,121],"composite":[10],"minimization":[11],"and":[12,31,102,119,126],"applicable":[13],"broad":[15],"of":[16,28,41],"machine":[17],"learning":[18],"problems.":[19],"In":[20,81],"this":[21,29],"article,":[22],"we":[23,72],"revisit":[24],"computational":[26,122],"complexity":[27],"algorithm":[30],"present":[32,120],"other":[33],"novel":[34],"results,":[35],"especially":[36],"on":[37],"convergence":[39,96],"rates":[40],"objective":[43,60,66,86],"function":[44,61,67,87],"values.":[45],"nonergodic":[47],"O(1/k)":[48],"rate":[49,76],"proved":[51,98],"PIGD":[53],"with":[54,77,99],"constant":[55],"step":[56,104],"size":[57,105],"when":[58],"coercive.":[63],"When":[64],"fails":[68],"to":[69,115],"promise":[70],"coercivity,":[71],"prove":[73],"sublinear":[75],"diminishing":[78],"parameters.":[80],"case":[83],"that":[84],"satisfies":[88],"Polyak-":[90],"Lojasiewicz":[91],"(P\u0141)":[92],"property,":[93],"linear":[95],"much":[100],"larger":[101],"general":[103],"than":[106],"previous":[108],"literature.":[109],"We":[110],"also":[111],"extend":[112],"our":[113],"results":[114],"multiblock":[117],"version":[118],"complexity.":[123],"Both":[124],"cyclic":[125],"stochastic":[127],"index":[128],"selection":[129],"strategies":[130],"are":[131],"considered.":[132]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}