{"id":"https://openalex.org/W4367603340","doi":"https://doi.org/10.1007/s40314-023-02307-9","title":"Stochastic perturbation of subgradient algorithm for nonconvex deep neural networks","display_name":"Stochastic perturbation of subgradient algorithm for nonconvex deep neural networks","publication_year":2023,"publication_date":"2023-05-01","ids":{"openalex":"https://openalex.org/W4367603340","doi":"https://doi.org/10.1007/s40314-023-02307-9"},"language":"en","primary_location":{"id":"doi:10.1007/s40314-023-02307-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40314-023-02307-9","pdf_url":null,"source":{"id":"https://openalex.org/S4210195884","display_name":"Computational and Applied Mathematics","issn_l":"1807-0302","issn":["1807-0302","2238-3603"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://doi.org/10.1007/s40314-023-02307-9","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5089447839","display_name":"Abdelkrim El Mouatasim","orcid":"https://orcid.org/0000-0001-9947-1408"},"institutions":[{"id":"https://openalex.org/I4210088687","display_name":"Universit\u00e9 Ibn Zohr","ror":"https://ror.org/006sgpv47","country_code":"MA","type":"education","lineage":["https://openalex.org/I4210088687"]}],"countries":["MA"],"is_corresponding":true,"raw_author_name":"A. El Mouatasim","raw_affiliation_strings":["Mathematical and Management Department, Ibnou Zohr University, FPO, Ouarzazate, Morocco"],"raw_orcid":"https://orcid.org/0000-0001-9947-1408","affiliations":[{"raw_affiliation_string":"Mathematical and Management Department, Ibnou Zohr University, FPO, Ouarzazate, Morocco","institution_ids":["https://openalex.org/I4210088687"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007518223","display_name":"Eduardo Souza de Cursi","orcid":"https://orcid.org/0000-0001-9184-2489"},"institutions":[{"id":"https://openalex.org/I4210105918","display_name":"Normandie Universit\u00e9","ror":"https://ror.org/01k40cz91","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918"]},{"id":"https://openalex.org/I62396329","display_name":"Universit\u00e9 de Rouen Normandie","ror":"https://ror.org/03nhjew95","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918","https://openalex.org/I62396329"]},{"id":"https://openalex.org/I88814501","display_name":"Institut National des Sciences Appliqu\u00e9es Rouen Normandie","ror":"https://ror.org/020ws7586","country_code":"FR","type":"education","lineage":["https://openalex.org/I4210105918","https://openalex.org/I88814501"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"J. E. Souza de Cursi","raw_affiliation_strings":["Laboratoire de M\u00e9canique de Normandie, Normandie Universit\u00e9, INSA Rouen Normandie, 685, Avenue de l\u2019Universit\u00e9 Saint-Etienne du Rouvray, Rouen, France","Laboratoire de M\u00e9canique de Normandie, Normandie Universit\u00e9, INSA Rouen Normandie, 685, Avenue de l'Universit\u00e9 Saint-Etienne du Rouvray, Rouen, France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de M\u00e9canique de Normandie, Normandie Universit\u00e9, INSA Rouen Normandie, 685, Avenue de l\u2019Universit\u00e9 Saint-Etienne du Rouvray, Rouen, France","institution_ids":["https://openalex.org/I62396329","https://openalex.org/I88814501","https://openalex.org/I4210105918"]},{"raw_affiliation_string":"Laboratoire de M\u00e9canique de Normandie, Normandie Universit\u00e9, INSA Rouen Normandie, 685, Avenue de l'Universit\u00e9 Saint-Etienne du Rouvray, Rouen, France","institution_ids":["https://openalex.org/I62396329","https://openalex.org/I88814501","https://openalex.org/I4210105918"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083730212","display_name":"Rachid Ellaia","orcid":null},"institutions":[{"id":"https://openalex.org/I126477371","display_name":"Mohammed V University","ror":"https://ror.org/00r8w8f84","country_code":"MA","type":"education","lineage":["https://openalex.org/I126477371"]},{"id":"https://openalex.org/I127336678","display_name":"Ecole Mohammadia d'Ing\u00e9nieurs","ror":"https://ror.org/00md3qm14","country_code":"MA","type":"education","lineage":["https://openalex.org/I127336678"]}],"countries":["MA"],"is_corresponding":false,"raw_author_name":"R. Ellaia","raw_affiliation_strings":["LERMA, Mohammadia School of Engineers, Mohammed V University in Rabat, Avenue Ibn Sina BP765, Agdal, Rabat, Morocco"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LERMA, Mohammadia School of Engineers, Mohammed V University in Rabat, Avenue Ibn Sina BP765, Agdal, Rabat, Morocco","institution_ids":["https://openalex.org/I127336678","https://openalex.org/I126477371"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089447839"],"corresponding_institution_ids":["https://openalex.org/I4210088687"],"apc_list":null,"apc_paid":null,"fwci":0.852,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.77859902,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"42","issue":"4","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":0.9998999834060669,"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":0.9998999834060669,"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/T10320","display_name":"Neural Networks and Applications","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/T12676","display_name":"Machine Learning and ELM","score":0.9993000030517578,"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/subgradient-method","display_name":"Subgradient method","score":0.9383812546730042},{"id":"https://openalex.org/keywords/mnist-database","display_name":"MNIST database","score":0.8757262229919434},{"id":"https://openalex.org/keywords/simultaneous-perturbation-stochastic-approximation","display_name":"Simultaneous perturbation stochastic approximation","score":0.7711610794067383},{"id":"https://openalex.org/keywords/convolutional-neural-network","display_name":"Convolutional neural network","score":0.6966800689697266},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6703269481658936},{"id":"https://openalex.org/keywords/stochastic-gradient-descent","display_name":"Stochastic gradient descent","score":0.5528959631919861},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5066609978675842},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.505209743976593},{"id":"https://openalex.org/keywords/stochastic-optimization","display_name":"Stochastic optimization","score":0.4986855983734131},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.4959281384944916},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.47244101762771606},{"id":"https://openalex.org/keywords/descent-direction","display_name":"Descent direction","score":0.44732117652893066},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.44729384779930115},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4250499904155731},{"id":"https://openalex.org/keywords/feedforward-neural-network","display_name":"Feedforward neural network","score":0.42391330003738403},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4039667844772339},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.4037418067455292},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.24274936318397522},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.23787924647331238},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.23597609996795654}],"concepts":[{"id":"https://openalex.org/C158968445","wikidata":"https://www.wikidata.org/wiki/Q7631150","display_name":"Subgradient method","level":2,"score":0.9383812546730042},{"id":"https://openalex.org/C190502265","wikidata":"https://www.wikidata.org/wiki/Q17069496","display_name":"MNIST database","level":3,"score":0.8757262229919434},{"id":"https://openalex.org/C2779880469","wikidata":"https://www.wikidata.org/wiki/Q17084424","display_name":"Simultaneous perturbation stochastic approximation","level":3,"score":0.7711610794067383},{"id":"https://openalex.org/C81363708","wikidata":"https://www.wikidata.org/wiki/Q17084460","display_name":"Convolutional neural network","level":2,"score":0.6966800689697266},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6703269481658936},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.5528959631919861},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5066609978675842},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.505209743976593},{"id":"https://openalex.org/C194387892","wikidata":"https://www.wikidata.org/wiki/Q1747770","display_name":"Stochastic optimization","level":2,"score":0.4986855983734131},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.4959281384944916},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.47244101762771606},{"id":"https://openalex.org/C116149140","wikidata":"https://www.wikidata.org/wiki/Q2070951","display_name":"Descent direction","level":4,"score":0.44732117652893066},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.44729384779930115},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4250499904155731},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.42391330003738403},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4039667844772339},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4037418067455292},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.24274936318397522},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.23787924647331238},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.23597609996795654},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s40314-023-02307-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40314-023-02307-9","pdf_url":null,"source":{"id":"https://openalex.org/S4210195884","display_name":"Computational and Applied Mathematics","issn_l":"1807-0302","issn":["1807-0302","2238-3603"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s40314-023-02307-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s40314-023-02307-9","pdf_url":null,"source":{"id":"https://openalex.org/S4210195884","display_name":"Computational and Applied Mathematics","issn_l":"1807-0302","issn":["1807-0302","2238-3603"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computational and Applied Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W1925449164","https://openalex.org/W1964755931","https://openalex.org/W1990626142","https://openalex.org/W1990787369","https://openalex.org/W2002011878","https://openalex.org/W2009086942","https://openalex.org/W2015861736","https://openalex.org/W2040870580","https://openalex.org/W2059900648","https://openalex.org/W2063238891","https://openalex.org/W2085145661","https://openalex.org/W2141125852","https://openalex.org/W2147800946","https://openalex.org/W2483920296","https://openalex.org/W2546302380","https://openalex.org/W2676080692","https://openalex.org/W2913634680","https://openalex.org/W2915912345","https://openalex.org/W2921019137","https://openalex.org/W2959400106","https://openalex.org/W2963131120","https://openalex.org/W2963273475","https://openalex.org/W2964159641","https://openalex.org/W3028452399","https://openalex.org/W3036100489","https://openalex.org/W3044864997","https://openalex.org/W3045233820","https://openalex.org/W3066853188","https://openalex.org/W3083378728","https://openalex.org/W3094547167","https://openalex.org/W3195917190","https://openalex.org/W4230946174","https://openalex.org/W4231109964","https://openalex.org/W4247050319","https://openalex.org/W4388297464"],"related_works":["https://openalex.org/W4364382936","https://openalex.org/W4367603340","https://openalex.org/W2110651519","https://openalex.org/W2012437081","https://openalex.org/W2000459133","https://openalex.org/W2734870733","https://openalex.org/W4286899070","https://openalex.org/W4221167213","https://openalex.org/W4323366756","https://openalex.org/W2118205332"],"abstract_inverted_index":{"Choosing":[0],"a":[1,5,46,60,111],"learning":[2,26,55,113],"rate":[3,27,56],"is":[4,57,74,101],"necessary":[6],"part":[7],"of":[8,21,48,66,125],"any":[9],"subgradient":[10,50],"method":[11,51,100],"optimization.":[12],"With":[13],"deeper":[14],"models":[15],"such":[16],"as":[17],"convolutional":[18,87,118],"neural":[19,84,88,119],"networks":[20,85],"image":[22,79,126],"classification,":[23],"fine-tuning":[24],"the":[25,49,54,93,123],"can":[28],"quickly":[29],"become":[30],"tedious,":[31],"and":[32,107,117,130],"it":[33],"does":[34],"not":[35],"always":[36],"result":[37],"in":[38,52,63],"optimal":[39],"convergence.":[40],"In":[41],"this":[42],"work,":[43],"we":[44],"suggest":[45],"variation":[47],"which":[53],"updated":[58],"by":[59],"control":[61],"step":[62],"each":[64,67],"iteration":[65],"epoch.":[68],"Stochastic":[69,104],"Perturbation":[70],"Subgradient":[71],"Algorithm":[72],"(SPSA)":[73],"our":[75,98],"approach":[76],"for":[77],"tackling":[78],"classification":[80,127],"issues":[81],"with":[82,110],"deep":[83],"including":[86,128],"networks.":[89],"Used":[90],"MNIST":[91],"dataset,":[92],"numerical":[94],"results":[95,124],"reveal":[96],"that":[97],"SPSA":[99,116],"faster":[102],"than":[103],"Gradient":[105],"Descent":[106],"its":[108],"variants":[109],"fixed":[112],"rate.":[114],"However":[115],"network":[120],"model":[121],"improve":[122],"loss":[129],"accuracy.":[131]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2023,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}