{"id":"https://openalex.org/W4366331128","doi":"https://doi.org/10.1109/access.2023.3268034","title":"Concentration Inequalities and Optimal Number of Layers for Stochastic Deep Neural Networks","display_name":"Concentration Inequalities and Optimal Number of Layers for Stochastic Deep Neural Networks","publication_year":2023,"publication_date":"2023-01-01","ids":{"openalex":"https://openalex.org/W4366331128","doi":"https://doi.org/10.1109/access.2023.3268034"},"language":"en","primary_location":{"id":"doi:10.1109/access.2023.3268034","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2023.3268034","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/6514899/10103873.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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 Access","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://ieeexplore.ieee.org/ielx7/6287639/6514899/10103873.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000098655","display_name":"Michele Caprio","orcid":"https://orcid.org/0000-0002-7569-097X"},"institutions":[{"id":"https://openalex.org/I79576946","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10","country_code":"US","type":"education","lineage":["https://openalex.org/I79576946"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michele Caprio","raw_affiliation_strings":["Department of Computer and Information Science, PRECISE Center, University of Pennsylvania, Philadelphia, PA, USA"],"raw_orcid":"https://orcid.org/0000-0002-7569-097X","affiliations":[{"raw_affiliation_string":"Department of Computer and Information Science, PRECISE Center, University of Pennsylvania, Philadelphia, PA, USA","institution_ids":["https://openalex.org/I79576946"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036208626","display_name":"Sayan Mukherjee","orcid":"https://orcid.org/0000-0002-2032-2707"},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]},{"id":"https://openalex.org/I4401726909","display_name":"Center for Scalable Data Analytics and Artificial Intelligence","ror":"https://ror.org/01t4ttr56","country_code":"DE","type":"education","lineage":["https://openalex.org/I4401726909","https://openalex.org/I78650965","https://openalex.org/I926574661"]},{"id":"https://openalex.org/I926574661","display_name":"Leipzig University","ror":"https://ror.org/03s7gtk40","country_code":"DE","type":"education","lineage":["https://openalex.org/I926574661"]}],"countries":["DE","US"],"is_corresponding":false,"raw_author_name":"Sayan Mukherjee","raw_affiliation_strings":["Center for Scalable Data Analytics and Artificial Intelligence, Universit&#x00E4;t Leipzig, Leipzig, Germany","Departments of Statistical Science, Mathematics, Computer Science, and Biostatistics & Bioinformatics, Duke University, Durham, NC, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Scalable Data Analytics and Artificial Intelligence, Universit&#x00E4;t Leipzig, Leipzig, Germany","institution_ids":["https://openalex.org/I926574661","https://openalex.org/I4401726909"]},{"raw_affiliation_string":"Departments of Statistical Science, Mathematics, Computer Science, and Biostatistics & Bioinformatics, Duke University, Durham, NC, USA","institution_ids":["https://openalex.org/I170897317"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1850,"currency":"USD","value_usd":1850},"apc_paid":{"value":1850,"currency":"USD","value_usd":1850},"fwci":0.4895,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.6925605,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"11","issue":null,"first_page":"38458","last_page":"38470"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","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/T11612","display_name":"Stochastic Gradient Optimization Techniques","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"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9988999962806702,"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/artificial-neural-network","display_name":"Artificial neural network","score":0.6955402493476868},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.6374489665031433},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6168369054794312},{"id":"https://openalex.org/keywords/feedforward-neural-network","display_name":"Feedforward neural network","score":0.6036279201507568},{"id":"https://openalex.org/keywords/probabilistic-logic","display_name":"Probabilistic logic","score":0.6024149656295776},{"id":"https://openalex.org/keywords/stochastic-neural-network","display_name":"Stochastic neural network","score":0.5876501202583313},{"id":"https://openalex.org/keywords/probabilistic-neural-network","display_name":"Probabilistic neural network","score":0.5578088164329529},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5521692633628845},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.5027081966400146},{"id":"https://openalex.org/keywords/recurrent-neural-network","display_name":"Recurrent neural network","score":0.3709483742713928},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.36397355794906616},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.363899827003479},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.33653223514556885},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3077823221683502},{"id":"https://openalex.org/keywords/time-delay-neural-network","display_name":"Time delay neural network","score":0.27054721117019653},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12970805168151855},{"id":"https://openalex.org/keywords/control","display_name":"Control (management)","score":0.09904533624649048}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6955402493476868},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.6374489665031433},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6168369054794312},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.6036279201507568},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.6024149656295776},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.5876501202583313},{"id":"https://openalex.org/C134342201","wikidata":"https://www.wikidata.org/wiki/Q7246859","display_name":"Probabilistic neural network","level":4,"score":0.5578088164329529},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5521692633628845},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.5027081966400146},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.3709483742713928},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.36397355794906616},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.363899827003479},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.33653223514556885},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3077823221683502},{"id":"https://openalex.org/C175202392","wikidata":"https://www.wikidata.org/wiki/Q2434543","display_name":"Time delay neural network","level":3,"score":0.27054721117019653},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12970805168151855},{"id":"https://openalex.org/C2775924081","wikidata":"https://www.wikidata.org/wiki/Q55608371","display_name":"Control (management)","level":2,"score":0.09904533624649048},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/access.2023.3268034","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2023.3268034","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/6514899/10103873.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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 Access","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/b00184ff-8e8e-4efc-84a7-0f1223bc6107","is_oa":true,"landing_page_url":"https://research.manchester.ac.uk/en/publications/b00184ff-8e8e-4efc-84a7-0f1223bc6107","pdf_url":"https://pure.manchester.ac.uk/ws/files/328581239/Concentration_Inequalities_and_Optimal_Number_of_Layers_for_Stochastic_Deep_Neural_Networks.pdf","source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Caprio, M & Mukherjee, S 2023, 'Concentration inequalities and optimal number of layers for stochastic deep neural networks', IEEE Access, pp. 38458 - 38470. https://doi.org/10.1109/ACCESS.2023.3268034","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:doaj.org/article:e56696bfa99b4f1fb594c98513defbdb","is_oa":true,"landing_page_url":"https://doaj.org/article/e56696bfa99b4f1fb594c98513defbdb","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Access, Vol 11, Pp 38458-38470 (2023)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1109/access.2023.3268034","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2023.3268034","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/6514899/10103873.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"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 Access","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.6200000047683716,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G1257848790","display_name":null,"funder_award_id":"DBI 1661386","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G1830701211","display_name":null,"funder_award_id":"RGP005","funder_id":"https://openalex.org/F4320320338","funder_display_name":"Human Frontier Science Program"},{"id":"https://openalex.org/G3740268834","display_name":null,"funder_award_id":"01IS18026B","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G5518200836","display_name":"HDR TRIPODS: Innovations in Data Science: Integrating Stochastic Modeling, Data Representations, and Algorithms","funder_award_id":"1934964","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5854142804","display_name":null,"funder_award_id":"MURI W911NF2010080","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G5865894165","display_name":null,"funder_award_id":"DMS 16-13261","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6483228630","display_name":null,"funder_award_id":"BMBF 01IS18026B","funder_id":"https://openalex.org/F4320321114","funder_display_name":"Bundesministerium f\u00fcr Bildung und Forschung"},{"id":"https://openalex.org/G6623739242","display_name":"CAREER: Evolution of Morphological Diversity in Primates as revealed by 3D Digital Data, Comprehensive Datasets, and Automated Phenotyping","funder_award_id":"1552848","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7525462396","display_name":null,"funder_award_id":"CCF-1934964","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7623462001","display_name":null,"funder_award_id":"ARO MURI W911NF2010080","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G7785050591","display_name":null,"funder_award_id":"BCS 1552848","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7797116693","display_name":null,"funder_award_id":"NSF BCS 1552848","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7926010854","display_name":"ABI Development: Collaborative Research: The first open access digital archive for high fidelity 3D data on morphological phenomes","funder_award_id":"1661386","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8073576747","display_name":null,"funder_award_id":"1661386","funder_id":"https://openalex.org/F4320337398","funder_display_name":"Division of Biological Infrastructure"},{"id":"https://openalex.org/G8201318391","display_name":null,"funder_award_id":"DMS 17-13012","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G853552336","display_name":null,"funder_award_id":"IIS 15-46331","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320308269","display_name":"Alexander von Humboldt-Stiftung","ror":"https://ror.org/012kf4317"},{"id":"https://openalex.org/F4320320338","display_name":"Human Frontier Science Program","ror":"https://ror.org/02ebx7v45"},{"id":"https://openalex.org/F4320321114","display_name":"Bundesministerium f\u00fcr Bildung und Forschung","ror":"https://ror.org/04pz7b180"},{"id":"https://openalex.org/F4320333591","display_name":"Multidisciplinary University Research Initiative","ror":null},{"id":"https://openalex.org/F4320337398","display_name":"Division of Biological Infrastructure","ror":"https://ror.org/04qn9mx93"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4366331128.pdf","grobid_xml":"https://content.openalex.org/works/W4366331128.grobid-xml"},"referenced_works_count":69,"referenced_works":["https://openalex.org/W604762380","https://openalex.org/W1534350895","https://openalex.org/W1587317356","https://openalex.org/W1753482797","https://openalex.org/W1990494141","https://openalex.org/W2007953285","https://openalex.org/W2019449965","https://openalex.org/W2040355924","https://openalex.org/W2062058089","https://openalex.org/W2091230209","https://openalex.org/W2128084896","https://openalex.org/W2133564696","https://openalex.org/W2135354436","https://openalex.org/W2160815625","https://openalex.org/W2331143823","https://openalex.org/W2605904100","https://openalex.org/W2618530766","https://openalex.org/W2753482910","https://openalex.org/W2787438119","https://openalex.org/W2914234208","https://openalex.org/W2919115771","https://openalex.org/W2943838153","https://openalex.org/W2950297419","https://openalex.org/W2951603627","https://openalex.org/W2962847160","https://openalex.org/W2962935523","https://openalex.org/W2962990163","https://openalex.org/W2964105630","https://openalex.org/W2964515685","https://openalex.org/W2969324140","https://openalex.org/W3007564264","https://openalex.org/W3043426275","https://openalex.org/W3049480465","https://openalex.org/W3089611792","https://openalex.org/W3094287490","https://openalex.org/W3097724114","https://openalex.org/W3104574805","https://openalex.org/W3128386044","https://openalex.org/W3148563252","https://openalex.org/W4212863985","https://openalex.org/W4245777296","https://openalex.org/W4250589301","https://openalex.org/W4287664849","https://openalex.org/W4289285448","https://openalex.org/W4293097209","https://openalex.org/W4293273089","https://openalex.org/W4310114983","https://openalex.org/W4321473434","https://openalex.org/W6635108766","https://openalex.org/W6637698695","https://openalex.org/W6679434410","https://openalex.org/W6679671054","https://openalex.org/W6683595889","https://openalex.org/W6695533872","https://openalex.org/W6729999211","https://openalex.org/W6736465782","https://openalex.org/W6744475501","https://openalex.org/W6751356287","https://openalex.org/W6757204547","https://openalex.org/W6759340446","https://openalex.org/W6759828284","https://openalex.org/W6762403029","https://openalex.org/W6763588080","https://openalex.org/W6767307204","https://openalex.org/W6769665925","https://openalex.org/W6772387883","https://openalex.org/W6781829578","https://openalex.org/W6783403180","https://openalex.org/W6849348740"],"related_works":["https://openalex.org/W2998088892","https://openalex.org/W2524120878","https://openalex.org/W2104714048","https://openalex.org/W2014323024","https://openalex.org/W1595652908","https://openalex.org/W2378845890","https://openalex.org/W2089093251","https://openalex.org/W2100563590","https://openalex.org/W2950022897","https://openalex.org/W2154718502"],"abstract_inverted_index":{"We":[0,51,67],"state":[1,53],"concentration":[2],"inequalities":[3],"for":[4,21,44,59],"the":[5,8,22,25,45,49,54,60],"output":[6,23],"of":[7,11,24,48,57,75],"hidden":[9],"layers":[10,58],"a":[12,72,76],"stochastic":[13,73],"deep":[14],"neural":[15,78],"network":[16,79],"(SDNN),":[17],"as":[18,20],"well":[19],"whole":[26],"SDNN.":[27],"These":[28],"results":[29],"allow":[30],"us":[31],"to":[32,39,71],"introduce":[33],"an":[34,63],"expected":[35],"classifier":[36],"(EC),":[37],"and":[38],"give":[40],"probabilistic":[41],"upper":[42],"bound":[43],"classification":[46],"error":[47],"EC.":[50],"also":[52],"optimal":[55,64],"number":[56],"SDNN":[61],"via":[62],"stopping":[65],"procedure.":[66],"apply":[68],"our":[69],"analysis":[70],"version":[74],"feedforward":[77],"with":[80],"ReLU":[81],"activation":[82],"function.":[83]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
