{"id":"https://openalex.org/W4399153159","doi":"https://doi.org/10.1088/2632-2153/ad524d","title":"The twin peaks of learning neural networks","display_name":"The twin peaks of learning neural networks","publication_year":2024,"publication_date":"2024-05-30","ids":{"openalex":"https://openalex.org/W4399153159","doi":"https://doi.org/10.1088/2632-2153/ad524d"},"language":"en","primary_location":{"id":"doi:10.1088/2632-2153/ad524d","is_oa":true,"landing_page_url":"https://doi.org/10.1088/2632-2153/ad524d","pdf_url":"https://iopscience.iop.org/article/10.1088/2632-2153/ad524d/pdf","source":{"id":"https://openalex.org/S4210200687","display_name":"Machine Learning Science and Technology","issn_l":"2632-2153","issn":["2632-2153"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning: Science and Technology","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://iopscience.iop.org/article/10.1088/2632-2153/ad524d/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050301474","display_name":"Elizaveta Demyanenko","orcid":"https://orcid.org/0009-0002-4366-6825"},"institutions":[{"id":"https://openalex.org/I71209653","display_name":"Bocconi University","ror":"https://ror.org/05crjpb27","country_code":"IT","type":"education","lineage":["https://openalex.org/I71209653"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Elizaveta Demyanenko","raw_affiliation_strings":["Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY"],"raw_orcid":"https://orcid.org/0009-0002-4366-6825","affiliations":[{"raw_affiliation_string":"Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY","institution_ids":["https://openalex.org/I71209653"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026008808","display_name":"Christoph Feinauer","orcid":"https://orcid.org/0000-0002-8941-7333"},"institutions":[{"id":"https://openalex.org/I71209653","display_name":"Bocconi University","ror":"https://ror.org/05crjpb27","country_code":"IT","type":"education","lineage":["https://openalex.org/I71209653"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Christoph Feinauer","raw_affiliation_strings":["Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY","institution_ids":["https://openalex.org/I71209653"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053636788","display_name":"Enrico M. Malatesta","orcid":"https://orcid.org/0000-0001-8558-6175"},"institutions":[{"id":"https://openalex.org/I71209653","display_name":"Bocconi University","ror":"https://ror.org/05crjpb27","country_code":"IT","type":"education","lineage":["https://openalex.org/I71209653"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Enrico M Malatesta","raw_affiliation_strings":["Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY"],"raw_orcid":"https://orcid.org/0000-0001-8558-6175","affiliations":[{"raw_affiliation_string":"Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY","institution_ids":["https://openalex.org/I71209653"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005071277","display_name":"Luca Saglietti","orcid":"https://orcid.org/0000-0002-5691-6662"},"institutions":[{"id":"https://openalex.org/I71209653","display_name":"Bocconi University","ror":"https://ror.org/05crjpb27","country_code":"IT","type":"education","lineage":["https://openalex.org/I71209653"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Luca Saglietti","raw_affiliation_strings":["Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computing Science, Bocconi University, Via Roentgen, 1, Milan, Milano, Lombardia, 20136, ITALY","institution_ids":["https://openalex.org/I71209653"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5053636788"],"corresponding_institution_ids":["https://openalex.org/I71209653"],"apc_list":{"value":1600,"currency":"GBP","value_usd":1962},"apc_paid":{"value":1600,"currency":"GBP","value_usd":1962},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.05758533,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"5","issue":"2","first_page":"025061","last_page":"025061"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9926000237464905,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9926000237464905,"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/T11948","display_name":"Machine Learning in Materials Science","score":0.9811000227928162,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.979200005531311,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/overfitting","display_name":"Overfitting","score":0.7384107112884521},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6136813759803772},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4980623722076416},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4562801122665405},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.45184987783432007},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.4175901412963867},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38134151697158813},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32627731561660767},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.27420103549957275},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10101768374443054}],"concepts":[{"id":"https://openalex.org/C22019652","wikidata":"https://www.wikidata.org/wiki/Q331309","display_name":"Overfitting","level":3,"score":0.7384107112884521},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6136813759803772},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4980623722076416},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4562801122665405},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.45184987783432007},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.4175901412963867},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38134151697158813},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32627731561660767},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.27420103549957275},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10101768374443054},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1088/2632-2153/ad524d","is_oa":true,"landing_page_url":"https://doi.org/10.1088/2632-2153/ad524d","pdf_url":"https://iopscience.iop.org/article/10.1088/2632-2153/ad524d/pdf","source":{"id":"https://openalex.org/S4210200687","display_name":"Machine Learning Science and Technology","issn_l":"2632-2153","issn":["2632-2153"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning: Science and Technology","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:f44fc0f9e9fa4b1c850ebc3d61b7321d","is_oa":true,"landing_page_url":"https://doaj.org/article/f44fc0f9e9fa4b1c850ebc3d61b7321d","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":"Machine Learning: Science and Technology, Vol 5, Iss 2, p 025061 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1088/2632-2153/ad524d","is_oa":true,"landing_page_url":"https://doi.org/10.1088/2632-2153/ad524d","pdf_url":"https://iopscience.iop.org/article/10.1088/2632-2153/ad524d/pdf","source":{"id":"https://openalex.org/S4210200687","display_name":"Machine Learning Science and Technology","issn_l":"2632-2153","issn":["2632-2153"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310320083","host_organization_name":"IOP Publishing","host_organization_lineage":["https://openalex.org/P4310320083","https://openalex.org/P4310311669"],"host_organization_lineage_names":["IOP Publishing","Institute of Physics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Machine Learning: Science and Technology","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.4699999988079071,"display_name":"Quality Education","id":"https://metadata.un.org/sdg/4"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4399153159.pdf"},"referenced_works_count":68,"referenced_works":["https://openalex.org/W1511560458","https://openalex.org/W1522301498","https://openalex.org/W1992068214","https://openalex.org/W1998160729","https://openalex.org/W2144902422","https://openalex.org/W2149298154","https://openalex.org/W2194775991","https://openalex.org/W2657631929","https://openalex.org/W2732724430","https://openalex.org/W2786622092","https://openalex.org/W2803921058","https://openalex.org/W2945976633","https://openalex.org/W2948488743","https://openalex.org/W2959995783","https://openalex.org/W2962772482","https://openalex.org/W2963518130","https://openalex.org/W2967536008","https://openalex.org/W2992043355","https://openalex.org/W2994588524","https://openalex.org/W3004368638","https://openalex.org/W3006943693","https://openalex.org/W3008127699","https://openalex.org/W3033802587","https://openalex.org/W3034704745","https://openalex.org/W3035703482","https://openalex.org/W3083720136","https://openalex.org/W3095486251","https://openalex.org/W3103509423","https://openalex.org/W3115509486","https://openalex.org/W3130674728","https://openalex.org/W3169055364","https://openalex.org/W3171727251","https://openalex.org/W3177828909","https://openalex.org/W3204187203","https://openalex.org/W4206410067","https://openalex.org/W4224035735","https://openalex.org/W4285396995","https://openalex.org/W4292779060","https://openalex.org/W4298135002","https://openalex.org/W4311450900","https://openalex.org/W4312933868","https://openalex.org/W4384918448","https://openalex.org/W4385245566","https://openalex.org/W4386875438","https://openalex.org/W4391901266","https://openalex.org/W6630440985","https://openalex.org/W6631190155","https://openalex.org/W6648413822","https://openalex.org/W6681302627","https://openalex.org/W6687483927","https://openalex.org/W6739901393","https://openalex.org/W6740483536","https://openalex.org/W6748230616","https://openalex.org/W6751575553","https://openalex.org/W6763725367","https://openalex.org/W6765920069","https://openalex.org/W6767329639","https://openalex.org/W6771233102","https://openalex.org/W6773778736","https://openalex.org/W6774616501","https://openalex.org/W6778747846","https://openalex.org/W6778883912","https://openalex.org/W6790857096","https://openalex.org/W6809885388","https://openalex.org/W6810595431","https://openalex.org/W6839847550","https://openalex.org/W6854866820","https://openalex.org/W6856459160"],"related_works":["https://openalex.org/W4362597605","https://openalex.org/W1574414179","https://openalex.org/W3009056573","https://openalex.org/W4297676672","https://openalex.org/W2922073769","https://openalex.org/W4281702477","https://openalex.org/W2490526372","https://openalex.org/W4376166922","https://openalex.org/W4378510483","https://openalex.org/W2075445622"],"abstract_inverted_index":{"Abstract":[0],"Recent":[1],"works":[2],"demonstrated":[3],"the":[4,11,31,41,52,59,69,78,91,102,111,114,119,124,129,139,144,148,155,161],"existence":[5],"of":[6,14,54,58,80,121,126,141,143],"a":[7,46,74,86,97,169,214],"double-descent":[8],"phenomenon":[9,50],"for":[10,90,110],"generalization":[12,162],"error":[13,163],"neural":[15,63],"networks,":[16],"where":[17,118],"highly":[18],"overparameterized":[19],"models":[20,196,205],"escape":[21],"overfitting":[22],"and":[23,51,56,128,165,186,203],"achieve":[24],"good":[25],"test":[26],"performance,":[27],"at":[28,154],"odds":[29],"with":[30,160,182],"standard":[32],"bias-variance":[33],"trade-off":[34],"described":[35],"by":[36,62],"statistical":[37],"learning":[38],"theory.":[39],"In":[40,65],"present":[42],"work,":[43],"we":[44,67,95,190],"explore":[45],"link":[47],"between":[48],"this":[49],"increase":[53],"complexity":[55],"sensitivity":[57],"function":[60,82],"represented":[61],"networks.":[64],"particular,":[66],"study":[68],"Boolean":[70,81],"mean":[71],"dimension":[72],"(BMD),":[73],"metric":[75],"developed":[76],"in":[77,113,158,179],"context":[79],"analysis.":[83],"Focusing":[84],"on":[85,101],"simple":[87],"teacher-student":[88],"setting":[89],"random":[92],"feature":[93],"model,":[94],"derive":[96],"theoretical":[98],"analysis":[99],"based":[100],"replica":[103],"method":[104],"that":[105,193,204,206],"yields":[106],"an":[107,151],"interpretable":[108],"expression":[109],"BMD,":[112],"high":[115],"dimensional":[116],"regime":[117],"number":[120,125],"data":[122],"points,":[123],"features,":[127],"input":[130],"size":[131],"grow":[132],"to":[133,198,210],"infinity.":[134],"We":[135],"find":[136,191],"that,":[137],"as":[138],"degree":[140],"overparameterization":[142],"network":[145],"is":[146,176],"increased,":[147],"BMD":[149,201],"reaches":[150],"evident":[152],"peak":[153],"interpolation":[156],"threshold,":[157],"correspondence":[159],"peak,":[164],"then":[166,177],"slowly":[167],"approaches":[168],"low":[170],"asymptotic":[171],"value.":[172],"The":[173],"same":[174],"phenomenology":[175],"traced":[178],"numerical":[180],"experiments":[181],"different":[183],"model":[184],"classes":[185],"training":[187],"setups.":[188],"Moreover,":[189],"empirically":[192],"adversarially":[194],"initialized":[195],"tend":[197],"show":[199],"higher":[200],"values,":[202],"are":[207],"more":[208],"robust":[209],"adversarial":[211],"attacks":[212],"exhibit":[213],"lower":[215],"BMD.":[216]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
