{"id":"https://openalex.org/W3012856280","doi":"https://doi.org/10.1162/neco_a_01388","title":"Critical Point-Finding Methods Reveal Gradient-Flat Regions of Deep Network Losses","display_name":"Critical Point-Finding Methods Reveal Gradient-Flat Regions of Deep Network Losses","publication_year":2021,"publication_date":"2021-04-22","ids":{"openalex":"https://openalex.org/W3012856280","doi":"https://doi.org/10.1162/neco_a_01388","mag":"3012856280","pmid":"https://pubmed.ncbi.nlm.nih.gov/34496389"},"language":"en","primary_location":{"id":"doi:10.1162/neco_a_01388","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01388","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2003.10397","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084043552","display_name":"Charles G. Frye","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Charles G. Frye","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, U.S.A. cfrye59@gmail.com","Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, U.S.A"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, U.S.A. cfrye59@gmail.com","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, U.S.A","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013953784","display_name":"James B. Simon","orcid":"https://orcid.org/0000-0003-4210-0731"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"James Simon","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience and Department of Physics, University of California, Berkeley, CA 94720, U.S.A. james.simon@berkeley.edu","Redwood Center for Theoretical Neuroscience and Department of Physics, University of California, Berkeley, CA 94720, U.S.A"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Department of Physics, University of California, Berkeley, CA 94720, U.S.A. james.simon@berkeley.edu","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Department of Physics, University of California, Berkeley, CA 94720, U.S.A","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011846368","display_name":"Neha S. Wadia","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Neha S. Wadia","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. neha.wadia@berkeley.edu","Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. neha.wadia@berkeley.edu","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032503764","display_name":"Andrew Ligeralde","orcid":"https://orcid.org/0000-0002-4777-2847"},"institutions":[{"id":"https://openalex.org/I4210145836","display_name":"Center for Theoretical Biological Physics","ror":"https://ror.org/04hfg7v94","country_code":"US","type":"facility","lineage":["https://openalex.org/I12912129","https://openalex.org/I181547552","https://openalex.org/I2801539370","https://openalex.org/I4210145836","https://openalex.org/I44461941","https://openalex.org/I74775410"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Andrew Ligeralde","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. ligeralde@berkeley.edu","Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. ligeralde@berkeley.edu","institution_ids":["https://openalex.org/I4210145836","https://openalex.org/I95457486"]},{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031274513","display_name":"Michael R. DeWeese","orcid":"https://orcid.org/0000-0003-2801-5768"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael R. DeWeese","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience, Helen Wills Neuroscience Institute, Department of Physics, and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. deweese@berkeley.edu","Redwood Center for Theoretical Neuroscience, Helen Wills Neuroscience Institute, Department of Physics, and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience, Helen Wills Neuroscience Institute, Department of Physics, and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A. deweese@berkeley.edu","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience, Helen Wills Neuroscience Institute, Department of Physics, and Biophysics Graduate Group, University of California, Berkeley, CA 94720, U.S.A","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040319820","display_name":"Kristofer E. Bouchard","orcid":"https://orcid.org/0000-0002-1974-4603"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kristofer E. Bouchard","raw_affiliation_strings":["Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, USA; and Biological Systems and Engineering Division and Computational Research Division, Lawrence Berkeley National Lab, Berkeley, CA 94720, U.S.A. kebouchard@lbl.gov"],"affiliations":[{"raw_affiliation_string":"Redwood Center for Theoretical Neuroscience and Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, USA; and Biological Systems and Engineering Division and Computational Research Division, Lawrence Berkeley National Lab, Berkeley, CA 94720, U.S.A. kebouchard@lbl.gov","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5084043552"],"corresponding_institution_ids":["https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0051816,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"33","issue":"6","first_page":"1469","last_page":"1497"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":1.0,"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":1.0,"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/T10036","display_name":"Advanced Neural Network Applications","score":0.9957000017166138,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.995199978351593,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/maxima-and-minima","display_name":"Maxima and minima","score":0.8845982551574707},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.7921925783157349},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.7234012484550476},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6185394525527954},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.6166917085647583},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6015023589134216},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.5527660250663757},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.493687242269516},{"id":"https://openalex.org/keywords/critical-point","display_name":"Critical point (mathematics)","score":0.48938387632369995},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4098418354988098},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.40943869948387146},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.40385979413986206},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.33215999603271484},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.30508536100387573},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.2760927677154541},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2637760043144226},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.19504877924919128},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.16277199983596802}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.8845982551574707},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.7921925783157349},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.7234012484550476},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6185394525527954},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.6166917085647583},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6015023589134216},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5527660250663757},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.493687242269516},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.48938387632369995},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4098418354988098},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.40943869948387146},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40385979413986206},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.33215999603271484},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.30508536100387573},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2760927677154541},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2637760043144226},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.19504877924919128},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.16277199983596802}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016571","descriptor_name":"Neural Networks, Computer","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":7,"locations":[{"id":"doi:10.1162/neco_a_01388","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01388","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},{"id":"pmid:34496389","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/34496389","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":"Neural computation","raw_type":null},{"id":"pmh:oai:arXiv.org:2003.10397","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.10397","pdf_url":"https://arxiv.org/pdf/2003.10397","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:escholarship.org:ark:/13030/qt62b696v9","is_oa":false,"landing_page_url":"https://escholarship.org/uc/item/62b696v9","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Neural Computation, vol 33, iss 6","raw_type":"article"},{"id":"pmh:oai:pubmedcentral.nih.gov:8919680","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/8919680","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Neural Comput","raw_type":"Text"},{"id":"doi:10.48550/arxiv.2003.10397","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2003.10397","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:3012856280","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2003.10397","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2003.10397","pdf_url":"https://arxiv.org/pdf/2003.10397","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":56,"referenced_works":["https://openalex.org/W388179047","https://openalex.org/W647818151","https://openalex.org/W1522301498","https://openalex.org/W1666670079","https://openalex.org/W1988675445","https://openalex.org/W2030521309","https://openalex.org/W2041792275","https://openalex.org/W2042860072","https://openalex.org/W2043938310","https://openalex.org/W2058194880","https://openalex.org/W2078626246","https://openalex.org/W2088415187","https://openalex.org/W2101234009","https://openalex.org/W2101961546","https://openalex.org/W2140035851","https://openalex.org/W2146502635","https://openalex.org/W2146989110","https://openalex.org/W2152703617","https://openalex.org/W2155894447","https://openalex.org/W2296319761","https://openalex.org/W2474090883","https://openalex.org/W2592651140","https://openalex.org/W2604901683","https://openalex.org/W2607928667","https://openalex.org/W2618381130","https://openalex.org/W2626325961","https://openalex.org/W2754835109","https://openalex.org/W2767286248","https://openalex.org/W2780951787","https://openalex.org/W2794854061","https://openalex.org/W2795934185","https://openalex.org/W2892943996","https://openalex.org/W2896083421","https://openalex.org/W2912713668","https://openalex.org/W2919651864","https://openalex.org/W2963384892","https://openalex.org/W2963487351","https://openalex.org/W2964160102","https://openalex.org/W2964161337","https://openalex.org/W2970971581","https://openalex.org/W2986941256","https://openalex.org/W2994927236","https://openalex.org/W3001454063","https://openalex.org/W3005768598","https://openalex.org/W3007253500","https://openalex.org/W3137695714","https://openalex.org/W4250589301","https://openalex.org/W6675354045","https://openalex.org/W6681435938","https://openalex.org/W6720643467","https://openalex.org/W6738685885","https://openalex.org/W6747847215","https://openalex.org/W6754787158","https://openalex.org/W6755150206","https://openalex.org/W6758207765","https://openalex.org/W6771084760"],"related_works":["https://openalex.org/W3159746799","https://openalex.org/W4226193833","https://openalex.org/W2608609325","https://openalex.org/W2593380010","https://openalex.org/W2777256551","https://openalex.org/W3095816591","https://openalex.org/W3046584731","https://openalex.org/W2797993462","https://openalex.org/W3035323570","https://openalex.org/W3105698610","https://openalex.org/W2912330043","https://openalex.org/W2887459604","https://openalex.org/W3104504839","https://openalex.org/W3033802587","https://openalex.org/W1971230344","https://openalex.org/W3009142328","https://openalex.org/W3135381856","https://openalex.org/W1733437702","https://openalex.org/W3171635672","https://openalex.org/W2344685289"],"abstract_inverted_index":{"Despite":[0],"the":[1,4,20,41,48,52,80,108,124,129,132,136,145,148,153],"fact":[2],"that":[3,61,79,135,165],"loss":[5,49,137],"functions":[6],"of":[7,30,47,96,131,147,155,171],"deep":[8],"neural":[9,62,172,182],"networks":[10],"are":[11,54],"highly":[12],"nonconvex,":[13],"gradient-based":[14],"optimization":[15],"algorithms":[16],"converge":[17,101],"to":[18,83,102,167],"approximately":[19,127,140],"same":[21],"performance":[22],"from":[23,90],"many":[24],"random":[25],"initial":[26],"points.":[27],"One":[28],"thread":[29],"work":[31],"has":[32,111],"focused":[33],"on":[34],"explaining":[35],"this":[36],"phenomenon":[37],"by":[38],"numerically":[39],"characterizing":[40],"local":[42,93],"curvature":[43],"near":[44,55],"critical":[45,87,169],"points":[46,88,170],"function,":[50],"where":[51,107],"gradients":[53],"zero.":[56],"Such":[57],"studies":[58],"have":[59],"reported":[60],"network":[63,173],"losses":[64,174],"enjoy":[65],"a":[66,91,112],"no-bad-local-minima":[67],"property,":[68],"in":[69,128,144,160,176],"disagreement":[70],"with":[71],"more":[72],"recent":[73],"theoretical":[74],"results.":[75],"We":[76,115,150],"report":[77],"here":[78],"methods":[81,179],"used":[82],"find":[84,168],"these":[85,117,156],"putative":[86],"suffer":[89],"bad":[92],"minima":[94],"problem":[95],"their":[97],"own:":[98],"they":[99,121],"often":[100],"or":[103,142],"pass":[104],"through":[105],"regions":[106,157],"gradient":[109,125],"norm":[110],"stationary":[113],"point.":[114],"call":[116],"gradient-flat":[118],"regions,":[119],"since":[120],"arise":[122],"when":[123],"is":[126,138],"kernel":[130],"Hessian,":[133],"such":[134],"locally":[139],"linear,":[141],"flat,":[143],"direction":[146],"gradient.":[149],"describe":[151],"how":[152],"presence":[154],"necessitates":[158],"care":[159],"both":[161],"interpreting":[162],"past":[163],"results":[164],"claimed":[166],"and":[175],"designing":[177],"second-order":[178],"for":[180],"optimizing":[181],"networks.":[183]},"counts_by_year":[],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}
