{"id":"https://openalex.org/W2971617498","doi":"https://doi.org/10.1137/17m1113898","title":"Gradient Descent Finds the Cubic-Regularized Nonconvex Newton Step","display_name":"Gradient Descent Finds the Cubic-Regularized Nonconvex Newton Step","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2971617498","doi":"https://doi.org/10.1137/17m1113898","mag":"2971617498"},"language":"en","primary_location":{"id":"doi:10.1137/17m1113898","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1113898","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/A5035869365","display_name":"Yair Carmon","orcid":"https://orcid.org/0000-0001-5731-8640"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Yair Carmon","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5061331093","display_name":"John C. Duchi","orcid":"https://orcid.org/0000-0003-0045-7185"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"John Duchi","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5035869365"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.9183,"has_fulltext":false,"cited_by_count":43,"citation_normalized_percentile":{"value":0.96683081,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":99},"biblio":{"volume":"29","issue":"3","first_page":"2146","last_page":"2178"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9995999932289124,"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":0.9995999932289124,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9976000189781189,"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/mathematics","display_name":"Mathematics","score":0.8482075929641724},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.6568539142608643},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6030541658401489},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.5844798684120178},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5704140663146973},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5632486939430237},{"id":"https://openalex.org/keywords/descent","display_name":"Descent (aeronautics)","score":0.5455675721168518},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5444203615188599},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5307222604751587},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.48486050963401794},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.47032392024993896},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.46335792541503906},{"id":"https://openalex.org/keywords/descent-direction","display_name":"Descent direction","score":0.4591675400733948},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38187554478645325},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32143524289131165},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.19397133588790894},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14133358001708984}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8482075929641724},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.6568539142608643},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6030541658401489},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.5844798684120178},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5704140663146973},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5632486939430237},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.5455675721168518},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5444203615188599},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5307222604751587},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.48486050963401794},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47032392024993896},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.46335792541503906},{"id":"https://openalex.org/C116149140","wikidata":"https://www.wikidata.org/wiki/Q2070951","display_name":"Descent direction","level":4,"score":0.4591675400733948},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38187554478645325},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32143524289131165},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.19397133588790894},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14133358001708984},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"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/C119599485","wikidata":"https://www.wikidata.org/wiki/Q43035","display_name":"Electrical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/17m1113898","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1113898","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6387985466","display_name":null,"funder_award_id":"NSF-CAREER-1553086","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/F4320333856","display_name":"Stanford Artificial Intelligence Lab-Toyota Center For AI Research","ror":null}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1483062147","https://openalex.org/W1965278079","https://openalex.org/W1971664328","https://openalex.org/W1988720110","https://openalex.org/W1994974865","https://openalex.org/W2006903949","https://openalex.org/W2009941369","https://openalex.org/W2040466831","https://openalex.org/W2051664786","https://openalex.org/W2063690908","https://openalex.org/W2082877410","https://openalex.org/W2110017196","https://openalex.org/W2130984546","https://openalex.org/W2156005216","https://openalex.org/W2157725068","https://openalex.org/W2546420264","https://openalex.org/W2759377581","https://openalex.org/W2809818748","https://openalex.org/W2919115771","https://openalex.org/W2963433607"],"related_works":["https://openalex.org/W1978576933","https://openalex.org/W1981914421","https://openalex.org/W3125203780","https://openalex.org/W1979925556","https://openalex.org/W2153649672","https://openalex.org/W2032898330","https://openalex.org/W2016763547","https://openalex.org/W1987483891","https://openalex.org/W2978731891","https://openalex.org/W2592651140"],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,34,66,76],"minimization":[3],"of":[4,85,91],"a":[5,11,20,55,83],"nonconvex":[6,94],"quadratic":[7],"form":[8],"regularized":[9],"by":[10],"cubic":[12],"term,":[13],"which":[14],"may":[15],"exhibit":[16],"saddle":[17],"points":[18,90],"and":[19,47],"suboptimal":[21],"local":[22],"minimum.":[23],"Nonetheless,":[24],"we":[25,58,70],"prove":[26],"that,":[27],"under":[28],"mild":[29],"assumptions,":[30],"gradient":[31,72],"descent":[32,73],"approximates":[33],"global":[35],"minimum":[36],"to":[37,54,74,87],"within":[38],"$\\varepsilon$":[39,46,52],"accuracy":[40],"in":[41],"$O(\\varepsilon^{-1}\\log(1/\\varepsilon))$":[42],"steps":[43,49],"for":[44,50],"large":[45],"$O(\\log(1/\\varepsilon))$":[48],"small":[51],"(compared":[53],"condition":[56],"number":[57],"define),":[59],"with":[60],"at":[61],"most":[62],"logarithmic":[63],"dependence":[64],"on":[65],"problem":[67],"dimension.":[68],"When":[69],"use":[71],"approximate":[75],"cubic-regularized":[77],"Newton":[78],"step,":[79],"our":[80],"result":[81],"implies":[82],"rate":[84],"convergence":[86],"second-order":[88],"stationary":[89],"general":[92],"smooth":[93],"functions.":[95]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":9},{"year":2019,"cited_by_count":3}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
