{"id":"https://openalex.org/W2935777426","doi":"https://doi.org/10.1109/icassp.2019.8683241","title":"Perturbed Projected Gradient Descent Converges to Approximate Second-order Points for Bound Constrained Nonconvex Problems","display_name":"Perturbed Projected Gradient Descent Converges to Approximate Second-order Points for Bound Constrained Nonconvex Problems","publication_year":2019,"publication_date":"2019-04-17","ids":{"openalex":"https://openalex.org/W2935777426","doi":"https://doi.org/10.1109/icassp.2019.8683241","mag":"2935777426"},"language":"en","primary_location":{"id":"doi:10.1109/icassp.2019.8683241","is_oa":false,"landing_page_url":"https://doi.org/10.1109/icassp.2019.8683241","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","raw_type":"proceedings-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/A5088593720","display_name":"Songtao Lu","orcid":"https://orcid.org/0000-0001-9256-9648"},"institutions":[{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]},{"id":"https://openalex.org/I4210101327","display_name":"Twin Cities Orthopedics","ror":"https://ror.org/01en4s460","country_code":"US","type":"healthcare","lineage":["https://openalex.org/I4210101327"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Songtao Lu","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, 55455, USA","Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, 55455, USA","institution_ids":["https://openalex.org/I130238516"]},{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA","institution_ids":["https://openalex.org/I4210101327","https://openalex.org/I130238516"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074721617","display_name":"Ziping Zhao","orcid":"https://orcid.org/0000-0002-8668-6263"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Ziping Zhao","raw_affiliation_strings":["Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Hong Kong","institution_ids":["https://openalex.org/I200769079"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002695199","display_name":"Kejun Huang","orcid":"https://orcid.org/0000-0002-6460-6365"},"institutions":[{"id":"https://openalex.org/I33213144","display_name":"University of Florida","ror":"https://ror.org/02y3ad647","country_code":"US","type":"education","lineage":["https://openalex.org/I33213144"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kejun Huang","raw_affiliation_strings":["Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 32611, USA","Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 32611, USA","institution_ids":["https://openalex.org/I33213144"]},{"raw_affiliation_string":"Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, USA","institution_ids":["https://openalex.org/I33213144"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100633783","display_name":"Mingyi Hong","orcid":"https://orcid.org/0000-0003-1263-9365"},"institutions":[{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]},{"id":"https://openalex.org/I4210101327","display_name":"Twin Cities Orthopedics","ror":"https://ror.org/01en4s460","country_code":"US","type":"healthcare","lineage":["https://openalex.org/I4210101327"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mingyi Hong","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, 55455, USA","Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, 55455, USA","institution_ids":["https://openalex.org/I130238516"]},{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of Minnesota Twin Cities, Minneapolis, MN, USA","institution_ids":["https://openalex.org/I4210101327","https://openalex.org/I130238516"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.1868,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.47102279,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"5356","last_page":"5360"},"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":0.9976000189781189,"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":0.9939000010490417,"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/saddle-point","display_name":"Saddle point","score":0.8316745758056641},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.690748929977417},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.6632567048072815},{"id":"https://openalex.org/keywords/gradient-method","display_name":"Gradient method","score":0.5560012459754944},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5548113584518433},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5461665987968445},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5227901339530945},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.5112249851226807},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.5057315826416016},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5003097057342529},{"id":"https://openalex.org/keywords/proximal-gradient-methods","display_name":"Proximal Gradient Methods","score":0.4756488502025604},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4717710018157959},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.46943119168281555},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.44118931889533997},{"id":"https://openalex.org/keywords/descent-direction","display_name":"Descent direction","score":0.43298470973968506},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.4238450527191162},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.37024742364883423},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3075166344642639},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17595112323760986},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07727494835853577},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.05471155047416687}],"concepts":[{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.8316745758056641},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.690748929977417},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.6632567048072815},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.5560012459754944},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5548113584518433},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5461665987968445},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5227901339530945},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.5112249851226807},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5057315826416016},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5003097057342529},{"id":"https://openalex.org/C10494615","wikidata":"https://www.wikidata.org/wiki/Q17086765","display_name":"Proximal Gradient Methods","level":4,"score":0.4756488502025604},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4717710018157959},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.46943119168281555},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.44118931889533997},{"id":"https://openalex.org/C116149140","wikidata":"https://www.wikidata.org/wiki/Q2070951","display_name":"Descent direction","level":4,"score":0.43298470973968506},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.4238450527191162},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.37024742364883423},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3075166344642639},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17595112323760986},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07727494835853577},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.05471155047416687},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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},{"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/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"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/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1109/icassp.2019.8683241","is_oa":false,"landing_page_url":"https://doi.org/10.1109/icassp.2019.8683241","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","raw_type":"proceedings-article"},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-98726","is_oa":false,"landing_page_url":"http://repository.hkust.edu.hk/ir/Record/1783.1-98726","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"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":"Conference paper"},{"id":"pmh:oai:repository.ust.hk:1783.1-98726","is_oa":false,"landing_page_url":"http://www.scopus.com/record/display.url?eid=2-s2.0-85068986430&origin=inward","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Conference paper"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Life below water","id":"https://metadata.un.org/sdg/14","score":0.6600000262260437}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W147998453","https://openalex.org/W1902027874","https://openalex.org/W1978804995","https://openalex.org/W2009941369","https://openalex.org/W2093522043","https://openalex.org/W2125772126","https://openalex.org/W2134618502","https://openalex.org/W2469230926","https://openalex.org/W2474090883","https://openalex.org/W2560848383","https://openalex.org/W2592651140","https://openalex.org/W2595304583","https://openalex.org/W2604130501","https://openalex.org/W2890842400","https://openalex.org/W2897032310","https://openalex.org/W2936906164","https://openalex.org/W2951781382","https://openalex.org/W2963075511","https://openalex.org/W2963092340","https://openalex.org/W2963202546","https://openalex.org/W2963349772","https://openalex.org/W2963417959","https://openalex.org/W2963446085","https://openalex.org/W2964106499","https://openalex.org/W3113425034","https://openalex.org/W3115592686","https://openalex.org/W4254313632","https://openalex.org/W6680025182","https://openalex.org/W6713348437","https://openalex.org/W6720643467","https://openalex.org/W6730290784","https://openalex.org/W6734280532","https://openalex.org/W6735923916","https://openalex.org/W6741721318","https://openalex.org/W6744060662","https://openalex.org/W6746050020","https://openalex.org/W6754512606","https://openalex.org/W6787706204"],"related_works":["https://openalex.org/W2978731891","https://openalex.org/W2592651140","https://openalex.org/W4224903686","https://openalex.org/W4295694585","https://openalex.org/W4387770141","https://openalex.org/W4200512840","https://openalex.org/W2372778757","https://openalex.org/W2899313272","https://openalex.org/W2916621068","https://openalex.org/W4295266778"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"a":[3,58,93],"gradient-based":[4],"method":[5],"for":[6,57],"bound":[7],"constrained":[8],"non-convex":[9],"problems":[10],"is":[11,55,82,100],"proposed.":[12],"By":[13],"leveraging":[14],"both":[15],"projected":[16,28],"gradient":[17,21,29,66],"descent":[18,30],"and":[19],"perturbed":[20,27],"descent,":[22],"the":[23,65,69,109,113,117,135],"proposed":[24,53],"algorithm,":[25],"named":[26],"(PP-GD),":[31],"converges":[32,127],"to":[33,102,130],"some":[34],"approximate":[35,43,104],"second-order":[36,44,105],"stationary":[37],"(SS2)":[38],"points":[39],"(which":[40],"satisfy":[41],"certain":[42],"necessary":[45],"conditions)":[46],"with":[47],"provable":[48],"convergence":[49],"rate":[50],"guarantees.":[51],"The":[52],"algorithm":[54],"suitable":[56],"large-scale":[59],"problem":[60],"since":[61],"it":[62],"only":[63],"uses":[64],"information":[67],"of":[68,112,137],"objective":[70,114],"function.":[71],"It":[72],"also":[73],"seamlessly":[74],"incorporates":[75],"variable":[76],"constraints":[77],"such":[78],"as":[79],"nonnegativity,":[80],"which":[81],"commonly":[83],"seen":[84],"in":[85,134],"many":[86],"practical":[87],"machine":[88],"learning":[89],"problems.":[90],"We":[91],"provide":[92],"concrete":[94],"theoretical":[95],"analysis":[96],"showing":[97],"that":[98,124],"PP-GD":[99,125],"able":[101],"obtain":[103],"solutions":[106],"by":[107],"extracting":[108],"negative":[110],"curvature":[111],"function":[115],"around":[116],"strict":[118,138],"saddle":[119,139],"points.":[120,140],"Numerical":[121],"results":[122],"demonstrate":[123],"indeed":[126],"faster":[128],"compared":[129],"other":[131],"first-order":[132],"methods":[133],"presence":[136]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2026-07-02T09:51:11.867554","created_date":"2025-10-10T00:00:00"}