{"id":"https://openalex.org/W7165630438","doi":"https://doi.org/10.48550/arxiv.2606.21683","title":"Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design","display_name":"Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design","publication_year":2026,"publication_date":"2026-06-19","ids":{"openalex":"https://openalex.org/W7165630438","doi":"https://doi.org/10.48550/arxiv.2606.21683"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.21683","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21683","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.21683","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5139216818","display_name":"Junren Chen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chen, Junren","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5051046818","display_name":"Arya Mazumdar","orcid":"https://orcid.org/0000-0003-4605-7996"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mazumdar, Arya","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10136","display_name":"Statistical Methods and Inference","score":0.5745000243186951,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10136","display_name":"Statistical Methods and Inference","score":0.5745000243186951,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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":0.2806999981403351,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.030500000342726707,"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/hessian-matrix","display_name":"Hessian matrix","score":0.7261000275611877},{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.656499981880188},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.6323000192642212},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.5656999945640564},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.45840001106262207},{"id":"https://openalex.org/keywords/population","display_name":"Population","score":0.4553999900817871}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7886999845504761},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.7261000275611877},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.656499981880188},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.6323000192642212},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5656999945640564},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5393999814987183},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.48170000314712524},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.45840001106262207},{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.4553999900817871},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4343000054359436},{"id":"https://openalex.org/C151956035","wikidata":"https://www.wikidata.org/wiki/Q1132755","display_name":"Logistic regression","level":2,"score":0.42989999055862427},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.41100001335144043},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3711000084877014},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.367000013589859},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.29019999504089355},{"id":"https://openalex.org/C48921125","wikidata":"https://www.wikidata.org/wiki/Q10861030","display_name":"Linear regression","level":2,"score":0.2612000107765198}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.21683","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21683","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.21683","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.21683","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,108],"consider":[1],"the":[2,12,43,55,60,117,134,137,153,156,171,183,193,217,233],"parameter":[3,18,106],"estimation":[4,13,45,99,235],"problem":[5],"in":[6,211,228],"logistic":[7,61,138],"regression":[8],"with":[9],"Gaussian":[10],"design:":[11],"of":[14,50,80,86,136,155,185],"a":[15,77,111,122,141,175,178,202,208],"fixed":[16],"unknown":[17],"$\u03b8^*\\in":[19],"\\mathbb{R}^d$":[20],"($\\|\u03b8^*\\|_2\\ge":[21],"1$)":[22],"from":[23,158],"$n$":[24],"i.i.d.":[25],"samples":[26],"$\\{(x_i,y_i)\\}_{i=1}^n$,":[27],"where":[28],"$x_i\\sim":[29],"N(0,I_d)$":[30],"and":[31,47,67,102,164,167,197,200,226],"$y_i|x_i":[32],"\\sim":[33],"{\\rm":[34],"Bernoulli}(1/(1+\\exp(-x_i^\\top":[35],"\u03b8^*)))$.":[36],"Our":[37],"main":[38,127],"aim":[39],"is":[40,130,174,232],"to":[41,76,116,131,242],"characterize":[42],"finite-sample":[44],"performance":[46],"convergence":[48,115],"behavior":[49],"gradient":[51,135,157],"descent":[52],"(GD)":[53],"on":[54,182,224],"maximum":[56],"likelihood":[57],"objective":[58],"(i.e.,":[59],"loss).":[62],"Under":[63],"small":[64,78],"$O(1)$":[65],"stepsize":[66],"$0$":[68],"initialization,":[69],"we":[70,150,190],"show":[71,132,169],"that":[72,96,133,148,170,206,216,227],"GD":[73,173],"linearly":[74],"converges":[75],"neighborhood":[79],"$\u03b8^*$":[81],"achieving":[82],"an":[83],"$\\ell_2$":[84],"error":[85,100,120,236],"order":[87],"$O(\\sqrt{\\|\u03b8^*\\|_2^5d/n})$.":[88],"This":[89,214],"substantially":[90],"goes":[91],"beyond":[92],"existing":[93,218],"theoretical":[94,245],"results":[95],"lack":[97],"non-asymptotic":[98,219],"rate":[101,210],"exhibit":[103,221],"much":[104],"slower":[105],"convergence.":[107],"also":[109],"establish":[110],"faster":[112],"local":[113],"linear":[114],"same":[118],"statistical":[119],"under":[121],"large":[123],"$\u0398(\\|\u03b8^*\\|_2)$":[124],"stepsize.":[125],"The":[126],"technical":[128],"component":[129],"loss":[139],"satisfies":[140],"certain":[142],"approximate":[143],"invertibility":[144],"condition":[145],"(AIC).":[146],"To":[147],"end,":[149],"uniformly":[151],"control":[152],"deviation":[154],"its":[159],"population":[160,172,186],"counterpart":[161],"by":[162,177],"covering":[163],"peeling":[165],"arguments,":[166],"then":[168],"contraction":[176],"delicate":[179],"analysis":[180],"based":[181],"eigenvalues":[184],"Hessian":[187],"matrices.":[188],"Finally,":[189],"build":[191],"upon":[192],"recent":[194],"work":[195],"Matsumoto":[196],"Mazumdar":[198],"(2025)":[199],"devise":[201],"novel":[203],"efficient":[204],"estimator":[205],"attains":[207],"sharper":[209],"high":[212],"dimensions.":[213],"indicates":[215],"guarantees":[220],"sub-optimal":[222],"dependence":[223],"$\\|\u03b8^*\\|_2$,":[225],"many":[229],"regimes":[230],"$\u0398(\\sqrt{\\|\u03b8^*\\|_2d/n})$":[231],"tight":[234],"rate.":[237],"Numerical":[238],"examples":[239],"are":[240],"provided":[241],"corroborate":[243],"our":[244],"results.":[246]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-24T00:00:00"}