{"id":"https://openalex.org/W7211990240","doi":"https://doi.org/10.48550/arxiv.2609.08021","title":"Bi-ZOL: Bilevel Zeroth-Order Learning with Nonsmooth Responses","display_name":"Bi-ZOL: Bilevel Zeroth-Order Learning with Nonsmooth Responses","publication_year":2026,"publication_date":"2026-09-07","ids":{"openalex":"https://openalex.org/W7211990240","doi":"https://doi.org/10.48550/arxiv.2609.08021"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.08021","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08021","pdf_url":"https://arxiv.org/pdf/2609.08021","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,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.08021","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5137187913","display_name":"Zhisen Jiang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jiang, Zhisen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5018168527","display_name":"Saverio Bolognani","orcid":"https://orcid.org/0000-0002-7935-1385","observed_orcids":["https://orcid.org/0000-0002-7935-1385"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bolognani, Saverio","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":true,"cited_by_count":1,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.7865999937057495,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10545","display_name":"Optimization and Variational Analysis","score":0.7865999937057495,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.1770000010728836,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.01850000023841858,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/bi-level-optimization","display_name":"bi-level optimization","score":0.9800000190734863},{"id":"https://openalex.org/keywords/derivative-free-optimization","display_name":"derivative-free optimization","score":0.9610000252723694},{"id":"https://openalex.org/keywords/locally-lipschitz-functions","display_name":"locally Lipschitz functions","score":0.7829999923706055},{"id":"https://openalex.org/keywords/clarke-subdifferential","display_name":"Clarke subdifferential","score":0.8410000205039978},{"id":"https://openalex.org/keywords/chain-rule","display_name":"chain rule","score":0.652999997138977},{"id":"https://openalex.org/keywords/frank-wolfe-algorithm","display_name":"Frank-Wolfe algorithm","score":0.843999981880188},{"id":"https://openalex.org/keywords/critical-points","display_name":"critical points","score":0.6510000228881836},{"id":"https://openalex.org/keywords/structural-bias","display_name":"structural bias","score":0.6179999709129333}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7138000130653381},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6787999868392944},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.6044999957084656},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.5968000292778015},{"id":"https://openalex.org/C3309286","wikidata":"https://www.wikidata.org/wiki/Q4907693","display_name":"Bilevel optimization","level":3,"score":0.5845999717712402},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5458999872207642},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5404000282287598},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.48590001463890076},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4797999858856201},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.39910000562667847},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.3783000111579895},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.36570000648498535},{"id":"https://openalex.org/C140779682","wikidata":"https://www.wikidata.org/wiki/Q210868","display_name":"Sampling (signal processing)","level":3,"score":0.32739999890327454},{"id":"https://openalex.org/C177769412","wikidata":"https://www.wikidata.org/wiki/Q278090","display_name":"Prior probability","level":3,"score":0.2847000062465668},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C2779231336","wikidata":"https://www.wikidata.org/wiki/Q7534724","display_name":"Sketch","level":2,"score":0.2533000111579895}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.08021","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08021","pdf_url":"https://arxiv.org/pdf/2609.08021","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,"listed_in":[],"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":"doi:10.48550/arxiv.2609.08021","doi":"https://doi.org/10.48550/arxiv.2609.08021","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.08021","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,"listed_in":[],"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.08021","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.08021","pdf_url":"https://arxiv.org/pdf/2609.08021","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,"listed_in":[],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7211990240.pdf","grobid_xml":"https://content.openalex.org/works/W7211990240.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W4399413626","https://openalex.org/W4403006692","https://openalex.org/W4403571156","https://openalex.org/W4416117110","https://openalex.org/W7123349775","https://openalex.org/W7126239887","https://openalex.org/W7128820432","https://openalex.org/W7131813967","https://openalex.org/W7133217789"],"related_works":["https://openalex.org/W4408965248","https://openalex.org/W4416132418","https://openalex.org/W4400064923","https://openalex.org/W4323651112","https://openalex.org/W7126239887","https://openalex.org/W4313484466","https://openalex.org/W4414969655","https://openalex.org/W4406763956","https://openalex.org/W4416434914","https://openalex.org/W7160998767"],"abstract_inverted_index":{"This":[0,98],"paper":[1],"studies":[2],"lower-level-constrained":[3],"bilevel":[4,74],"optimization":[5],"in":[6],"a":[7,47,66,121,134],"response-oracle":[8,181],"setting,":[9,29],"where":[10],"lower-level":[11],"model":[12],"information":[13],"is":[14,21,105,141],"unavailable":[15],"and":[16,34,88,129,150,171],"the":[17,30,56,63,73,79,85,95,110,117],"induced":[18],"response":[19,32,87,96],"mapping":[20],"locally":[22],"Lipschitz":[23],"but":[24],"potentially":[25],"nonsmooth.":[26],"In":[27],"this":[28],"classical":[31],"Jacobian":[33],"reduced":[35,58,69],"hypergradient":[36,103],"may":[37],"fail":[38],"to":[39,93,133],"exist.":[40],"We":[41,114],"propose":[42],"Bilevel":[43],"Zeroth-Order":[44],"Learning":[45],"(Bi-ZOL),":[46],"structure-guided":[48],"zeroth-order":[49,90,177],"method":[50],"for":[51,143,152],"finding":[52],"stationary":[53,137],"points":[54],"of":[55,61,65],"nonsmooth":[57],"problem.":[59],"Instead":[60],"estimating":[62],"gradient":[64],"fully":[67],"smoothed":[68],"hyperobjective,":[70],"Bi-ZOL":[71,118,166],"separates":[72],"chain-rule":[75,112],"structure:":[76],"it":[77],"keeps":[78],"exact":[80],"upper-level":[81],"partial":[82],"gradients":[83],"at":[84],"queried":[86],"uses":[89],"sampling":[91],"only":[92],"estimate":[94],"Jacobian.":[97],"construction":[99],"yields":[100],"an":[101],"approximate":[102],"that":[104,116,165],"more":[106],"directly":[107],"aligned":[108],"with":[109],"Clarke":[111],"subdifferential.":[113],"show":[115,164],"direction":[119],"admits":[120],"partial-smoothing":[122],"interpretation,":[123],"quantify":[124],"its":[125],"pointwise":[126],"structural":[127],"bias,":[128],"prove":[130],"finite-time":[131],"convergence":[132],"$(\u03b4,\u03b5)$-Bi-ZOL":[135],"Frank--Wolfe":[136],"point.":[138],"The":[139],"bias":[140],"$O(\u03b4)$":[142],"piecewise":[144,153],"$C^{1,1}$":[145],"responses":[146,155],"under":[147,179],"local":[148],"regularity":[149],"vanishes":[151],"affine":[154],"on":[156,160],"active-cell":[157],"neighborhoods.":[158],"Experiments":[159],"incentive-based":[161],"tracking":[162],"problems":[163],"achieves":[167],"smaller":[168],"stationarity":[169],"gaps":[170],"lower":[172],"hyperobjective":[173],"values":[174],"than":[175],"vanilla":[176],"smoothing":[178],"comparable":[180],"budgets.":[182]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}
