{"id":"https://openalex.org/W7220466417","doi":"https://doi.org/10.48550/arxiv.2610.02791","title":"Finding Stationary Points under Higher-Order Smoothness: Trapping Algorithms and First-Order Lower Bounds","display_name":"Finding Stationary Points under Higher-Order Smoothness: Trapping Algorithms and First-Order Lower Bounds","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220466417","doi":"https://doi.org/10.48550/arxiv.2610.02791"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.02791","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02791","pdf_url":"https://arxiv.org/pdf/2610.02791","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/2610.02791","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157424367","display_name":"Huanjian Zhou","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Zhou, Huanjian","raw_affiliation_strings":["The University of Tokyo"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The University of Tokyo","institution_ids":["https://openalex.org/I74801974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072744508","display_name":"Masashi Sugiyama","orcid":"https://orcid.org/0000-0001-6658-6743","observed_orcids":["https://orcid.org/0000-0001-6658-6743"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sugiyama, Masashi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5157628609","display_name":"Taiji Suzuki","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I4210126580","display_name":"RIKEN Center for Advanced Intelligence Project","ror":"https://ror.org/03ckxwf91","country_code":"JP","type":"facility","lineage":["https://openalex.org/I4210110652","https://openalex.org/I4210126580"]},{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Suzuki, Taiji","raw_affiliation_strings":["RIKEN Center for Advanced Intelligence Project","The University of Tokyo"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"RIKEN Center for Advanced Intelligence Project","institution_ids":["https://openalex.org/I4210126580"]},{"raw_affiliation_string":"The University of Tokyo","institution_ids":["https://openalex.org/I74801974"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.6826000213623047,"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":0.6826000213623047,"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":0.22020000219345093,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.043800000101327896,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/approximate-stationary-points","display_name":"approximate stationary points","score":0.8970000147819519},{"id":"https://openalex.org/keywords/query-complexity","display_name":"query complexity","score":0.9160000085830688},{"id":"https://openalex.org/keywords/kkt-points","display_name":"KKT points","score":0.6259999871253967},{"id":"https://openalex.org/keywords/constrained-optimization","display_name":"constrained optimization","score":0.48100000619888306}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7250000238418579},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6865000128746033},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5254999995231628},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.515999972820282},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.4966999888420105},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.48910000920295715},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48100000619888306},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4796999990940094},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.453000009059906},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.44040000438690186},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4228000044822693},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41819998621940613},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.4002000093460083},{"id":"https://openalex.org/C13533509","wikidata":"https://www.wikidata.org/wiki/Q1064349","display_name":"Deterministic algorithm","level":2,"score":0.3765000104904175},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3560999929904938},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.35429999232292175},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3294000029563904},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.3176000118255615},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.3109999895095825},{"id":"https://openalex.org/C153658351","wikidata":"https://www.wikidata.org/wiki/Q746264","display_name":"Constant function","level":3,"score":0.30230000615119934},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.29679998755455017},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.29660001397132874},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2856999933719635},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.28529998660087585},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.2768999934196472},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C134466208","wikidata":"https://www.wikidata.org/wiki/Q1520657","display_name":"Weight function","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.2637999951839447},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.25189998745918274},{"id":"https://openalex.org/C66690126","wikidata":"https://www.wikidata.org/wiki/Q2914302","display_name":"Concave function","level":3,"score":0.25110000371932983},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.25}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.02791","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02791","pdf_url":"https://arxiv.org/pdf/2610.02791","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.2610.02791","doi":"https://doi.org/10.48550/arxiv.2610.02791","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02791","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.02791","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02791","pdf_url":"https://arxiv.org/pdf/2610.02791","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":[{"id":"https://openalex.org/G2207295377","display_name":"\u5927\u898f\u6a21\u57fa\u76e4\u30e2\u30c7\u30eb\u306e\u9769\u65b0\u7684\u539f\u7406\u89e3\u660e\u3068\u65b9\u6cd5\u8ad6\u306e\u65b0\u5c55\u958b","funder_award_id":"24K02905","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320320709","display_name":"National Research Foundation Singapore","ror":"https://ror.org/03cpyc314"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"},{"id":"https://openalex.org/F4320338075","display_name":"Core Research for Evolutional Science and Technology","ror":"https://ror.org/00097mb19"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220466417.pdf","grobid_xml":"https://content.openalex.org/works/W7220466417.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4390097971","https://openalex.org/W7160321266","https://openalex.org/W7163636643"],"related_works":["https://openalex.org/W2766533971","https://openalex.org/W4296151198","https://openalex.org/W7221756104","https://openalex.org/W7214957700","https://openalex.org/W4414938305","https://openalex.org/W2765150911","https://openalex.org/W7221813324","https://openalex.org/W4297899866","https://openalex.org/W7213523973","https://openalex.org/W2999758721"],"abstract_inverted_index":{"We":[0],"study":[1],"the":[2,77,136,178],"query":[3,47,103],"complexity":[4],"of":[5],"finding":[6],"approximate":[7,119,126],"stationary":[8,120],"points":[9,121,128],"under":[10],"higher-order":[11],"smoothness":[12],"using":[13,107,113],"only":[14,18],"function":[15,108],"values":[16,109],"or":[17],"gradients.":[19,114],"Our":[20],"results":[21],"cover":[22],"different":[23],"dimensions,":[24],"oracle":[25],"models,":[26],"randomization,":[27],"and":[28,34,53,110,125],"constraints.":[29],"For":[30,63,82,95,143,155],"constant":[31],"dimension":[32],"$d$":[33],"$C^{p,\u03b2_p}$":[35,96],"objectives,":[36,66,158],"with":[37,46,98],"$\u03bd:=p+\u03b2_p$,":[38],"we":[39,85,147,181],"give":[40],"deterministic":[41,78,102],"function-value":[42],"gradient":[43,71],"trapping":[44],"algorithms":[45],"complexities":[48,104],"$O(\\varepsilon^{-(d-1)})$":[49],"on":[50,55,73,122,129],"$\\mathbb":[51,74,123],"R^d$":[52,124],"$O(\\varepsilon^{-(d-1)/\u03bd})$":[54],"$[0,1]^d$.":[56,130],"High-order":[57],"line":[58],"integration":[59,68],"enables":[60],"gradient-only":[61,79,153],"trapping.":[62],"two-dimensional":[64],"$C^{1,1}$":[65,179],"randomized":[67,149],"yields":[69],"$O(\\varepsilon^{-4/3})$":[70],"queries":[72],"R^2$,":[75],"breaking":[76],"$\u0398(\\varepsilon^{-2})$":[80],"barrier.":[81],"general":[83],"dimension,":[84,146],"introduce":[86],"a":[87],"trapping-or-descent":[88],"framework":[89],"that":[90],"lazily":[91],"reuses":[92],"boundary":[93],"models.":[94],"objectives":[97],"$p\\ge2$,":[99],"it":[100],"achieves":[101],"$O(d^p+d^{p-1+\u03b2_p/(\u03bd-1)}":[105],"\\varepsilon^{-\u03bd/(\u03bd-1)})$":[106,112],"$O(d^{p-1}+d^{p-2+\u03b2_p/(\u03bd-1)}":[111],"These":[115],"bounds":[116,151,167],"apply":[117],"to":[118,172],"KKT":[127],"Their":[131],"dimension-independent":[132],"accuracy":[133],"exponent":[134,139],"matches":[135],"classical":[137],"AR$p$":[138],"$(p+1)/p$":[140],"when":[141],"$\u03b2_p=1$.":[142],"sufficiently":[144],"high":[145],"prove":[148],"lower":[150,160,185],"for":[152,168,175,187],"methods.":[154],"$C^{1,1}\\cap":[156],"C^{p,1}$":[157],"our":[159],"bound":[161,186],"is":[162],"$\u03a9(\\varepsilon^{-(3/2+1/(2p))})$,":[163],"matching":[164],"known":[165],"upper":[166],"$p=2$":[169],"and,":[170],"up":[171],"logarithmic":[173],"factors,":[174],"$p=3$.":[176],"Without":[177],"assumption,":[180],"obtain":[182],"an":[183],"$\u03a9(\\varepsilon^{-5/3})$":[184],"every":[188],"fixed":[189],"$p\\ge3$.":[190]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-06T00:00:00"}