{"id":"https://openalex.org/W7225613143","doi":"https://doi.org/10.5281/zenodo.24069886","title":"Randomized smoothing at high confidence: the Gaussian is optimal for every noise law","display_name":"Randomized smoothing at high confidence: the Gaussian is optimal for every noise law","publication_year":2026,"publication_date":"2026-11-08","ids":{"openalex":"https://openalex.org/W7225613143","doi":"https://doi.org/10.5281/zenodo.24069886"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24069886","doi":"https://doi.org/10.5281/zenodo.24069886","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069886","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24069886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041315596","display_name":"Yitzchak Shmalo","orcid":"https://orcid.org/0000-0002-2886-3049","observed_orcids":["https://orcid.org/0000-0002-2886-3049"]},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Shmalo, Yitzchak","raw_affiliation_strings":["Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041315596"],"corresponding_institution_ids":["https://openalex.org/I197251160"],"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9390000104904175,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9390000104904175,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.01360000018030405,"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/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.008100000210106373,"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/randomized-smoothing","display_name":"randomized smoothing","score":0.9959999918937683},{"id":"https://openalex.org/keywords/certified-robustness","display_name":"certified robustness","score":0.8669999837875366},{"id":"https://openalex.org/keywords/gaussian-noise","display_name":"Gaussian noise","score":0.4970000088214874},{"id":"https://openalex.org/keywords/noise-law","display_name":"noise law","score":0.4129999876022339},{"id":"https://openalex.org/keywords/second-moment","display_name":"second moment","score":0.4129999876022339},{"id":"https://openalex.org/keywords/adversarial-robustness","display_name":"adversarial robustness","score":0.5910000205039978}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.741100013256073},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.6539000272750854},{"id":"https://openalex.org/C4199805","wikidata":"https://www.wikidata.org/wiki/Q2725903","display_name":"Gaussian noise","level":2,"score":0.6003999710083008},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4580000042915344},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.4440000057220459},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4124000072479248},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.3968000113964081},{"id":"https://openalex.org/C29265498","wikidata":"https://www.wikidata.org/wiki/Q7047719","display_name":"Noise measurement","level":3,"score":0.3488999903202057},{"id":"https://openalex.org/C95623464","wikidata":"https://www.wikidata.org/wiki/Q1096149","display_name":"Classifier (UML)","level":2,"score":0.3481999933719635},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3215999901294708},{"id":"https://openalex.org/C22679943","wikidata":"https://www.wikidata.org/wiki/Q159375","display_name":"Standard deviation","level":2,"score":0.31369999051094055},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.30979999899864197},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C2909318450","wikidata":"https://www.wikidata.org/wiki/Q15222032","display_name":"Unbiased Estimation","level":3,"score":0.28060001134872437},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.2727000117301941},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.27149999141693115},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.2709999978542328},{"id":"https://openalex.org/C2781170535","wikidata":"https://www.wikidata.org/wiki/Q30587856","display_name":"Noisy data","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C44249647","wikidata":"https://www.wikidata.org/wiki/Q208498","display_name":"Confidence interval","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C2984485829","wikidata":"https://www.wikidata.org/wiki/Q100766421","display_name":"Noise level","level":3,"score":0.26010000705718994},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.2540000081062317},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24069886","doi":"https://doi.org/10.5281/zenodo.24069886","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069886","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.24069887","doi":"https://doi.org/10.5281/zenodo.24069887","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069887","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.24069886","doi":"https://doi.org/10.5281/zenodo.24069886","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24069886","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3103443469","https://openalex.org/W7215070903","https://openalex.org/W3110717408","https://openalex.org/W4399421130","https://openalex.org/W3047877360","https://openalex.org/W4396986644","https://openalex.org/W3207297971","https://openalex.org/W2953334130","https://openalex.org/W7167700682","https://openalex.org/W2979313717"],"abstract_inverted_index":{"Randomized":[0],"smoothing":[1],"certifies":[2,71],"a":[3,7,11],"classifier":[4,9,25],"by":[5],"averaging":[6],"base":[8,24],"over":[10],"noise":[12,91,99],"law":[13],"$q$":[14],"on":[15],"$\\mathbb{R}^d$.":[16],"We":[17,41],"consider":[18],"certificates":[19],"that":[20,29,43],"hold":[21],"for":[22,47,51,89],"every":[23,84],"and":[26,53],"require":[27],"only":[28,116],"the":[30,39,62,94,107,117],"top":[31],"class":[32],"has":[33],"probability":[34],"at":[35,83,97],"least":[36],"$1-\\alpha$":[37],"under":[38],"noise.":[40],"prove":[42],"if":[44],"$\\mathbb{E}\\|\\eta\\|_2^2\\le":[45],"d\\sigma^2$":[46],"$\\eta\\sim":[48],"q$,":[49],"then":[50],"$0<\\alpha<1/4$":[52],"$d\\ge":[54],"1012\\,\\alpha^{-2}\\log(16/\\alpha)$":[55],"no":[56,122],"such":[57],"certificate":[58],"exceeds":[59],"$8\\sigma\\sqrt{\\log(16/\\alpha)}\\,d^{1/p-1/2}$":[60],"in":[61],"$\\ell_p$":[63],"norm,":[64],"$2\\le":[65],"p\\le\\infty$.":[66],"The":[67,113],"Gaussian":[68],"$N(0,\\sigma^2":[69],"I)$":[70],"$\\sigma\\Phi^{-1}(1-\\alpha)\\,d^{1/p-1/2}$,":[72],"so":[73],"it":[74],"is":[75,110],"optimal":[76],"up":[77],"to":[78,101],"an":[79],"absolute":[80],"constant":[81],"factor":[82],"confidence":[85],"level.":[86],"Earlier":[87],"bounds":[88],"general":[90],"laws":[92],"allow":[93],"certified":[95],"radius":[96],"fixed":[98],"level":[100],"grow":[102],"like":[103],"$\\alpha^{-1/2}$":[104],"as":[105],"$\\alpha\\to0$;":[106],"correct":[108],"rate":[109],"$(\\log":[111],"1/\\alpha)^{1/2}$.":[112],"proof":[114],"uses":[115],"second":[118],"moment":[119],"of":[120],"$q$:":[121],"product":[123],"structure,":[124],"symmetry,":[125],"density":[126],"or":[127],"log-concavity.":[128]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}