In this paper, we study private optimization problems for non-smooth convex functions \\(F(x)=\\mathbb{E}_i f_i(x)\\) on \\(\\mathbb{R}^d\\). We show that modifying the exponential mechanism by adding an \\(\\ell_2^2\\) regularizer to \\(F(x)\\) and sampling from \\(\\pi(x)\\propto \\exp(-k(F(x)+\\mu\\|x\\|_2^2\/2))\\) recovers both the known optimal empirical risk and population loss under \\((\u03f5,\u03b4)\\)-DP. Furthermore, we show how to implement this mechanism using \\(O\u02dc(nmin(d,n))\\) queries to \\(f_i(x)\\) for the DP-SCO where \\(n\\) is the number of samples\/users and \\(d\\) is the ambient dimension. We also give a (nearly) matching lower bound \\(\u03a9\u02dc(nmin(d,n))\\) on the number of evaluation queries.<\/p>\n
Our results utilize the following tools that are of independent interest: (1) We prove Gaussian Differential Privacy (GDP) of the exponential mechanism if the loss function is strongly convex and the perturbation is Lipschitz. Our privacy bound is optimal<\/em> as it includes the privacy of Gaussian mechanism as a special case and is proved using the isoperimetric inequality for strongly log-concave measures. (2) We show how to sample from \\(exp(\u2212F(x)\u2212\u03bc\u2225x\u22252^2\/2)\\) for \\(G\\)-Lipschitz \\(F\\) with \\(\u03b7\\) error in total variation (TV) distance using \\(O\u02dc((G^2\/\u03bc)log^2(d\/\u03b7))\\) unbiased queries to \\(F(x)\\). This is the first sampler whose query complexity has <em>polylogarithmic dependence<\/em> on both dimension \\(d\\) and accuracy \\(\u03b7\\).<\/p>\n","protected":false},"excerpt":{"rendered":" In this paper, we study private optimization problems for non-smooth convex functions on . We show that modifying the exponential mechanism by adding an regularizer to and sampling from recovers both the known optimal empirical risk and population loss under -DP. Furthermore, we show how to implement this mechanism using queries to for the DP-SCO […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13546,13558],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[260479],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-823309","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-research-area-security-privacy-cryptography","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2022-3-1","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":0,"msr_main_download":"","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"url","viewUrl":"false","id":"false","title":"https:\/\/arxiv.org\/abs\/2203.00263","label_id":"243109","label":0}],"msr_related_uploader":"","msr_attachments":[],"msr-author-ordering":[{"type":"user_nicename","value":"Sivakanth Gopi","user_id":37830,"rest_url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=Sivakanth Gopi"},{"type":"text","value":"Yin Tat Lee","user_id":0,"rest_url":false},{"type":"text","value":"Daogao Liu","user_id":0,"rest_url":false}],"msr_impact_theme":[],"msr_research_lab":[199565],"msr_event":[850618],"msr_group":[437022,761911,793670],"msr_project":[556311],"publication":[],"video":[],"download":[],"msr_publication_type":"inproceedings","related_content":{"projects":[{"ID":556311,"post_title":"Project Laplace","post_name":"project-laplace","post_type":"msr-project","post_date":"2020-02-26 12:27:22","post_modified":"2024-06-04 13:24:35","post_status":"publish","permalink":"https:\/\/www.microsoft.com\/en-us\/research\/project\/project-laplace\/","post_excerpt":"The broad goal of Project Laplace is to enable privacy-preserving data analysis and machine learning using differential privacy.","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-project\/556311"}]}}]},"_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/823309"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-research-item"}],"version-history":[{"count":6,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/823309\/revisions"}],"predecessor-version":[{"id":850654,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-item\/823309\/revisions\/850654"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=823309"}],"wp:term":[{"taxonomy":"msr-content-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-content-type?post=823309"},{"taxonomy":"msr-research-highlight","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-highlight?post=823309"},{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=823309"},{"taxonomy":"msr-publication-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-publication-type?post=823309"},{"taxonomy":"msr-product-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-product-type?post=823309"},{"taxonomy":"msr-focus-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-focus-area?post=823309"},{"taxonomy":"msr-platform","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-platform?post=823309"},{"taxonomy":"msr-download-source","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-download-source?post=823309"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=823309"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=823309"},{"taxonomy":"msr-field-of-study","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-field-of-study?post=823309"},{"taxonomy":"msr-conference","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-conference?post=823309"},{"taxonomy":"msr-journal","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-journal?post=823309"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=823309"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=823309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}