eprintid: 48670 rev_number: 10 eprint_status: archive userid: 1482 importid: 105 dir: disk0/00/04/86/70 datestamp: 2024-02-22 10:55:49 lastmod: 2024-11-04 13:31:51 status_changed: 2024-11-04 13:31:51 type: monograph metadata_visibility: show creators_name: Lalanne, Clément creators_name: Gadat, Sébastien creators_id: clement.lalanne@tse-fr.eu creators_id: sebastien.gadat@tse-fr.eu creators_idrefppn: 080889433 creators_affiliation: Toulouse School of Economics creators_affiliation: Toulouse School of Economics creators_halaffid: 1002422 creators_halaffid: 1002422 title: Privately Learning Smooth Distributions on the Hypercube by Projections ispublished: pub subjects: subjects_ECO abstract: Fueled by the ever-increasing need for statistics that guarantee the privacy of their training sets, this article studies the centrally-private estimation of Sobolev-smooth densities of probability over the hypercube in dimension d. The contributions of this article are two-fold : firstly, it generalizes the one-dimensional results of (Lalanne et al., 2023b) to non-integer levels of smoothness and to a high-dimensional setting, which is important for two reasons : it is more suited for modern learning tasks, and it allows understanding the relations between privacy, dimensionality and smoothness, which is a central question with differential privacy. Secondly, this article presents a private strategy of estimation that is data-driven (usually referred to as adaptive in Statistics) in order to privately choose an estimator that achieves a good bias-variance trade-off among a finite family of private projection estimators without prior knowledge of the ground-truth smoothness β. This is achieved by adapting the Lepskii method for private selection, by adding a new penalization term that makes the estimation privacy-aware. date: 2024-02 date_type: published publisher: TSE Working Paper official_url: http://tse-fr.eu/pub/129117 faculty: tse divisions: tse language: en has_fulltext: TRUE view_date_year: 2024 full_text_status: public monograph_type: working_paper series: TSE Working Paper volume: 24-1505 place_of_pub: Toulouse pages: 46 institution: Université Toulouse Capitole department: Toulouse School of Economics book_title: TSE Working Paper oai_identifier: oai:tse-fr.eu:129117 harvester_local_overwrite: department harvester_local_overwrite: pending harvester_local_overwrite: creators_idrefppn harvester_local_overwrite: creators_halaffid harvester_local_overwrite: pages harvester_local_overwrite: creators_id harvester_local_overwrite: institution harvester_local_overwrite: place_of_pub oai_lastmod: 2024-04-29T08:29:27Z oai_set: tse site: ut1 citation: Lalanne, Clément and Gadat, Sébastien (2024) Privately Learning Smooth Distributions on the Hypercube by Projections. TSE Working Paper, n. 24-1505, Toulouse document_url: https://publications.ut-capitole.fr/id/eprint/48670/1/wp_tse_1505.pdf