About the Cost of Central Privacy in Density Estimation

Published: 31 Aug 2023, Last Modified: 31 Aug 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: We study non-parametric density estimation for densities in Lipschitz and Sobolev spaces, and under central privacy. In particular, we investigate regimes where the privacy budget is not supposed to be constant. We consider the classical definition of central differential privacy, but also the more recent notion of central concentrated differential privacy. We recover the result of Barber & Duchi (2014) stating that histogram estimators are optimal against Lipschitz distributions for the L2 risk and, under regular differential privacy, we extend it to other norms and notions of privacy. Then, we investigate higher degrees of smoothness, drawing two conclusions: First, and contrary to what happens with constant privacy budget (Wasserman & Zhou, 2010), there are regimes where imposing privacy degrades the regular minimax risk of estimation on Sobolev densities. Second, so-called projection estimators are near-optimal against the same classes of densities in this new setup with pure differential privacy, but contrary to the constant privacy budget case, it comes at the cost of relaxation. With zero concentrated differential privacy, there is no need for relaxation, and we prove that the estimation is optimal.
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: Since the last revision, the changes are the following : - We deanonymized the article. - We changed to blue color that was used to identify revision changes to black. - We added the acknowledgements that specify our funding and that thank the reviewers.
Supplementary Material: pdf
Assigned Action Editor: ~Gautam_Kamath1
Submission Number: 1306