Privately Learning Smooth Distributions on the Hypercube by Projections

Clément Lalanne (1) , Sébastien Gadat (1, 2)

1 TSE-R - Toulouse School of Economics
2 IUF - Institut universitaire de France

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., 2023) 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.

Mots clés: Nonpara- metric estimation; Differential privacy; Statistics

Domaines: Statistiques [math.ST]

Clément Lalanne, Sébastien Gadat. Privately Learning Smooth Distributions on the Hypercube by Projections. 2024. ⟨hal-04549279⟩