Semiparametric privacy-constrained inference
by
Amphi L. Schwartz
1R3, RDC
We study the problem of non-parametric density estimation for densities in Sobolev spaces, under the additional constraint that only privatized data are allowed to be published and available for inference. More precisely, we focus on the estimation of a functional of the density via plug-in estimators. For this purpose, we adapt recent results from the minimax theory under the framework of local α-differential privacy. We first build a nonparametric projection estimator based on spline wavelets and add suitably scaled Laplace noise to empirical wavelet coefficients to fulfill the privacy requirement. Under some regularity assumptions of the functional, we derive convergence rates in expectation for the corresponding plug-in estimators and show that these are optimal up to a logarithmic factor. We observe different regimes in the rate, depending on the size of α and the regularity of the functional. We also provide a Lepski-type estimator that adapts to the Sobolev regularity of the density.