Towards multi-purpose locally differentially-private synthetic data release via spline wavelet plug-in estimation

We develop plug-in estimators for locally differentially private semi-parametric estimation via spline wavelets. The approach leads to optimal rates of convergence for a large class of estimation problems that are characterized by (differentiable) functionals $\mathrm{\Lambda }(f)$ of the true data generating density $f$. The crucial feature of the locally private data $Z_1,\dots ,Z_n$ we generate is that it does not depend on the particular functional $\mathrm{\Lambda }$ (or the unknown density $f$) the analyst wants to estimate. Hence, the synthetic data can be generated and stored a priori and can subsequently be used by any number of analysts to estimate many vastly different functionals of interest at the provably optimal rate. In principle, this removes a long standing practical limitation in statistics of differential privacy, namely, that optimal privacy mechanisms need to be tailored towards the specific estimation problem at hand.