Debiasing Functions of Private Statistics in Postprocessing
Flavio Calmon, Elbert Du, Cynthia Dwork, Brian Finley, Grigory Franguridi
TL;DR
This work develops a deconvolution-based framework to obtain unbiased post-processing estimators for functions of private statistics under differential privacy, primarily using Laplace noise. It provides closed-form unbiased estimators for a broad class of twice-differentiable tempered functions (notably including 1/q) and extends to non-tempered cases via domain bounds and polynomial extensions, enabling unbiased estimation even when n is unknown. The authors apply these results to private mean estimation, slowly scaling per-record DP, and polynomials under general noise distributions, demonstrating practical advantages and proposing avenues for multivariate and non-Laplace extensions. Overall, the paper advances unbiased post-processing techniques for DP, with concrete mechanisms and theoretical guarantees that improve utility in privacy-preserving data analysis.
Abstract
Given a differentially private unbiased estimate $\tilde{q}=q(D) +ν$ of a statistic $q(D)$, we wish to obtain unbiased estimates of functions of $q(D)$, such as $1/q(D)$, solely through post-processing of $\tilde{q}$, with no further access to the confidential dataset $D$. To this end, we adapt the deconvolution method used for unbiased estimation in the statistical literature, deriving unbiased estimators for a broad family of twice-differentiable functions when the privacy-preserving noise $ν$ is drawn from the Laplace distribution (Dwork et al., 2006). We further extend this technique to a more general class of functions, deriving approximately optimal estimators that are unbiased for values in a user-specified interval (possibly extending to $\pm \infty$). We use these results to derive an unbiased estimator for private means when the size $n$ of the dataset is not publicly known. In a numerical application, we find that a mechanism that uses our estimator to return an unbiased sample size and mean outperforms a mechanism that instead uses the previously known unbiased privacy mechanism for such means (Kamath et al., 2023). We also apply our estimators to develop unbiased transformation mechanisms for per-record differential privacy, a privacy concept in which the privacy guarantee is a public function of a record's value (Seeman et al., 2024). Our mechanisms provide stronger privacy guarantees than those in prior work (Finley et al., 2024) by using Laplace, rather than Gaussian, noise. Finally, using a different approach, we go beyond Laplace noise by deriving unbiased estimators for polynomials under the weak condition that the noise distribution has sufficiently many moments.