Debiasing Functions of Private Statistics in Postprocessing

TL;DR

This paper develops methods for post-processing differentially private estimates to produce unbiased estimators of functions of the original statistic, improving privacy and accuracy in various applications.

Contribution

It introduces a deconvolution-based approach for unbiased estimation of functions of private statistics, extending to general functions and noise distributions beyond Laplace.

Findings

Unbiased estimators for functions of private statistics derived using deconvolution.

Improved private mean estimation when dataset size is unknown.

Enhanced privacy guarantees using Laplace noise in per-record privacy mechanisms.

Abstract

Given a differentially private unbiased estimate $\tilde{q}=q(D) +\nu$ 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 $\nu$ 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.