Debiasing Functions of Private Statistics in Postprocessing

2025-02-18

AI Summary

In differential privacy, applying nonlinear transformations to noisy statistics (e.g., Laplace-mechanism outputs) induces systematic bias due to the lack of closed-form unbiased estimators. Method: This paper proposes a post-processing-only unbiased estimation framework: (i) it introduces statistical deconvolution to the DP post-processing setting for the first time; (ii) it constructs an unbiased estimator for general twice-differentiable functions; (iii) it designs the first mechanism that yields an unbiased estimate of the private mean even when the sample size is unknown; and (iv) it extends the framework to per-record DP and polynomial function estimation. Contributions/Results: Theoretical analysis guarantees unbiasedness under finite-moment noise. Experiments demonstrate significant improvements over Kamath et al. (2023) in jointly estimating the private mean and sample size, and stronger privacy guarantees than Finley et al. (2024) under per-record DP.

Technology Category

• Machine Learning: Privacy
• Natural Language Processing: Ethics — Bias, Fairness, Transparency & Privacy
• Computer Vision: Bias, Fairness & Privacy

Application Category

• Security and Privacy: Data transparency and provenance
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

Given a differentially private unbiased estimate $ ilde{q}=q(D) + u$ 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 $ ilde{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 $ u$ 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.

Problem

• Unbiased estimation of functions from private statistics.
• Extending deconvolution for Laplace noise applications.
• Enhancing privacy guarantees using Laplace noise mechanisms.

Innovation

• Debiasing via deconvolution method
• Unbiased estimators for functions
• Laplace noise for privacy