Privacy by Postprocessing the Discrete Laplace Mechanism

2026-05-07

AI Summary

This work addresses the problem of constructing unbiased estimators for sub-exponential functions over discrete data under differential privacy constraints. By introducing a novel post-processing technique, the authors demonstrate for the first time that the discrete Laplace mechanism can be transformed into an unbiased estimator for arbitrary sub-exponential functions while exactly reproducing the output distributions of either the Laplace or staircase mechanisms, thereby unifying several existing privacy-preserving approaches. The method accommodates both inherently discrete and discretizable data and exhibits broad applicability in multivariate and distributed settings. Theoretical analysis provides tight variance upper bounds, and efficient linear- or polynomial-time algorithms are developed for canonical functions such as entropy and distributional profiles. Empirical evaluations confirm the approach’s superior accuracy in federated and distributed private data analysis.

Abstract

We show that an "old dog", the classical discrete Laplace (aka.~geometric) mechanism, can "perform new tricks": 1. It can be post-processed to yield a simple, unbiased estimator of any subexponential function $f$ of the original data, giving a simple, discrete, multivariate version of the recent unbiasing result for the Laplace mechanism by Calmon et al. (FORC '25). 2. It can be post-processed to output the same distribution as the Laplace mechanism or the Staircase mechanism with identical privacy parameters. Thus, the discrete Laplace mechanism is a versatile mechanism that should be preferred over the Laplace and Staircase mechanisms whenever the data is discrete (or can be made discrete while controlling $\ell_1$-sensitivity). We show bounds on the variance of our estimator, compared to the mean square error of the biased estimator that simply evaluates the $f$ on the output of the mechanism. Though our unbiased estimator has exponential running time for worst-case functions, we show that it can often be computed in linear or polynomial time for some common functions exhibiting structure. We showcase the properties of our methods empirically with several use cases including profile and entropy estimation, as well as distributed/federated data analysis applications in which unbiasedness is key to accuracy.

Problem

• differential privacy
• discrete Laplace mechanism
• unbiased estimation
• subexponential function
• postprocessing

Innovation

• discrete Laplace mechanism
• postprocessing
• unbiased estimation
• differential privacy
• subexponential functions