Privacy by Postprocessing the Discrete Laplace Mechanism

TL;DR

This paper demonstrates that the classical discrete Laplace mechanism can be post-processed to produce unbiased estimators for functions of data and replicate other mechanisms, enhancing privacy-preserving data analysis for discrete data.

Contribution

It introduces a method to post-process the discrete Laplace mechanism for unbiased estimation and distribution matching, improving privacy and utility in discrete data scenarios.

Findings

Unbiased estimators can be derived from the discrete Laplace mechanism.

The method can replicate the Laplace and Staircase mechanisms with the same privacy parameters.

Empirical results show improved accuracy in profile, entropy estimation, and federated 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.