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.

翻译：我们证明了一个“老方法”——经典的离散拉普拉斯（亦称几何）机制——能够“玩出新花样”：1. 对其进行后处理可得到原始数据任意次指数函数$f$的简单无偏估计量，这对应了Calmon等人（FORC '25）近期针对拉普拉斯机制去偏结果的一个简单、离散、多变量版本。2. 对其进行后处理可输出与拉普拉斯机制或阶梯机制相同的分布，且具有相同的隐私参数。因此，当数据为离散数据（或可通过控制$\ell_1$敏感性实现离散化）时，离散拉普拉斯机制是一种更应优先于拉普拉斯和阶梯机制的多功能机制。我们给出了估计量方差的界限，并与简单对机制输出计算$f$的有偏估计量的均方误差进行了比较。尽管对于最坏情况函数，我们的无偏估计量需要指数级运行时间，但研究表明，对于某些具有结构特性的常见函数，它通常可以在线性或多项式时间内计算。我们通过多个用例（包括轮廓估计和熵估计，以及无偏性对精度至关重要的分布式/联邦数据分析应用）实证展示了我们方法的特性。