Privately Answering Counting Queries with Generalized Gaussian Mechanisms
Abstract
We consider the problem of answering $k$ counting (i.e. sensitivity1) queries about a database with $(\epsilon, \delta)$differential privacy. We give a mechanism such that if the true answers to the queries are the vector $d$, the mechanism outputs answers $\tilde{d}$ with the $\ell_\infty$error guarantee: $$\mathcal{E}\left[\tilde{d}  d_\infty\right] = O\left(\frac{\sqrt{k \log \log \log k \log(1/\delta)}}{\epsilon}\right).$$ This reduces the multiplicative gap between the best known upper and lower bounds on $\ell_\infty$error from $O(\sqrt{\log \log k})$ to $O(\sqrt{\log \log \log k})$. Our main technical contribution is an analysis of the family of mechanisms of the following form for answering counting queries: Sample $x$ from a \textit{Generalized Gaussian}, i.e. with probability proportional to $\exp((x_p/\sigma)^p)$, and output $\tilde{d} = d + x$. This family of mechanisms offers a tradeoff between $\ell_1$ and $\ell_\infty$error guarantees and may be of independent interest. For $p = O(\log \log k)$, this mechanism already matches the previous best known $\ell_\infty$error bound. We arrive at our main result by composing this mechanism for $p = O(\log \log \log k)$ with the sparse vector mechanism, generalizing a technique of Steinke and Ullman.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.01457
 arXiv:
 arXiv:2010.01457
 Bibcode:
 2020arXiv201001457G
 Keywords:

 Computer Science  Data Structures and Algorithms