Privately Answering Counting Queries with Generalized Gaussian Mechanisms

TL;DR

This paper introduces a new differentially private mechanism using Generalized Gaussian noise to answer counting queries with improved error bounds, narrowing the gap between upper and lower bounds for privacy-preserving data analysis.

Contribution

The paper presents a novel mechanism based on Generalized Gaussian distributions that improves the error bounds for answering counting queries under differential privacy.

Findings

Achieves $ ilde{O}(rac{ oot ext{log} ext{log} ext{log}k}{epsilon})$ error bound.

Reduces the gap between known upper and lower bounds for $ ext{l}_ ext{infinity}$-error.

Introduces a mechanism that balances $ ext{l}_1$ and $ ext{l}_ ext{infinity}$ errors, potentially of independent interest.

Abstract

We consider the problem of answering $k$ counting (i.e. sensitivity-1) 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.