Improved Differentially Private Euclidean Distance Approximation

TL;DR

This paper introduces a more accurate and private method for estimating Euclidean distances between vectors using differentially private sketches, combining advanced Johnson-Lindenstrauss constructions with Laplace or Gaussian mechanisms.

Contribution

It presents a novel differentially private estimator for Euclidean distance that improves variance and privacy guarantees over previous methods, especially for small delta values.

Findings

Laplace mechanism yields lower variance than Gaussian for small delta

Proposed estimator achieves pure differential privacy

Utilizes optimized Johnson-Lindenstrauss constructions for efficiency

Abstract

This work shows how to privately and more accurately estimate Euclidean distance between pairs of vectors. Input vectors $x$ and $y$ are mapped to differentially private sketches $x'$ and $y'$, from which one can estimate the distance between $x$ and $y$. Our estimator relies on the Sparser Johnson-Lindenstrauss constructions by Kane \& Nelson (Journal of the ACM 2014), which for any $0<\alpha,\beta<1/2$ have optimal output dimension $k=\Theta(\alpha^{-2}\log(1/\beta))$ and sparsity $s=O(\alpha^{-1}\log(1/\beta))$. We combine the constructions of Kane \& Nelson with either the Laplace or the Gaussian mechanism from the differential privacy literature, depending on the privacy parameters $\varepsilon$ and $\delta$. We also suggest a differentially private version of Fast Johnson-Lindenstrauss Transform (FJLT) by Ailon \& Chazelle (SIAM Journal of Computing 2009) which offers a tradeoff in speed for variance for certain parameters. We answer an open question by Kenthapadi et al.~(Journal of Privacy and Confidentiality 2013) by analyzing the privacy and utility guarantees of an estimator for Euclidean distance, relying on Laplacian rather than Gaussian noise. We prove that the Laplace mechanism yields lower variance than the Gaussian mechanism whenever $\delta<\beta^{O(1/\alpha)}$. Thus, our work poses an improvement over the work of Kenthapadi et al.~by giving a more efficient estimator with lower variance for sufficiently small $\delta$. Our sketch also achieves \emph{pure} differential privacy as a neat side-effect of the Laplace mechanism rather than the \emph{approximate} differential privacy guarantee of the Gaussian mechanism, which may not be sufficiently strong for some settings.