arXiv subjects

Nina Mesing Stausholm

Publications and source records attributed to Nina Mesing Stausholm.

3 recordsLinked to original sources

arXiv2022

Improved Differentially Private Euclidean Distance Approximation

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 $0<α,β<1/2$ have optimal output dimension $k=Θ(α^{-2}\log(1/β))$ and sparsity $s=O(α^{-1}\log(1/β))$. 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 $δ$. 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 $δ<β^{O(1/α)}$. 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 $δ$. 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.

cs.DS↗

arXiv2022

Infinitely Divisible Noise in the Low Privacy Regime

Federated learning, in which training data is distributed among users and never shared, has emerged as a popular approach to privacy-preserving machine learning. Cryptographic techniques such as secure aggregation are used to aggregate contributions, like a model update, from all users. A robust technique for making such aggregates differentially private is to exploit infinite divisibility of the Laplace distribution, namely, that a Laplace distribution can be expressed as a sum of i.i.d. noise shares from a Gamma distribution, one share added by each user. However, Laplace noise is known to have suboptimal error in the low privacy regime for $\varepsilon$-differential privacy, where $\varepsilon > 1$ is a large constant. In this paper we present the first infinitely divisible noise distribution for real-valued data that achieves $\varepsilon$-differential privacy and has expected error that decreases exponentially with $\varepsilon$.

cs.LG↗

arXiv2020

Efficient Differentially Private $F_0$ Linear Sketching

A powerful feature of linear sketches is that from sketches of two data vectors, one can compute the sketch of the difference between the vectors. This allows us to answer fine-grained questions about the difference between two data sets. In this work, we consider how to construct sketches for weighted $F_0$, i.e., the summed weights of the elements in the data set, that are small, differentially private, and computationally efficient. Let a weight vector $w\in(0,1]^u$ be given. For $x\in\{0,1\}^u$ we are interested in estimating $\Vert x\circ w\Vert_1$ where $\circ$ is the Hadamard product (entrywise product). Building on a technique of Kushilevitz et al.~(STOC 1998), we introduce a sketch (depending on $w$) that is linear over GF(2), mapping a vector $x\in \{0,1\}^u$ to $Hx\in\{0,1\}^τ$ for a matrix $H$ sampled from a suitable distribution $\mathcal{H}$. Differential privacy is achieved by using randomized response, flipping each bit of $Hx$ with probability $p<1/2$. We show that for every choice of $0<β< 1$ and $\varepsilon=O(1)$ there exists $p<1/2$ and a distribution $\mathcal{H}$ of linear sketches of size $τ= O(\log^2(u)\varepsilon^{-2}β^{-2})$ such that: 1) For random $H\sim\mathcal{H}$ and noise vector $φ$, given $Hx + φ$ we can compute an estimate of $\Vert x\circ w\Vert_1$ that is accurate within a factor $1\pmβ$, plus additive error $O(\log(u)\varepsilon^{-2}β^{-2})$, with probability $1-1/u$, and 2) For every $H\sim\mathcal{H}$, $Hx + φ$ is $\varepsilon$-differentially private over the randomness in $φ$. The special case $w=(1,\dots,1)$ is unweighted $F_0$. Our results both improve the efficiency of existing methods for unweighted $F_0$ estimating and extend to a weighted generalization. We also give a distributed streaming implementation for estimating the size of the union between two input streams.

cs.DS↗