A Central Limit Theorem for Differentially Private Query Answering

• URL: http://arxiv.org/abs/2103.08721v1
• Date: Mon, 15 Mar 2021 21:06:25 GMT
• Title: A Central Limit Theorem for Differentially Private Query Answering
• Authors: Jinshuo Dong, Weijie J. Su, Linjun Zhang
• Abstract summary: We show the product of the privacy parameter and the $ell$-loss of the mechanism is lower bounded by the dimension. Our findings are corroborated by numerical experiments.
• Score: 23.015107368002884
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Perhaps the single most important use case for differential privacy is to privately answer numerical queries, which is usually achieved by adding noise to the answer vector. The central question, therefore, is to understand which noise distribution optimizes the privacy-accuracy trade-off, especially when the dimension of the answer vector is high. Accordingly, extensive literature has been dedicated to the question and the upper and lower bounds have been matched up to constant factors [BUV18, SU17]. In this paper, we take a novel approach to address this important optimality question. We first demonstrate an intriguing central limit theorem phenomenon in the high-dimensional regime. More precisely, we prove that a mechanism is approximately Gaussian Differentially Private [DRS21] if the added noise satisfies certain conditions. In particular, densities proportional to $\mathrm{e}^{-\|x\|_p^\alpha}$, where $\|x\|_p$ is the standard $\ell_p$-norm, satisfies the conditions. Taking this perspective, we make use of the Cramer--Rao inequality and show an "uncertainty principle"-style result: the product of the privacy parameter and the $\ell_2$-loss of the mechanism is lower bounded by the dimension. Furthermore, the Gaussian mechanism achieves the constant-sharp optimal privacy-accuracy trade-off among all such noises. Our findings are corroborated by numerical experiments.

Related papers

• Optimized Tradeoffs for Private Prediction with Majority Ensembling [59.99331405291337]
  We introduce the Data-dependent Randomized Response Majority (DaRRM) algorithm. DaRRM is parameterized by a data-dependent noise function $gamma$, and enables efficient utility optimization over the class of all private algorithms. We show that DaRRM provably enjoys a privacy gain of a factor of 2 over common baselines, with fixed utility.
  arXiv  Detail & Related papers  (2024-11-27T00:48:48Z)
• Some Constructions of Private, Efficient, and Optimal $K$-Norm and Elliptic Gaussian Noise [54.34628844260993]
  Differentially private computation often begins with a bound on some $d$-dimensional statistic's sensitivity. For pure differential privacy, the $K$-norm mechanism can improve on this approach using a norm tailored to the statistic's sensitivity space. This paper solves both problems for the simple statistics of sum, count, and vote.
  arXiv  Detail & Related papers  (2023-09-27T17:09:36Z)
• Differential Privacy with Higher Utility by Exploiting Coordinate-wise Disparity: Laplace Mechanism Can Beat Gaussian in High Dimensions [9.20186865054847]
  In a differentially private additive noise mechanism, independent and identically distributed (i.i.d.) noise samples are added to each coordinate of the response. We study the i.n.i.d. Gaussian and Laplace mechanisms and obtain the conditions under which these mechanisms guarantee privacy.
  arXiv  Detail & Related papers  (2023-02-07T14:54:20Z)
• General Gaussian Noise Mechanisms and Their Optimality for Unbiased Mean Estimation [58.03500081540042]
  A classical approach to private mean estimation is to compute the true mean and add unbiased, but possibly correlated, Gaussian noise to it. We show that for every input dataset, an unbiased mean estimator satisfying concentrated differential privacy introduces approximately at least as much error.
  arXiv  Detail & Related papers  (2023-01-31T18:47:42Z)
• Brownian Noise Reduction: Maximizing Privacy Subject to Accuracy Constraints [53.01656650117495]
  There is a disconnect between how researchers and practitioners handle privacy-utility tradeoffs. Brownian mechanism works by first adding Gaussian noise of high variance corresponding to the final point of a simulated Brownian motion. We complement our Brownian mechanism with ReducedAboveThreshold, a generalization of the classical AboveThreshold algorithm.
  arXiv  Detail & Related papers  (2022-06-15T01:43:37Z)
• Optimal Algorithms for Mean Estimation under Local Differential Privacy [55.32262879188817]
  We show that PrivUnit achieves the optimal variance among a large family of locally private randomizers. We also develop a new variant of PrivUnit based on the Gaussian distribution which is more amenable to mathematical analysis and enjoys the same optimality guarantees.
  arXiv  Detail & Related papers  (2022-05-05T06:43:46Z)
• Differential privacy for symmetric log-concave mechanisms [0.0]
  Adding random noise to database query results is an important tool for achieving privacy. We provide a sufficient and necessary condition for $(epsilon, delta)$-differential privacy for all symmetric and log-concave noise densities.
  arXiv  Detail & Related papers  (2022-02-23T10:20:29Z)
• A unified interpretation of the Gaussian mechanism for differential privacy through the sensitivity index [61.675604648670095]
  We argue that the three prevailing interpretations of the GM, namely $(varepsilon, delta)$-DP, f-DP and R'enyi DP can be expressed by using a single parameter $psi$, which we term the sensitivity index. $psi$ uniquely characterises the GM and its properties by encapsulating its two fundamental quantities: the sensitivity of the query and the magnitude of the noise perturbation.
  arXiv  Detail & Related papers  (2021-09-22T06:20:01Z)
• Non-Euclidean Differentially Private Stochastic Convex Optimization [15.302167005107135]
  We show that noisy gradient descent (SGD) algorithms attain the optimal excess risk in low-dimensional regimes. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.
  arXiv  Detail & Related papers  (2021-03-01T19:48:44Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.