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1. Central Limit Theorem and Uncertainty Principles for Differentially Private Query Answering

   Jinshuo Dong, Weijie Su ( January 2021 , Advances in neural information processing systems)

   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 is, therefore, to understand which noise distribution optimizes the privacy-accuracy trade-off, especially when the dimension of the answer vector is high. Accordingly, an extensive literature has been dedicated to the question and the upper and lower bounds have been successfully matched up to constant factors (Bun et al.,2018; Steinke & Ullman, 2017). 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 (Dong et al., 2021) if the added noise satisfies certain conditions. In particular, densities proportional to exp(-||x||_p^alpha), where ||x||_p is the standard l_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 privacy parameter and the l_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. 

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2. Optimizing fitness-for-use of differentially private linear queries

   https://doi.org/10.14778/3467861.3467864  

   Xiao, Yingtai ; Ding, Zeyu ; Wang, Yuxin ; Zhang, Danfeng ; Kifer, Daniel ( June 2021 , Proceedings of the VLDB Endowment)

   null (Ed.)

   In practice, differentially private data releases are designed to support a variety of applications. A data release is fit for use if it meets target accuracy requirements for each application. In this paper, we consider the problem of answering linear queries under differential privacy subject to per-query accuracy constraints. Existing practical frameworks like the matrix mechanism do not provide such fine-grained control (they optimize total error, which allows some query answers to be more accurate than necessary, at the expense of other queries that become no longer useful). Thus, we design a fitness-for-use strategy that adds privacy-preserving Gaussian noise to query answers. The covariance structure of the noise is optimized to meet the fine-grained accuracy requirements while minimizing the cost to privacy. 

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3. Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions

   Brown, Gavin ; Hopkins, Samuel B. ; Smith, Adam ( July 2023 , 36th Annual Conference on Learning Theory (COLT 2023))

   We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known to achieve this guarantee. Given n samples from a (sub-)Gaussian distribution with unknown mean μ and covariance Σ, our (ε,δ)-differentially private estimator produces μ~ such that ∥μ−μ~∥Σ≤α as long as n≳dα2+dlog1/δ√αε+dlog1/δε. The Mahalanobis error metric ∥μ−μ^∥Σ measures the distance between μ^ and μ relative to Σ; it characterizes the error of the sample mean. Our algorithm runs in time O~(ndω−1+nd/ε), where ω<2.38 is the matrix multiplication exponent. We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above. Our stable covariance estimator can be turned to private covariance estimation for unrestricted subgaussian distributions. With n≳d3/2 samples, our estimate is accurate in spectral norm. This is the first such algorithm using n=o(d2) samples, answering an open question posed by Alabi et al. (2022). With n≳d2 samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV distance. Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently. 

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4. Sharp Noisy Binary Search with Monotonic Probabilities

   https://doi.org/10.4230/LIPIcs.ICALP.2024.75  

   Gretta, Lucas ; Price, Eric ( January 2024 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl ; Grohe, Martin ; Puppis, Gabriele ; Svensson, Ola (Ed.)

   We revisit the noisy binary search model of [Karp and Kleinberg, 2007], in which we have n coins with unknown probabilities p_i that we can flip. The coins are sorted by increasing p_i, and we would like to find where the probability crosses (to within ε) of a target value τ. This generalized the fixed-noise model of [Burnashev and Zigangirov, 1974], in which p_i = 1/2 ± ε, to a setting where coins near the target may be indistinguishable from it. It was shown in [Karp and Kleinberg, 2007] that Θ(1/ε² log n) samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability 1-δ from 1/C_{τ, ε} ⋅ (log₂ n + O(log^{2/3} n log^{1/3} 1/(δ) + log 1/(δ))) samples, where C_{τ, ε} is the optimal such constant achievable. For δ > n^{-o(1)} this is within 1 + o(1) of optimal, and for δ ≪ 1 it is the first bound within constant factors of optimal. 

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5. Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions

   Brown, G. ; Hopkins, S. ; Smith, A. ( July 2023 , The Thirty Sixth Annual Conference on Learning Theory (COLT))

   We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation with nearly optimal sample complexity. Only exponential-time estimators were previously known to achieve this guarantee. Given n samples from a (sub-)Gaussian distribution with unknown mean μ and covariance Σ, our (ϵ,δ)-differentially private estimator produces μ~ such that ∥μ−μ~∥Σ≤α as long as n≳dα2+dlog1/δ√αϵ+dlog1/δϵ. The Mahalanobis error metric ∥μ−μ^∥Σ measures the distance between μ^ and μ relative to Σ; it characterizes the error of the sample mean. Our algorithm runs in time O~(ndω−1+nd/\eps), where ω<2.38 is the matrix multiplication exponent.We adapt an exponential-time approach of Brown, Gaboardi, Smith, Ullman, and Zakynthinou (2021), giving efficient variants of stable mean and covariance estimation subroutines that also improve the sample complexity to the nearly optimal bound above.Our stable covariance estimator can be turned to private covariance estimation for unrestricted subgaussian distributions. With n≳d3/2 samples, our estimate is accurate in spectral norm. This is the first such algorithm using n=o(d2) samples, answering an open question posed by Alabi et al. (2022). With n≳d2 samples, our estimate is accurate in Frobenius norm. This leads to a fast, nearly optimal algorithm for private learning of unrestricted Gaussian distributions in TV distance.Duchi, Haque, and Kuditipudi (2023) obtained similar results independently and concurrently. 

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