The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy

Cai, T. Tony; Wang, Yichen; Zhang, Linjun

doi:10.1214/21-aos2058

Cited by 46 publications

(25 citation statements)

References 24 publications

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“…Therefore, the Laplace mechanism that we used in 3.1 will not provide reasonable deviation bound. Actually, the same limitations also hold for the truncated mean estimator studied in [7], where the authors deal with a truncation parameter T of order log n since their data are sub-Gaussian, whereas under the current assumptions, T should be taken of the order of √ n. In general, it seems that the usual estimators of the mean under the existence of only low moments (median of means, truncated mean, Catoni's estimators [6,8,11]) can not be adapted in a straightforward way in order to obtain differentially private estimators that admit a sub-Gaussian error plus a term of smaller order.…”

Section: Discussion On Mean Estimationmentioning

confidence: 78%

Section: Motivationmentioning

confidence: 99%

“…In light of [7], one may naturally wonder how differential privacy will affect the deviation bounds discussed above. The statistical minimax rates established in [7] show that the rates of convergence of differentially private mean estimators are described by two terms. The first one correspond to the usual parametric 1 √ n convergence, while the second one is driven by the differential privacy tuning parameters and and is of the order 1 n. Consequently for large n differential privacy does not come at the expense of slower statistical convergence rates.…”

Section: Motivationmentioning

confidence: 99%

“…Even though the machine learning community has been very prolific in developing differentially private algorithms for complex settings such as multi-armed bandit problems [27,29,33], high-dimensional regression [21,32] and deep learning [1,23], some basic statistical questions are only starting to be understood. For example, the first statistical minimax rates of convergence under differential privacy were recently established in [7,13]. Some earlier work framing differential privacy in traditional statistics terms include [9,20,24,31,34].…”

Section: Introductionmentioning

confidence: 99%

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Differentially private sub-Gaussian location estimators

Avella-Medina,

Brunel

2019

Preprint

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We tackle the problem of estimating a location parameter with differential privacy guarantees and sub-Gaussian deviations. Recent work in statistics has focused on the study of estimators that achieve sub-Gaussian type deviations even for heavy tailed data. We revisit some of these estimators through the lens of differential privacy and show that a naive application of the Laplace mechanism can lead to sub-optimal results. We design two private algorithms for estimating the median that lead to estimators with sub-Gaussian type errors. Unlike most existing differentially private median estimators, both algorithms are well defined for unbounded random variables that are not even required to have finite moments. We then turn to the problem of sub-Gaussian mean estimation and show that under heavy tails natural differentially private alternatives lead to strictly worse deviations than their non-private sub-Gaussian counterparts. This is in sharp contrast with recent results that show that from an asymptotic perspective the cost of differential privacy is negligible.

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Section: Discussion On Mean Estimationmentioning

confidence: 78%

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Differentially private sub-Gaussian location estimators

Avella-Medina,

Brunel

2019

Preprint

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“…However, the design and development of optimal statistical estimation procedures under differential privacy is still at its beginnings. A few first contributions in that direction are Butucea et al (2019); Cai, Wang and Zhang (2019); Wainwright (2013a,b, 2014); Rohde and Steinberger (2019a); Smith (2008Smith ( , 2011; Wasserman and Zhou (2010); Ye and Barg (2017).…”

mentioning

confidence: 99%

Interactive versus non-interactive locally differentially private estimation: Two elbows for the quadratic functional

Butucea¹,

Rohde²,

Steinberger³

2020

Preprint

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Local differential privacy has recently received increasing attention from the statistics community as a valuable tool to protect the privacy of individual data owners without the need of a trusted third party. Similar to the classic notion of randomized response, the idea is that data owners randomize their true information locally and only release the perturbed data. Many different protocols for such local perturbation procedures can be designed. In all the estimation problems studied in the literature so far, however, no significant difference in terms of minimax risk between purely non-interactive protocols and protocols that allow for some amount of interaction between individual data providers could be observed. In this paper we show that for estimating the integrated square of a density, sequentially interactive procedures improve substantially over the best possible non-interactive procedure in terms of minimax rate of estimation. In particular, in the non-interactive scenario we identify an elbow in the minimax rate at s = 3 4 , whereas in the sequentially interactive scenario the elbow is at s = 1 2 . This is markedly different from both, the case of direct observations, where the elbow is well known to be at s = 1 4 , as well as from the case where Laplace noise is added to the original data, where an elbow at s = 9 4 is obtained. The fact that a particular locally differentially private, but interactive, mechanism improves over the simple non-interactive one is also of great importance for practical implementations of local differential privacy.

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Privacy-preserving parametric inference for spatial autoregressive model

Wang,

Song

2024

TEST

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The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy

Cited by 46 publications

References 24 publications

Differentially private sub-Gaussian location estimators

Differentially private sub-Gaussian location estimators

Interactive versus non-interactive locally differentially private estimation: Two elbows for the quadratic functional

Privacy-preserving parametric inference for spatial autoregressive model

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