Geometrizing rates of convergence under local differential privacy constraints

Rohde, Angelika; Steinberger, Lukas

doi:10.1214/19-aos1901

Cited by 32 publications

(38 citation statements)

References 17 publications

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“…To address this shortcoming, the local differential privacy constraint [see, for example, 21, 12, and the references therein] was introduced to provide a setting where analysis must be carried out in such a way that each raw data point is only ever seen by the original data holder. The simplest example of a locally differentially private mechanism is the randomised response [35] used with binary data, but mechanisms have also been developed for tasks such as classification [3], generalised linear modelling [12], empirical risk minimisation [33], density estimation [5], functional estimation [27] and goodness-of-fit testing [4].…”

Section: Introductionmentioning

confidence: 99%

Strongly universally consistent nonparametric regression and classification with privatised data

Berrett

Györfi

Walk

2021

Electron. J. Statist.

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In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X 1 , Y 1 ), . . . , (Xn, Yn), taking values in R d × R, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector X i and to the value of its response variable Y i . Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent, and we further establish an upper bound on the rate of convergence. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

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Section: Introductionmentioning

confidence: 99%

Strongly universally consistent nonparametric regression and classification with privatised data

Berrett

Györfi

Walk

2021

Electron. J. Statist.

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“…Evidently, the privacy condition (1) becomes more restrictive for smaller values of the two parameters α and β. Although Definition 2.1 smoothly bridges the cases β = 0 and β > 0, the classical anonymization techniques used for β = 0 and β > 0 are essentially different: In the case β = 0, Laplace perturbation as well as randomization techniques as considered in [11,21] can be used. In the case β > 0, adding appropriately scaled Gaussian noise has been suggested in [17].…”

Section: Definition Of Approximate Differential Privacymentioning

confidence: 99%

“…At least, using the approach suggested in Subsection 2.3 (with its specializations considered in Propositions 3.3 and 3.4) we relieved ourselves from the drawback of the Laplace method that one can privatize only one functional of the form f (t) for one single t that has to be fixed even before the anonymization. Note that this drawback is, for instance, also present in the mechanisms suggested in [21]. From this point of view, (α, β)differential privacy with strictly positive β via one of these approaches should be preferred.…”

Section: Adaptation To Unknown Smoothnessmentioning

confidence: 99%

On density estimation at a fixed point under local differential privacy

Kroll

2021

Electron. J. Statist.

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We consider non-parametric density estimation in the framework of local, both pure and approximate, differential privacy. In contrast to centralized privacy scenarios with a trusted curator, in the local setup anonymization must be guaranteed already on the individual data owners' side and must therefore precede any data mining tasks. Thus, the published anonymized data should be compatible with as many statistical procedures as possible. We consider different mechanisms to establish pure and approximate differential privacy, respectively. We obtain minimax type results over Sobolev classes indexed by a smoothness parameter s > 1/2 for the mean squared error at a fixed point. In particular, we show that appropriately defined kernel density estimators can attain the optimal rate of convergence if the bandwidth parameter is correctly specified. Notably, the optimal convergence rate in terms of the sample size n is n −(2s−1)/(2s+1) under pure differential privacy and thus deteriorated to the rate n −(2s−1)/(2s) which holds both without privacy restrictions and under approximate differential privacy. Since the optimal choice of the bandwidth parameter depends on the smoothness s and is thus not accessible in practise, adaptive methods for bandwidth selection are necessary and must, in the local privacy framework, be performed based on the anonymized data only. We address this problem by means of variants of Lepski's method tailored to the privacy setups at hand and obtain general oracle inequalities for private kernel density estimators. In the Sobolev case, the resulting adaptive estimators attain the optimal rates of convergence at least up to logarithmic factors. On the side, we discuss some critical issues related with the notion of approximate differential privacy.

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“…Although this originates in cryptography, there is a growing statistical literature that aims to explore the constraints of this framework and provide procedures that make optimal use of available data (e.g. Wasserman & Zhou 2010, Duchi et al 2018, Rohde & Steinberger 2020, Cai et al 2021. Work in this area is split between central models of privacy, where there is a third party trusted to collect and analyse data before releasing privatised results, and local models of privacy, where data are randomised before collection.…”

Section: Introductionmentioning

confidence: 99%

“…Duchi et al 2018), nonparametric estimation problems (e.g. Rohde & Steinberger 2020, and change point analysis (e.g. Berrett & Yu 2021), to name but a few.…”

Section: Introductionmentioning

confidence: 99%

On robustness and local differential privacy

Li¹,

Berrett²,

Chen³

2022

Preprint

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It is of soaring demand to develop statistical analysis tools that are robust against contamination as well as preserving individual data owners' privacy. In spite of the fact that both topics host a rich body of literature, to the best of our knowledge, we are the first to systematically study the connections between the optimality under Huber's contamination model and the local differential privacy (LDP) constraints.In this paper, we start with a general minimax lower bound result, which disentangles the costs of being robust against Huber's contamination and preserving LDP. We further study three concrete examples: a two-point testing problem, a potentially-diverging mean estimation problem and a nonparametric density estimation problem. For each problem, we demonstrate procedures that are optimal in the presence of both contamination and LDP constraints, comment on the connections with the state-of-the-art methods that are only studied under either contamination or privacy constraints, and unveil the connections between robustness and LDP via partially answering whether LDP procedures are robust and whether robust procedures can be efficiently privatised. Overall, our work showcases a promising prospect of joint study for robustness and local differential privacy.

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Geometrizing rates of convergence under local differential privacy constraints

Cited by 32 publications

References 17 publications

Strongly universally consistent nonparametric regression and classification with privatised data

Strongly universally consistent nonparametric regression and classification with privatised data

On density estimation at a fixed point under local differential privacy

On robustness and local differential privacy

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