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August 2020 Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids
Cristina Butucea, Amandine Dubois, Martin Kroll, Adrien Saumard
Bernoulli 26(3): 1727-1764 (August 2020). DOI: 10.3150/19-BEJ1165


We address the problem of non-parametric density estimation under the additional constraint that only privatised data are allowed to be published and available for inference. For this purpose, we adopt a recent generalisation of classical minimax theory to the framework of local $\alpha$-differential privacy and provide a lower bound on the rate of convergence over Besov spaces $\mathcal{B}^{s}_{pq}$ under mean integrated $\mathbb{L}^{r}$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. In order to fulfill the privacy requirement, we suggest adding suitably scaled Laplace noise to empirical wavelet coefficients. Upper bounds within (at most) a logarithmic factor are derived under the assumption that $\alpha$ stays bounded as $n$ increases: A linear but non-adaptive wavelet estimator is shown to attain the lower bound whenever $p\geq r$ but provides a slower rate of convergence otherwise. An adaptive non-linear wavelet estimator with appropriately chosen smoothing parameters and thresholding is shown to attain the lower bound within a logarithmic factor for all cases.


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Cristina Butucea. Amandine Dubois. Martin Kroll. Adrien Saumard. "Local differential privacy: Elbow effect in optimal density estimation and adaptation over Besov ellipsoids." Bernoulli 26 (3) 1727 - 1764, August 2020.


Received: 1 February 2019; Revised: 1 July 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193941
MathSciNet: MR4091090
Digital Object Identifier: 10.3150/19-BEJ1165

Keywords: adaptive estimation , Besov classes of functions , Density estimation , Local differential privacy , lower bounds , Minimax rates , wavelet thresholding

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability


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Vol.26 • No. 3 • August 2020
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