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L_1 density estimation from privatised data

by   László Györfi, et al.

We revisit the classical problem of nonparametric density estimation, but impose local differential privacy constraints. Under such constraints, the original data X_1,…,X_n, taking values in ℝ^d, cannot be directly observed, and all estimators are functions of the randomised output of a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and in this work we propose to add Laplace distributed noise to a discretisation of the location of a vector X_i. Based on these randomised data, we design a novel estimator of the density function, which can be viewed as a privatised version of the well-studied histogram density estimator. Our theoretical results include universal pointwise consistency and strong universal L_1-consistency. In addition, a convergence rate over classes of Lipschitz functions is derived, which is complemented by a matching minimax lower bound.


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