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Title: Local differential privacy: Elbow effect in optimal density estimation and adaptation Authors:  Cristina Butucea - University Paris-Est Marne (France)
Amandine Dubois - CREST-ENSAI (France) [presenting]
Martin Kroll - Universitaet Mannheim (Germany)
Adrien Saumard - Crest-Ensai (France)
Abstract: The problem of non-parametric density estimation is addressed 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 $B^s_{pq}$ under mean integrated $L_r$-risk. This lower bound is deteriorated compared to the standard setup without privacy, and reveals a twofold elbow effect. 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.