On density estimation at a fixed point under local differential privacy

Abstract

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.