Optimal locally private estimation under lp loss for 1< p< 2
Abstract: We consider the minimax estimation problem of a discrete distribution with support size k under locally differential privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number o measures the privacy level of a privatization scheme. In our previous work (IEEE Trans. Inform. Theory, 2018) we proposed a family of new privatization schemes and the corresponding estimator. We also proved that our scheme and estimator are order optimal in the regime e k under both l 2 (mean square) and l1 loss. In this paper, we sharpen this result by showing asymptotic optimality of the proposed scheme under the l p p loss for all 1 d p d 2. More precisely, we show that for any p P r1, 2s and any k and o, the ratio between the worst case lpp estimation loss of our scheme and the optimal value approaches 1 as the number of samples tends to infinity. The lower bound on the minimax risk of private estimation that we establish as a part of the proof is valid for any loss function l p p, p e 1.
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