Asymptotically optimal private estimation under mean square loss

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 $ε$ measures the privacy level of a privatization scheme. In our previous work (arXiv:1702.00610), 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 $e^ε \ll k$ under both $\ell_2^2$ and $\ell_1$ loss. In other words, for a large number of samples the worst-case estimation loss of our scheme was shown to differ from the optimal value by at most a constant factor. In this paper, we eliminate this gap by showing asymptotic optimality of the proposed scheme and estimator under the $\ell_2^2$ (mean square) loss. More precisely, we show that for any $k$ and $ε,$ the ratio between the worst-case estimation loss of our scheme and the optimal value approaches $1$ as the number of samples tends to infinity.