Name
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 $\epsilon$ 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 $e^{\epsilon} \ll k$ 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 $\epsilon,$ 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. |
Year | Venue | DocType |
|---|---|---|
2017 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1708.00059 | 0 | 0.34 |
References | Authors | |
0 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Min Ye | 1 | 18 | 6.10 |
| Alexander Barg | 2 | 910 | 85.90 |