%0 Journal Article %1 kamath2020private %A Kamath, Gautam %A Singhal, Vikrant %A Ullman, Jonathan %D 2020 %K differential-privacy inequality privacy robustness stats %T Private Mean Estimation of Heavy-Tailed Distributions %U http://arxiv.org/abs/2002.09464 %X We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. Roughly speaking, in the univariate case, we show that $n = \Thetałeft(1\alpha^2 + 1\alpha^kk-1\varepsilon\right)$ samples are necessary and sufficient to estimate the mean to $\alpha$-accuracy under $\varepsilon$-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all $k 2$. We also give algorithms for the multivariate setting whose sample complexity is a factor of $O(d)$ larger than the univariate case.

@article{kamath2020private, abstract = {We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. Roughly speaking, in the univariate case, we show that $n = \Theta\left(\frac{1}{\alpha^2} + \frac{1}{\alpha^{\frac{k}{k-1}}\varepsilon}\right)$ samples are necessary and sufficient to estimate the mean to $\alpha$-accuracy under $\varepsilon$-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all $k \geq 2$. We also give algorithms for the multivariate setting whose sample complexity is a factor of $O(d)$ larger than the univariate case.}, added-at = {2020-02-26T14:14:06.000+0100}, author = {Kamath, Gautam and Singhal, Vikrant and Ullman, Jonathan}, biburl = {https://www.bibsonomy.org/bibtex/24d787c75384b647ab57657c92beb0893/kirk86}, description = {[2002.09464] Private Mean Estimation of Heavy-Tailed Distributions}, interhash = {62a16b3d96269c0154f42fd27e4b5174}, intrahash = {4d787c75384b647ab57657c92beb0893}, keywords = {differential-privacy inequality privacy robustness stats}, note = {cite arxiv:2002.09464}, timestamp = {2020-02-26T14:14:06.000+0100}, title = {Private Mean Estimation of Heavy-Tailed Distributions}, url = {http://arxiv.org/abs/2002.09464}, year = 2020 }