Related papers: Private Mean Estimation of Heavy-Tailed Distributions

Private Mean Estimation of Heavy-Tailed Distributions

URL: http://arxiv.org/abs/2002.09464v3
Date: Tue, 16 Feb 2021 17:06:17 GMT
Title: Private Mean Estimation of Heavy-Tailed Distributions
Authors: Gautam Kamath, Vikrant Singhal, Jonathan Ullman
Abstract summary: We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. We show that $n = Thetaleft(frac1alpha2 + frac1alphafrackk-1varepsilonright)$ samples are necessary and sufficient to estimate the mean to $alpha$-accuracy under $varepsilon$-differential privacy, or any of its common relaxations.
Score: 10.176795938619417
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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.

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