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# Private Mean Estimation of Heavy-Tailed Distributions

*Proceedings of Thirty Third Conference on Learning Theory*, PMLR 125:2204-2235, 2020.

#### 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.