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# On Mean Estimation for General Norms with Statistical Queries

*Proceedings of the Thirty-Second Conference on Learning Theory*, PMLR 99:2158-2172, 2019.

#### Abstract

We study the problem of mean estimation for high-dimensional distributions given access to a statistical query oracle. For a normed space $X = (\mathbb{R}^d, \|\cdot\|_X)$ and a distribution supported on vectors $x \in \mathbb{R}^d$ with $\|x\|_{X} \leq 1$, the task is to output an estimate $\hat{\mu} \in \mathbb{R}^d$ which is $\varepsilon$-close in the distance induced by $\|\cdot\|_X$ to the true mean of the distribution. We obtain sharp upper and lower bounds for the statistical query complexity of this problem when the the underlying norm is \emph{symmetric} as well as for Schatten-$p$ norms, answering two questions raised by Feldman, Guzmán, and Vempala (SODA 2017).