Fast Mean Estimation with Sub-Gaussian Rates

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Yeshwanth Cherapanamjeri, Nicolas Flammarion, Peter L. Bartlett ;
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:786-806, 2019.

Abstract

We propose an estimator for the mean of a random vector in $\mathbb{R}^d$ that can be computed in time $O(n^{3.5}+n^2d)$ for $n$ i.i.d. samples and that has error bounds matching the sub-Gaussian case. The only assumptions we make about the data distribution are that it has finite mean and covariance; in particular, we make no assumptions about higher-order moments. Like the polynomial time estimator introduced by Hopkins (2018), which is based on the sum-of-squares hierarchy, our estimator achieves optimal statistical efficiency in this challenging setting, but it has a significantly faster runtime and a simpler analysis.

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