Mean Estimation with Sub-Gaussian Rates in Polynomial Time

Samuel B. Hopkins

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arXiv · 1809.07425

Mean Estimation with Sub-Gaussian Rates in Polynomial Time

Samuel B. Hopkins

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Abstract

We study polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. We assume only that the random vector $X$ has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that $X$ is Gaussian or sub-Gaussian. We offer the first polynomial time algorithm to estimate the mean with sub-Gaussian-size confidence intervals under such mild assumptions. Our algorithm is based on a new semidefinite programming relaxation of a high-dimensional median. Previous estimators which assumed only existence of finitely-many moments of $X$ either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring time exponential in the dimension.

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Samuel B. Hopkins. 2018-09-19. Mean Estimation with Sub-Gaussian Rates in Polynomial Time. https://arxiv.org/abs/1809.07425

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