Sub-Gaussian Mean Estimation in Polynomial Time.
arXiv: Statistics Theory(2018)
摘要
study polynomial time algorithms for estimating the mean of a random vector $X$ in $mathbb{R}^d$ from $n$ independent samples $X_1,ldots,X_n$ when $X$ may be heavy-tailed. assume only that $X$ has finite mean $mu$ and covariance $Sigma$. 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. In particular, for confidence $delta u003e 0$, the empirical mean has confidence intervals with radius of order $sqrt{text{Tr} Sigma / delta n}$ rather than $sqrt{text{Tr} Sigma /n } + sqrt{ lambda_{max}(Sigma) log (1/delta) / n}$ from the Gaussian case. We offer the first polynomial time algorithm to estimate the mean with sub-Gaussian 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 $O(1)$ moments of $X$ either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring $exp(d)$ time.
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