The Geometric Median and Applications to Robust Mean Estimation

TL;DR

This paper investigates the statistical and numerical properties of the geometric median, providing bounds and inequalities that improve robust mean estimation, especially for heavy-tailed distributions, with practical optimization and numerical insights.

Contribution

It offers new bounds for the geometric median's deviation from the mean, dimension-independent results for certain distributions, and practical stopping rules for numerical approximation.

Findings

Dimension-independent bounds for specific distribution classes

Exponential deviation inequalities for sample median

Effective stopping rules for numerical median approximation

Abstract

This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension d; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself, and propose a simple stopping rule applicable to any optimization method which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.