The Geometric Median and Applications to Robust Mean Estimation
The Geometric Median and Applications to Robust Mean Estimation
复制标题
DOI：
10.1137/23m1592420
复制
发表时间：
2023-07
期刊：
影响因子：
--
通讯作者：
Stanislav Minsker;Nate Strawn
中科院分区：
文献类型：
--
作者：
Stanislav Minsker;Nate Strawn
本文致力于几何中位数的统计和数值特性，及其应用于通过平均原理中位数的稳健平均值估计问题。我们的主要理论结果包括（a）在r^d中绝对连续分布的平均值和中值之间的距离的上限，以及这些界限的特定分布类别的示例不取决于环境维度d； （b）样品中位数的距离与总体版本之间的距离的指数偏差不等式，这再次仅取决于痕量型数量，而不取决于环境维度。作为推论，我们推断出（几何）估计器的（几何）中位数的界限，该估计值适用于大量的重尾分布。最后，我们解决了数值近似的误差，这是任何统计估计程序的重要实践方面。我们证明，通过几何中位数最小化的目标函数满足“局部二次增长”条件，该条件允许一个人将目标函数的次要范围转换为中位数本身的数值近似值的相应边界，并提出了一个简单的停止规则适用对于任何可产生明确错误保证的优化方法。我们以数值实验为总结，包括应用于标准普尔500数据的日志归还平均值的估计。
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.