论文标题
在非阳性曲率空间中均值的几何中间的指数浓度
Exponential Concentration for Geometric-Median-of-Means in Non-Positive Curvature Spaces
论文作者
论文摘要
在欧几里得空间中，已知经验平均值载体作为种群平均值的估计量具有多项式浓度，除非对基本概率措施施加了强大的尾巴假设。中位数锦标赛的想法被认为是克服经验均值向量的次级优势的一种方式。在本文中，为了解决经验平均值在更一般环境中的次优性能，我们考虑具有通用度量的通用波兰空间，该空间被允许是非紧凑和无限量的。我们讨论了相关人群特征平均值的估计，为此，我们将现有的均值概念扩展到了这个一般环境。我们设计了几种与基础度量的几何形状相关的新概念和不平等，并使用它们研究了中位数中位数的扩展概念的浓度特性作为人口特雷希特均值的估计量。我们表明，新的估计器仅在第二矩条件下在基础分布的第二矩条件下实现指数浓度，而经验特征平均值则具有多项式浓度。我们将研究重点放在具有非阳性亚历山德罗夫曲率的空间上，因为它们的收敛速度比具有正弯曲的空间较慢。我们注意到，这是第一项引发非偶数中值概念在具有通用度量的非向量空间中均值的概念的非反应浓度不平等的工作。
In Euclidean spaces, the empirical mean vector as an estimator of the population mean is known to have polynomial concentration unless a strong tail assumption is imposed on the underlying probability measure. The idea of median-of-means tournament has been considered as a way of overcoming the sub-optimality of the empirical mean vector. In this paper, to address the sub-optimal performance of the empirical mean in a more general setting, we consider general Polish spaces with a general metric, which are allowed to be non-compact and of infinite-dimension. We discuss the estimation of the associated population Frechet mean, and for this we extend the existing notion of median-of-means to this general setting. We devise several new notions and inequalities associated with the geometry of the underlying metric, and using them we study the concentration properties of the extended notions of median-of-means as the estimators of the population Frechet mean. We show that the new estimators achieve exponential concentration under only a second moment condition on the underlying distribution, while the empirical Frechet mean has polynomial concentration. We focus our study on spaces with non-positive Alexandrov curvature since they afford slower rates of convergence than spaces with positive curvature. We note that this is the first work that derives non-asymptotic concentration inequalities for extended notions of the median-of-means in non-vector spaces with a general metric.
