For sets $\mathcal Q$ and $\mathcal Y$, the generalized Fr\'echet mean $m \in \mathcal Q$ of a random variable $Y$, which has values in $\mathcal Y$, is any minimizer of $q\mapsto \mathbb E[\mathfrak c(q,Y)]$, where $\mathfrak c \colon \mathcal Q \times \mathcal Y \to \mathbb R$ is a cost function. There are little restrictions to $\mathcal Q$ and $\mathcal Y$. In particular, $\mathcal Q$ can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fr\'echet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fr\'echet means, we do not require a finite diameter of the $\mathcal Q$ or $\mathcal Y$. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.

翻译：对于集合$\mathcal Q$和$\mathcal Y$，随机变量$Y$（取值于$\mathcal Y$）的广义Fréchet均值$m \in \mathcal Q$是使得$q\mapsto \mathbb E[\mathfrak c(q,Y)]$最小化的任意元素，其中$\mathfrak c \colon \mathcal Q \times \mathcal Y \to \mathbb R$为代价函数。该定义对$\mathcal Q$和$\mathcal Y$的结构限制极少，尤其允许$\mathcal Q$为非欧几里得度量空间。本文给出了经验广义Fréchet均值的收敛速率，并提供了依概率收敛与依期望收敛的条件。与以往关于Fréchet均值的研究不同，我们无需假设$\mathcal Q$或$\mathcal Y$具有有限直径，而是引入一种称为四重不等式的条件。该不等式推广了常见的代价函数Lipschitz条件，已知在Hadamard空间中成立。我们进一步证明，该不等式对Hadamard度量的特定幂次也以适当形式成立。