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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 anyminimizer 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 arelittle restrictions to $\mathcal Q$ and $\mathcal Y$. In particular, $\mathcalQ$ can be a non-Euclidean metric space. We provide convergence rates for theempirical generalized Fr\'echet mean. Conditions for rates in probability andrates in expectation are given. In contrast to previous results on Fr\'echetmeans, we do not require a finite diameter of the $\mathcal Q$ or $\mathcal Y$.Instead, we assume an inequality, which we call quadruple inequality. Itgeneralizes an otherwise common Lipschitz condition on the cost function. Thisquadruple inequality is known to hold in Hadamard spaces. We show that it alsoholds in a suitable way for certain powers of a Hadamard-metric.

Convergence Rates for the Generalized Fr\'echet Mean via the Quadruple Inequality