  Expected risk minimization (ERM) is at the core of many machine learning
systems. This means that the risk inherent in a loss distribution is summarized
using a single number - its average. In this paper, we propose a general
approach to construct risk measures which exhibit a desired tail sensitivity
and may replace the expectation operator in ERM. Our method relies on the
specification of a reference distribution with a desired tail behaviour, which
is in a one-to-one correspondence to a coherent upper probability. Any risk
measure, which is compatible with this upper probability, displays a tail
sensitivity which is finely tuned to the reference distribution. As a concrete
example, we focus on divergence risk measures based on f-divergence ambiguity
sets, which are a widespread tool used to foster distributional robustness of
machine learning systems. For instance, we show how ambiguity sets based on the
Kullback-Leibler divergence are intricately tied to the class of subexponential
random variables. We elaborate the connection of divergence risk measures and
rearrangement invariant Banach norms.
