Representation of Quasi-Monotone Functionals by Families of Separating Hyperplanes

TL;DR

This paper characterizes the conditions under which quasi-monotone functionals' level sets can be uniquely represented by continuous functionals, exploring their regularity and relevance to property elicitation in machine learning and finance.

Contribution

It provides a novel characterization of level set representations for quasi-monotone functionals and links this to property elicitation problems in various fields.

Findings

Unique representation conditions established

Dependence of functionals on level parameters analyzed

Connections to property elicitation in ML and finance elucidated

Abstract

We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how regularly these functionals depend on the parameterizing level. Finally, we show how this question relates to the recent problem of property elicitation that simultaneously attracted interest in machine learning, statistical evaluation of forecasts, and finance.