Worst-Case Risk Quantification under Distributional Ambiguity using Kernel Mean Embedding in Moment Problem

TL;DR

This paper introduces a novel method for worst-case risk quantification under distributional ambiguity using kernel mean embedding, providing a nonparametric approach with theoretical guarantees and practical application in stochastic control.

Contribution

It formulates a generalized moment problem with ambiguity sets defined in a reproducing kernel Hilbert space, offering a tractable approximation with theoretical validation.

Findings

Successfully characterizes worst-case constraint violation probability

Provides a nonparametric, kernel-based approach with theoretical guarantees

Demonstrates effectiveness in stochastic control system applications

Abstract

In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constraint violation probability in the context of a constrained stochastic control system.