Worst-Case Risk Quantification under Distributional Ambiguity using Kernel Mean Embedding in Moment Problem
This work addresses risk assessment under uncertainty for applications like stochastic control, but it appears incremental as it builds on existing kernel methods.
The paper tackles the problem of quantifying worst-case risk under distributional ambiguity by using kernel mean embedding to formulate a generalized moment problem with nonparametric constraints, and it demonstrates the method's application in characterizing worst-case constraint violation probabilities for a stochastic control system.
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.