Distributionally Robust Decision Making Leveraging Conditional Distributions

TL;DR

This paper introduces CKDRO, a novel method for decision making under conditional distributions using kernel DRO and RKHS, effectively handling auxiliary information and demonstrating superior performance in generation scheduling.

Contribution

The paper develops a new CKDRO framework that leverages kernel methods and RKHS for robust decision making with conditional distributions, addressing a gap in existing DRO approaches.

Findings

CKDRO outperforms benchmarks in generation scheduling.

The ambiguity set in RKHS can be viewed as a Wasserstein-like ball.

The method effectively incorporates auxiliary information into DRO.

Abstract

Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision-making with some auxiliary information, and DRO in the context of conditional distribution is not straightforward. We propose a conditional kernel distributionally robust optimization (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). In particular, we consider problems where there is a correlation between the unknown variable y and an auxiliary observable variable x. Given past data of the two variables and a queried auxiliary variable, CKDRO represents the conditional distribution P(y|x) as the conditional mean operator in the RKHS space and quantifies the ambiguity set in the RKHS as well, which depends on the size of the dataset as well as the query point. To justify the use of RKHS, we demonstrate that the ambiguity set defined in RKHS can be viewed as a ball under a metric that is similar to the Wasserstein metric. The DRO is then dualized and solved via a finite dimensional convex program. The proposed CKDRO approach is applied to a generation scheduling problem and shows that the result of CKDRO is superior to common benchmarks in terms of quality and robustness.