Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning

Abstract

Many decision problems in science, engineering and economics are affected by uncertain parameters whose distribution is only indirectly observable through samples. The goal of data-driven decision-making is to learn a decision from finitely many training samples that will perform well on unseen test samples. This learning task is difficult even if all training and test samples are drawn from the same distributionespecially if the dimension of the uncertainty is large relative to the training sample size. Wasserstein distributionally robust optimization seeks data-driven decisions that perform well under the most adverse distribution within a certain Wasserstein distance from a nominal distribution constructed from the training samples. In this tutorial we will argue that this approach has many conceptual and computational benefits. Most prominently, the optimal decisions can often be computed by solving tractable convex optimization problems, and they enjoy rigorous out-of-sample and asymptotic consistency guarantees. We will also show that Wasserstein distributionally robust optimization has interesting ramifications for statistical learning and motivates new approaches for fundamental learning tasks such as classification, regression, maximum likelihood estimation or minimum mean square error estimation, among others.

GeneraLight: Improving Environment Generalization of Traffic Signal Control via Meta Reinforcement Learning

“…MAESN [3] adds noise as a part of the state to enhance the exploration ability of policy. Some people think robust optimization [9,28] is also a possible solution. EPOPT [5] searches the environment in train set to find some hard environments, and optimize the policy on the hard environments.…”

Sharing the value‐at‐risk under distributional ambiguity

“…If confident estimations about mean and covariance are to be incorporated with shape information, it is also helpful to choose from the family of elliptical distributions that generalizes multivariate normal distribution and multivariate t ‐distribution, among others. We refer to Kuhn, Mohajerin Esfahani, Nguyen, and Shafieezadeh‐Abadeh (2019) on a tutorial of Wasserstein distributionally robust optimization models.…”

Data-Driven Distributionally Robust Stochastic Control of Energy Storage for Wind Power Ramp Management Using the Wasserstein Metric

“…In other words, the proposed controller design approach is an offline method unlike, for example, receding horizon control. Note also that the proposed reformulation method does not require the direct calculation of Wasserstein distances, which is #P-hard [34]. This is an advantage of Wasserstein DRO that enables us to obtain an optimal solution through the dual form (7) or (9) without explicitly computing Wasserstein distances.…”