Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

Abstract: We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs---in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.

• The paper introduces a novel DRO framework that leverages the Wasserstein metric to construct uncertainty sets from finite data samples.
• The authors derive finite convex reformulations that enable polynomial-time solutions for worst-case expectation problems under specific loss function structures.
• The approach provides performance guarantees that mitigate overfitting and support practical applications in fields such as finance and risk management.

Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: An Expert Analysis

The paper "Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations" by Peyman Mohajerin Esfahani and Daniel Kuhn addresses the challenges inherent in stochastic programming where uncertainty in parameter distributions must be accounted for using finite datasets. The authors propose a robust optimization approach founded on the Wasserstein metric, which constructs uncertainty sets around empirical data distributions. Unlike classical stochastic programming that may suffer from overfitting and inefficacy in large-scale problems, this approach provides finite-sample performance guarantees and computational tractability for several practical cases.

The primary contribution lies in reformulating distributionally robust optimization problems under Wasserstein ambiguity sets as finite convex programs. This advancement hinges on recent measure concentration results that ensure the Wasserstein ball, when properly sized, contains the true data-generating distribution with high confidence. Specifically, the authors derive conditions under which worst-case expectations over Wasserstein balls can be efficiently solved, often reducible to linear programs when specific structural assumptions about the loss functions—such as piecewise affinities—are made.

The paper systematically segments its theoretical proposals into coherent findings:

1. Performance Guarantees: By leveraging measure concentration results, the paper ensures that solutions derived from Wasserstein ambiguity sets provide upper confidence bounds for the true, out-of-sample cost with a quantifiable degree of reliability. This is crucial as it shields the decision-maker from potential overfitting, typical of strategies that solely depend on empirical distributions for optimization.
2. Tractable Reformulations: One of the most salient results is the derivation of convex reformulations for many worst-case problems under mild assumptions. Specifically, for functions manifestable as pointwise maxima of concave functions, the distributionally robust problem can be addressed via polynomial-time solvable convex programs. The paper extends these tractability insights to loss functions with separable structures and certain convex forms.
3. Explicit Linear Programs for Special Cases: The tractability of the proposed method is demonstrated mathematically, with explicit formulations for piecewise affine loss optimization, uncertainty quantification, and two-stage stochastic programming. These scenarios often arise in financial portfolio optimization and risk management contexts, underscoring the immediate applicability of the work in real-world scenarios.
4. Alternative Challenges and Solutions: The authors acknowledge scenarios where extremal distributions—those achieving the worst-case—aids in decision-making. The constructive procedure for generating such distributions facilitates practical stress tests. The paper acknowledges circumstances (e.g., non-existence of worst-case distributions in some setups) where theoretical adjustments are needed, illustrating a comprehensive understanding of the nuanced complexities in robust optimization.

The implications of utilizing Wasserstein balls in robust optimization are profound. From a practical perspective, professionals in fields like finance can now proceed with decisions backed by more realistic models of uncertainty, with tangible guarantees against data perturbation impacts. Theoretically, this opens new research channels on probability metrics for ambiguity set construction, laying ground for discovering more robust, computationally feasible solutions to complex decision problems under uncertainty.

The authors speculate on future developments by distinguishing their approach's advantages over other metric-based methods such as Kullback-Leibler or total variation metrics, which fail to secure the data-driven true distribution confidence with the same level of assurance. The paper invites further exploration into non-convex optimization domains and underscores the flexibility of Wasserstein-based metrics in shaping future robust optimization methodologies.

In summary, this paper not only provides a sound theoretical framework and mathematical underpinning for Wasserstein-based distributionally robust optimization but also exemplifies its practical relevance in decision-making problems dominated by uncertainty. It lays a foundation for continued explorations in robust optimization models and further enriches methodologies in operations research, control systems, and financial engineering.