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

TL;DR

This paper develops a data-driven distributionally robust optimization framework using 1-W Wasserstein balls centered at the empirical distribution to guard against distributional misspecification in stochastic programs. It proves that worst-case expectations over such balls can be computed via finite-dimensional convex programs for broad loss classes, and provides systematic ways to construct extremal distributions. Leveraging measure concentration, it derives finite-sample guarantees that the DRO solution upper-bounds out-of-sample costs with high confidence and establishes asymptotic consistency as sample size grows. The authors demonstrate tractability through explicit LP/convex reformulations for mean-risk portfolio problems, uncertainty quantification, and two-stage stochastic programs, and validate the approach with extensive numerical experiments on portfolios and UQ tasks. Overall, the method offers strong out-of-sample performance guarantees with scalable computation in continuous uncertainty spaces.

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.

Figures (10)

• Figure 1: Example of a worst-case expectation problem without a worst-case distribution
• Figure 2: Representative distributions in balls centered at $\widehat{\mathds{P}}_N$ induced by different metrics
• Figure 3: Representing the indicator function of a convex set and its complement as a pointwise maximum of concave functions
• Figure 4: Optimal portfolio composition as a function of the Wasserstein radius $\varepsilon$ averaged over 200 simulations; the portfolio weights are depicted in ascending order, i.e., the weight of asset 1 at the bottom (dark blue area) and that of asset 10 at the top (dark red area)
• Figure 5: Out-of-sample performance $J(\widehat{x}_N(\varepsilon))$ (left axis, solid line and shaded area) and reliability $\mathds{P}^N[J(\widehat{x}_N(\varepsilon)) \le \widehat{J}_N(\varepsilon)]$ (right axis, dashed line) as a function of the Wasserstein radius $\varepsilon$ and estimated on the basis of 200 simulations

Theorems & Definitions (46)

• Definition 3.1: Wasserstein metric ref:KantRub-58
• Theorem 3.2: Kantorovich-Rubinstein ref:KantRub-58
• Theorem 3.4: Measure concentration ref:FouGui-14
• Theorem 3.5: Finite sample guarantee
• proof
• Theorem 3.6: Asymptotic consistency
• Lemma 3.7: Convergence of distributions
• proof
• proof : Proof of Theorem \ref{['thm:convergence']}
• Example 1: Necessity of regularity conditions