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Wasserstein Distributionally Robust Regret Optimization

Lukas-Benedikt Fiechtner, Jose Blanchet

TL;DR

The paper studies ex-ante Wasserstein Distributionally Robust Regret Optimization (DRRO) as an alternative to DRO to mitigate over-conservatism while hedging model misspecification. It establishes that, under smoothness and unique-optimum conditions, DRRO behaves like ERM up to first order, and is exactly equivalent to ERM for quadratic losses, with explicit regret characterizations. It proves NP-hardness for general max-affine loss classes, motivating a convex relaxation that provides strong empirical performance and bounds the ex-ante regret. The authors develop exact and relaxation-based algorithms, including finite convex reformulations and hill-climbing initialization to accelerate solving, and demonstrate via Newsvendor experiments that DRRO can achieve robust yet growth-friendly decisions beyond traditional DRO. Overall, the work provides theoretical insights, computational strategies, and practical evidence that DRRO can achieve better trade-offs between robustness and upside potential than DRO alone.

Abstract

Distributionally Robust Optimization (DRO) is a popular framework for decision-making under uncertainty, but its adversarial nature can lead to overly conservative solutions. To address this, we study ex-ante Distributionally Robust Regret Optimization (DRRO), focusing on Wasserstein-based ambiguity sets which are popular due to their links to regularization and machine learning. We provide a systematic analysis of Wasserstein DRRO, paralleling known results for Wasserstein DRO. Under smoothness and regularity conditions, we show that Wasserstein DRRO coincides with Empirical Risk Minimization (ERM) up to first-order terms, and exactly so in convex quadratic settings. We revisit the Wasserstein DRRO newsvendor problem, where the loss is the maximum of two linear functions of demand and decision. Extending [25], we show that the regret can be computed by maximizing two one-dimensional concave functions. For more general loss functions involving the maximum of multiple linear terms in multivariate random variables and decision vectors, we prove that computing the regret and thus also the DRRO policy is NP-hard. We then propose a convex relaxation for these more general Wasserstein DRRO problems and demonstrate its strong empirical performance. Finally, we provide an upper bound on the optimality gap of our relaxation and show it improves over recent alternatives.

Wasserstein Distributionally Robust Regret Optimization

TL;DR

The paper studies ex-ante Wasserstein Distributionally Robust Regret Optimization (DRRO) as an alternative to DRO to mitigate over-conservatism while hedging model misspecification. It establishes that, under smoothness and unique-optimum conditions, DRRO behaves like ERM up to first order, and is exactly equivalent to ERM for quadratic losses, with explicit regret characterizations. It proves NP-hardness for general max-affine loss classes, motivating a convex relaxation that provides strong empirical performance and bounds the ex-ante regret. The authors develop exact and relaxation-based algorithms, including finite convex reformulations and hill-climbing initialization to accelerate solving, and demonstrate via Newsvendor experiments that DRRO can achieve robust yet growth-friendly decisions beyond traditional DRO. Overall, the work provides theoretical insights, computational strategies, and practical evidence that DRRO can achieve better trade-offs between robustness and upside potential than DRO alone.

Abstract

Distributionally Robust Optimization (DRO) is a popular framework for decision-making under uncertainty, but its adversarial nature can lead to overly conservative solutions. To address this, we study ex-ante Distributionally Robust Regret Optimization (DRRO), focusing on Wasserstein-based ambiguity sets which are popular due to their links to regularization and machine learning. We provide a systematic analysis of Wasserstein DRRO, paralleling known results for Wasserstein DRO. Under smoothness and regularity conditions, we show that Wasserstein DRRO coincides with Empirical Risk Minimization (ERM) up to first-order terms, and exactly so in convex quadratic settings. We revisit the Wasserstein DRRO newsvendor problem, where the loss is the maximum of two linear functions of demand and decision. Extending [25], we show that the regret can be computed by maximizing two one-dimensional concave functions. For more general loss functions involving the maximum of multiple linear terms in multivariate random variables and decision vectors, we prove that computing the regret and thus also the DRRO policy is NP-hard. We then propose a convex relaxation for these more general Wasserstein DRRO problems and demonstrate its strong empirical performance. Finally, we provide an upper bound on the optimality gap of our relaxation and show it improves over recent alternatives.

Paper Structure

This paper contains 29 sections, 14 theorems, 129 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and Problem Statement
  3. Sensitivity of the Value Function
  4. Modeling Assumptions and Computational Tractability
  5. NP-Hardness
  6. Algorithmic Solutions for the Max-Affine Loss Model Class
  7. Exact algorithms
  8. Convex relaxations
  9. Experiments
  10. Newsvendor Problem
  11. Multi-Item Newsvendor Problem
  12. Conclusion
  13. Appendix
  14. DRRO for the Univariate Newsvendor Problem
  15. Reformulations of h(beta; theta) as Finite Convex Programs
  16. ...and 14 more sections

Key Result

Theorem 1

Suppose that $\mathbb{E}_{\mathbb{P}_0}|X|^p < \infty$. Then for the newsvendor loss $\ell(\theta, x) = b\theta - s\,\min (\theta, x)$ and Wasserstein distributional uncertainty sets, the function $h(\beta; \theta)$ in eq:g_def is concave on the intervals $[0, \theta]$ and $[\theta, \infty)$.

Figures (9)

  • Figure 1: Difference in expected profit between ERM and DRO policy for all normal distributions in Wasserstein ball. Positive differences (red) indicate the gap by which ERM strategy outperforms the DRO in an out-of-sample environment corresponding by the pair $(\mu, \sigma)$. Negative differences represent the corresponding gap (blue) when the DRO strategy outperforms the ERM strategy.
  • Figure 2: Comparison of order quantities and regret for ERM, DRO, and DRRO with respect to $\delta$.
  • Figure 3: Distributions attaining the worst case regret for the DRRO order quantity.
  • Figure 4: Comparison of order quantities and regret for ERM, DRO, DRRO, and DRRO relaxation with respect to $\delta$ when $b=0.1$ and $s=2$ and $N=1000$.
  • Figure 5: Comparison of order quantities and regret for ERM, DRO, DRRO, and DRRO relaxation with respect to $\delta$ when $b=1.5$, $s=2$, and $N=1000$.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 1
  • Lemma 2
  • Corollary 2
  • Theorem 6
  • ...and 18 more