Paper: arXiv 2504.06381

Authors: Brandon Tam, Silvana M. Pesenti

Abstract

We study distributionally robust optimization (DRO) problems with uncertainty sets consisting of high-dimensional random vectors that are close in the multivariate Wasserstein distance to a reference random vector. We give conditions when the images of these sets under scalar-valued aggregation functions are contained in and contain uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This provides lower and upper bounds for the solution to general multivariate DRO problems that are computationally tractable. Furthermore, we generalize the results to uncertainty sets characterized by Bregman-Wasserstein divergences, which allows for asymmetric deviations from the reference random vector. Moreover, for DRO problems with risk measure criterion in the class of signed Choquet integrals, we derive semi-analytic formulae for the upper and lower bounds and the distribution that attains these bounds.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework for high-dimensional distributionally robust optimization (DRO), deriving closed-form bounds using Wasserstein distances and Bregman divergences. While mathematically sophisticated and well-written, it focuses primarily on theoretical derivations and proofs with limited empirical validation or real-world backtesting included in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Compute tractable bounds for high-dimensional Multivariate DRO] --> B[Inputs: Reference Random Vector & Wasserstein / Bregman-Wasserstein Uncertainty Sets]
    B --> C[Methodology: Map multivariate sets to univariate sets via scalar aggregation functions]
    C --> D[Computational Process: Derive lower & upper bounds using univariate Wasserstein metrics]
    D --> E[Specialization: Apply signed Choquet integrals for risk measure criteria]
    E --> F[Outcomes: Semi-analytic formulae for optimal bounds & worst-case/best-case distributions]

Cited References (69)

  1. Model Aggregation for Risk Evaluation and Robust Optimization (2025)
  2. Rings of Functions with Certain L ipschitz Properties (1970)
  3. Characterization, Robustness and Aggregation of Signed C hoquet Integrals (2020)
  4. Optimal Transport Divergences Induced by Scoring Functions — → 2311.12183
  5. Theory of Capacities (1954)
  6. Reverse Sensitivity Analysis for Risk Modelling (2022)
  7. Robust Distortion Risk Measures (2024)
  8. Preference robust distortion risk measures (2026) — link
and 61 more
  1. The Isotonic Regression Problem and Its Dual (1972)
  2. Fairness through awareness (2012)
  3. Lipschitz regularity of deep neural networks: analysis and efficient estimation (2018)
  4. Exactly computing the local L ipschitz constant of R e LU networks (2020)
  5. Optimal Payoff Under B regman- W asserstein Divergence Constraints (2024)
  6. Bregman- W asserstein Divergence: Geometry and Applications (2025)
  7. M ahalanobis Distance (1999)
  8. Simulation Based Inference with the Generalized K ullback- L iebler Divergence (2023)
  9. Statistical Inference Under Order Restrictions: The Theory and Application of Isotonic Regression (1972)
  10. Conditional Expectation Given a Sigma Lattice and Applications (1965)
  11. Approximated Convex Envelope of a Function (1994)
  12. Isotone Optimization in R : Pool-Adjacent-Violators Algorithm ( PAVA ) and Active Set Methods (2009)
  13. Advances in Prospect Theory: Cumulative Representation of Uncertainty (1992)
  14. W asserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning (2019)
  15. Kullback- L eibler Barycentre of Stochastic Processes (2026)
  16. Sugli Estremi Dei Momenti Delle Funzioni Di Ripartizione Doppia (1956)
  17. On projection robust optimal transport: Sample complexity and model misspecification (2021)
  18. Coherent Measures of Risk (1999)
  19. Convex Measures of Risk and Trading Constraints (2002)
  20. Generalized Deviations in Risk Analysis (2006)
  21. On Law Invariant Coherent Risk Measures (2001)
  22. Robustness and Sensitivity Analysis of Risk Measurement Procedures (2010)
  23. Aggregation-Robustness and Model Uncertainty of Regulatory Risk Measures (2015)
  24. Robustness Regions for Measures of Risk Aggregation (2016)
  25. Outperforming a Benchmark with $\alpha$- B regman W asserstein divergence — → 2603.20580
  26. Linear Programming Under Uncertainty (1955)
  27. On minimizing a convex function subject to linear inequalities (1955)
  28. Convex analysis (1970)
  29. C onvexity and O ptimization in B anach S paces (2012)
  30. Lectures on stochastic programming: modeling and theory (2021)
  31. The Optimizer’s Curse: Skepticism and Postdecision Surprise in Decision Analysis (2006)
  32. The M arkowitz Optimization Enigma: Is ‘optimized’ optimal? (1989)
  33. A min-max solution of an inventory problem (1958)
  34. Robust Convex Optimization (1998)
  35. Robust Solutions of Uncertain Linear Programs (1999)
  36. Topics in Optimal Transportation (1973)
  37. Statistical Aspects of W asserstein Distances (2019)
  38. Prescribing a System of Random Variables by Conditional Distributions (1970)
  39. Nonparametric Validation of Similar Distributions and Assessment of Goodness of Fit (1998)
  40. Distributionally Robust Stochastic Optimization with W asserstein Distance (2023)
  41. Quantifying Distributional Model Risk via Optimal Transport (2019)
  42. On M onge’s Problem for B regman-like Cost Functions (2007)
  43. The L ipschitz constant of self-attention (2021)
  44. Uncertainty propagation and dynamic robust risk measures — → 2308.12856
  45. Portfolio optimization within a W asserstein ball (2023)
  46. Optimizing distortion riskmetrics with distributional uncertainty (2024)
  47. Ambiguity Set and Learning via B regman and W asserstein (2017)
  48. Robust Reinforcement Learning with Dynamic Distortion Risk Measures — → 2409.10096
  49. The F r' e chet Distance Between Multivariate Normal Distributions (1982)
  50. Sinkhorn Distances: Lightspeed Computation of Optimal Transport (2013)
  51. Distributionally Robust Optimization Under Distorted Expectations (2023)
  52. Frameworks and Results in Distributionally Robust Optimization (2022)
  53. Robustness (2008)
  54. On a Formula for the L 2 W asserstein Metric between Measures on E uclidean and H ilbert Spaces (1990)
  55. Data-driven chance constrained programs over W asserstein balls (2024)
  56. On distributionally robust chance constrained programs with W asserstein distance (2021)
  57. Data-driven risk-averse stochastic optimization with W asserstein metric (2018)
  58. N on- A dditive M easure and I ntegral (1994)
  59. Data-driven distributionally robust optimization using the W asserstein metric: Performance guarantees and tractable reformulations (2018)
  60. Robustifying convex risk measures for linear portfolios: A nonparametric approach (2014)
  61. I nf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures (2022)

4 of 69 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.