# Unifying Distributionally Robust Optimization via Optimal Transport Theory

arXiv: https://arxiv.org/abs/2308.05414 · HTML: https://mlsanity.com/paper/2308.05414/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 88 references that are in the map.
- First posted on arXiv 10 Aug 2023.

## Builds on (most cited first, 30 of 44)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Data-Driven Robust Optimization](https://mlsanity.com/paper/1401.0212/) (2013)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [From Data to Decisions: Distributionally Robust Optimization is Optimal](https://mlsanity.com/paper/1704.04118/) (2017)
- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization](https://mlsanity.com/paper/1605.09349/) (2016)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [On Linear Optimization over Wasserstein Balls](https://mlsanity.com/paper/2004.07162/) (2020)
- [Fast Distributionally Robust Learning with Variance Reduced Min-Max Optimization](https://mlsanity.com/paper/2104.13326/) (2021)
- [Holistic Robust Data-Driven Decisions](https://mlsanity.com/paper/2207.09560/) (2022)
- [On Distributionally Robust Chance Constrained Programs with Wasserstein Distance](https://mlsanity.com/paper/1806.07418/) (2018)
- [Generalised Lipschitz Regularisation Equals Distributional Robustness](https://mlsanity.com/paper/2002.04197/) (2020)
- [Likelihood Robust Optimization for Data-driven Problems](https://mlsanity.com/paper/1307.6279/) (2013)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/2205.00362/) (2022)

## Built on by (most cited first, 13 of 13)

- [DRO: A Python Library for Distributionally Robust Optimization in Machine Learning](https://mlsanity.com/paper/2505.23565/) (2025)
- [LLM Embeddings Improve Test-time Adaptation to Tabular Y|X-Shifts](https://mlsanity.com/paper/2410.07395/) (2024)
- [Nested Stochastic Algorithm for Generalized Sinkhorn distance-Regularized Distributionally Robust Optimization](https://mlsanity.com/paper/2503.22923/) (2025)
- [Proximal optimal transport divergences](https://mlsanity.com/paper/2505.12097/) (2025)
- [Distributionally Robust Optimization via Generative Ambiguity Modeling](https://mlsanity.com/paper/2602.08976/) (2026)
- [Optimal Transport Regularized Divergences: Application to Adversarial Robustness](https://mlsanity.com/paper/2309.03791/) (2023)
- [Stability Evaluation via Distributional Perturbation Analysis](https://mlsanity.com/paper/2405.03198/) (2024)
- [Data Curation Matters: Model Collapse and Spurious Shift Performance Prediction from Training on Uncurated Text Embeddings](https://mlsanity.com/paper/2506.17989/) (2025)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)
- [From Dual Tracking to Clipping: Provably Faster Distributionally Robust Multi-Objective Optimization](https://mlsanity.com/paper/2605.05660/) (2026)
- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)
- [Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport](https://mlsanity.com/paper/2609.22240/) (2026)
- [Adversarial Training for Deep Hedging in Nonstationary Markets](https://mlsanity.com/paper/2610.07162/) (2026)

## Abstract (arXiv)

In recent years, two prominent paradigms have shaped distributionally robust optimization (DRO), modeling distributional ambiguity through φ-divergences and Wasserstein distances, respectively. While the former focuses on ambiguity in likelihood ratios, the latter emphasizes ambiguity in outcomes and uses a transportation cost function to capture geometric structure in the outcome space. This paper proposes a unified framework that bridges these approaches by leveraging optimal transport (OT) with conditional moment constraints. Our formulation enables adversarial distributions to jointly perturb likelihood ratios and outcomes, yielding a generalized OT coupling between the nominal and perturbed distributions. We further establish key duality results and develop tractable reformulations that highlight the practical power of our unified approach.