# Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters

arXiv: https://arxiv.org/abs/2203.12136 · HTML: https://mlsanity.com/paper/2203.12136/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 71 of its 138 references that are in the map.
- First posted on arXiv 23 Mar 2022.

## Builds on (most cited first, 30 of 71)

- [Communication-Efficient Learning of Deep Networks from Decentralized Data](https://mlsanity.com/paper/1602.05629/) (2016)
- [Advances and Open Problems in Federated Learning](https://mlsanity.com/paper/1912.04977/) (2019)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (2019)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Agnostic Federated Learning](https://mlsanity.com/paper/1902.00146/) (2019)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [A Field Guide to Federated Optimization](https://mlsanity.com/paper/2107.06917/) (2021)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Large-Scale Methods for Distributionally Robust Optimization](https://mlsanity.com/paper/2010.05893/) (2020)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Distributionally Robust Federated Averaging](https://mlsanity.com/paper/2102.12660/) (2021)
- [Robust Federated Learning: The Case of Affine Distribution Shifts](https://mlsanity.com/paper/2006.08907/) (2020)
- [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 and Generalization in Kernel Methods](https://mlsanity.com/paper/1905.10943/) (2019)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [What Do We Mean by Generalization in Federated Learning?](https://mlsanity.com/paper/2110.14216/) (2021)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Non-convex Distributionally Robust Optimization: Non-asymptotic Analysis](https://mlsanity.com/paper/2110.12459/) (2021)
- [Wasserstein Dictionary Learning: Optimal Transport-based unsupervised non-linear dictionary learning](https://mlsanity.com/paper/1708.01955/) (2017)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)

## Built on by (most cited first, 3 of 3)

- [On the Generalization of Wasserstein Robust Federated Learning](https://mlsanity.com/paper/2206.01432/) (2022)
- [FDR-SVM: A Federated Distributionally Robust Support Vector Machine via a Mixture of Wasserstein Balls Ambiguity Set](https://mlsanity.com/paper/2410.03877/) (2024)
- [Multi-Source Wasserstein Distributionally Robust Graph Learning](https://mlsanity.com/paper/2608.19914/) (2026)

## Abstract (arXiv)

In many applications in statistics and machine learning, the availability of data samples from multiple possibly heterogeneous sources has become increasingly prevalent. On the other hand, in distributionally robust optimization, we seek data-driven decisions which perform well under the most adverse distribution from a nominal distribution constructed from data samples within a certain discrepancy of probability distributions. However, it remains unclear how to achieve such distributional robustness in model learning and estimation when data samples from multiple sources are available. In this work, we propose constructing the nominal distribution in optimal transport-based distributionally robust optimization problems through the notion of Wasserstein barycenter as an aggregation of data samples from multiple sources. Under specific choices of the loss function, the proposed formulation admits a tractable reformulation as a finite convex program, with powerful finite-sample and asymptotic guarantees. As an illustrative example, we demonstrate with the problem of distributionally robust sparse inverse covariance matrix estimation for zero-mean Gaussian random vectors that our proposed scheme outperforms other widely used estimators in both the low- and high-dimensional regimes.