# Nonlinear Distributionally Robust Optimization

arXiv: https://arxiv.org/abs/2306.03202 · HTML: https://mlsanity.com/paper/2306.03202/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 62 references that are in the map.
- First posted on arXiv 5 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [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)
- [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)
- [Distributionally Robust Logistic Regression](https://mlsanity.com/paper/1509.09259/) (2015)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2109.11926/) (2021)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Distributionally Robust Mean-Variance Portfolio Selection with Wasserstein Distances](https://mlsanity.com/paper/1802.04885/) (2018)
- [First-order Conditions for Optimization in the Wasserstein Space](https://mlsanity.com/paper/2209.12197/) (2022)
- [Linearly Convergent Frank-Wolfe with Backtracking Line-Search](https://mlsanity.com/paper/1806.05123/) (2018)
- [Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization](https://mlsanity.com/paper/1911.03539/) (2019)
- [Mean-Covariance Robust Risk Measurement](https://mlsanity.com/paper/2112.09959/) (2021)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [A Particle-Based Algorithm for Distributional Optimization on textitConstrained Domains via Variational Transport and Mirror Descent](https://mlsanity.com/paper/2208.00587/) (2022)

## Abstract (arXiv)

This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative-based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the entropic and variance risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and "sufficient statistic", compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.