# From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective

arXiv: https://arxiv.org/abs/2410.14008 · HTML: https://mlsanity.com/paper/2410.14008/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 17 Oct 2024.

## Builds on (most cited first, 11 of 11)

- [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)
- [Data-Driven Robust Optimization](https://mlsanity.com/paper/1401.0212/) (2013)
- [Regularization via Mass Transportation](https://mlsanity.com/paper/1710.10016/) (2017)
- [From Data to Decisions: Distributionally Robust Optimization is Optimal](https://mlsanity.com/paper/1704.04118/) (2017)
- [Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization](https://mlsanity.com/paper/1605.09349/) (2016)
- [Confidence Regions in Wasserstein Distributionally Robust Estimation](https://mlsanity.com/paper/1906.01614/) (2019)
- [Holistic Robust Data-Driven Decisions](https://mlsanity.com/paper/2207.09560/) (2022)
- [Statistical Limit Theorems in Distributionally Robust Optimization](https://mlsanity.com/paper/2303.14867/) (2023)
- [Smoothed f-Divergence Distributionally Robust Optimization](https://mlsanity.com/paper/2306.14041/) (2023)
- [Efficient Data-Driven Optimization with Noisy Data](https://mlsanity.com/paper/2102.04363/) (2021)

## Built on by (most cited first, 1 of 1)

- [Bulk-Calibrated Credal Ambiguity Sets: Fast, Tractable Decision Making under Out-of-Sample Contamination](https://mlsanity.com/paper/2601.21324/) (2026)

## Abstract (arXiv)

We establish a connection between distributionally robust optimization (DRO) and classical robust statistics. We demonstrate that this connection arises naturally in the context of estimation under data corruption, where the goal is to construct ``minimal'' confidence sets for the unknown data-generating distribution. Specifically, we show that a DRO ambiguity set, based on the Kullback-Leibler divergence and total variation distance, is uniformly minimal, meaning it represents the smallest confidence set that contains the unknown distribution with at a given confidence power. Moreover, we prove that when parametric assumptions are imposed on the unknown distribution, the ambiguity set is never larger than a confidence set based on the optimal estimator proposed by Huber. This insight reveals that the commonly observed conservatism of DRO formulations is not intrinsic to these formulations themselves but rather stems from the non-parametric framework in which these formulations are employed.