# Smoothed f-Divergence Distributionally Robust Optimization

arXiv: https://arxiv.org/abs/2306.14041 · HTML: https://mlsanity.com/paper/2306.14041/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 91 references that are in the map.
- First posted on arXiv 24 Jun 2023.

## Builds on (most cited first, 26 of 26)

- [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)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (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)
- [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)
- [Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality](https://mlsanity.com/paper/2009.04382/) (2020)
- [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)
- [Acceleration with a Ball Optimization Oracle](https://mlsanity.com/paper/2003.08078/) (2020)
- [Robust Sample Average Approximation](https://mlsanity.com/paper/1408.4445/) (2014)
- [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)
- [Distributional Robustness with IPMs and links to Regularization and GANs](https://mlsanity.com/paper/2006.04349/) (2020)
- [Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory](https://mlsanity.com/paper/2011.06186/) (2020)
- [The Empirical Likelihood Approach to Quantifying Uncertainty in Sample Average Approximation](https://mlsanity.com/paper/1604.02573/) (2016)
- [Formulation and properties of a divergence used to compare probability measures without absolute continuity](https://mlsanity.com/paper/1911.07422/) (2019)
- [Generalization Bounds with Minimal Dependency on Hypothesis Class via Distributionally Robust Optimization](https://mlsanity.com/paper/2106.11180/) (2021)
- [Learning and Decision-Making with Data: Optimal Formulations and Phase Transitions](https://mlsanity.com/paper/2109.06911/) (2021)
- [Sensitivity to Serial Dependency of Input Processes: A Robust Approach](https://mlsanity.com/paper/1606.06645/) (2016)
- [The discrete moment problem with nonconvex shape constraints](https://mlsanity.com/paper/1708.02079/) (2017)
- [Worst-Case Expected Shortfall with Univariate and Bivariate Marginals](https://mlsanity.com/paper/1701.04167/) (2017)
- [Tractable Reformulations of Distributionally Robust Two-stage Stochastic Programs with ∞-Wasserstein Distance](https://mlsanity.com/paper/1908.08454/) (2019)
- [Orthounimodal Distributionally Robust Optimization: Representation, Computation and Multivariate Extreme Event Applications](https://mlsanity.com/paper/2111.07894/) (2021)
- [A Distributionally Robust Optimization Framework for Extreme Event Estimation](https://mlsanity.com/paper/2301.01360/) (2023)

## Built on by (most cited first, 7 of 7)

- [Unifying Distributionally Robust Optimization via Optimal Transport Theory](https://mlsanity.com/paper/2308.05414/) (2023)
- [Nested Stochastic Algorithm for Generalized Sinkhorn distance-Regularized Distributionally Robust Optimization](https://mlsanity.com/paper/2503.22923/) (2025)
- [Optimal Transport Regularized Divergences: Application to Adversarial Robustness](https://mlsanity.com/paper/2309.03791/) (2023)
- [From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective](https://mlsanity.com/paper/2410.14008/) (2024)
- [A Dual Optimization View to Empirical Risk Minimization with f-Divergence Regularization](https://mlsanity.com/paper/2508.03314/) (2025)
- [Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences](https://mlsanity.com/paper/2603.27871/) (2026)
- [Adversarial Training for Deep Hedging in Nonstationary Markets](https://mlsanity.com/paper/2610.07162/) (2026)

## Abstract (arXiv)

In data-driven optimization, sample average approximation (SAA) is known to suffer from the so-called optimizer's curse that causes an over-optimistic evaluation of the solution performance. We argue that a special type of distributionallly robust optimization (DRO) formulation offers theoretical advantages in correcting for this optimizer's curse compared to simple ``margin'' adjustments to SAA and other DRO approaches: It attains a statistical bound on the out-of-sample performance, for a wide class of objective functions and distributions, that is nearly tightest in terms of exponential decay rate. This DRO uses an ambiguity set based on a Kullback Leibler (KL) divergence smoothed by the Wasserstein or L'evy-Prokhorov (LP) distance via a suitable distance optimization. Computationally, we also show that such a DRO, and its generalized versions using smoothed f-divergence, are not harder than DRO problems based on f-divergence or Wasserstein distances, rendering our DRO formulations both statistically optimal and computationally viable.