From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective
This work addresses the problem of conservatism in DRO formulations for researchers and practitioners in robust optimization and statistics, revealing it stems from non-parametric frameworks rather than inherent limitations.
The paper connects distributionally robust optimization (DRO) to robust statistics by showing that DRO ambiguity sets, based on Kullback-Leibler divergence and total variation distance, are uniformly minimal confidence sets for unknown data-generating distributions under data corruption, and are never larger than those based on Huber's optimal estimator in parametric settings.
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.