揭示分布鲁棒优化与经典稳健统计的联系，指出其保守性源于非参数框架而非方法本身。
From Distributional Robustness to Robust Statistics: A Confidence Sets Perspective
- 从置信集视角建立分布鲁棒优化与稳健统计的关联
- 基于KL散度和总变差的距离，构造最小置信集，具有统一最优性
- 在参数假设下，其结果比Huber最优估计的置信集更小，适合稳健推断场景
我们建立了分布鲁棒优化（DRO）与经典稳健统计之间的联系。在数据污染下的估计问题中，目标是为未知数据生成分布构造‘最小’置信集。具体而言，我们证明了基于KL散度和总变差距离的DRO模糊集具有统一最小性，即在给定置信水平下，它是最小的包含未知分布的置信集。此外，当对未知分布施加参数假设时，该模糊集不会大于Huber提出的最优估计所对应的置信集。这一发现表明，通常观察到的DRO保守性并非源于方法本身，而是源于其采用的非参数框架。
原文摘要 · Abstract (English)
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.
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