Statistical learning under distribution shift is challenging when neither prior knowledge nor fully accessible data from the target distribution is available. Distributionally robust learning (DRL) aims to control the worst-case statistical performance within an uncertainty set of candidate distributions, but how to properly specify the set remains challenging. To enable distributional robustness without being overly conservative, in this paper, we propose a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate. More specifically, we assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. At the population level, we provide a solution to the shape-constrained optimization problem that does not involve the isotonic constraint. At the sample level, we provide consistency results for an empirical estimator of the target in a range of different settings. Empirical studies on both synthetic and real data examples demonstrate the improved accuracy of the proposed shape-constrained approach.

翻译：当既无先验知识又无法完全获取目标分布数据时，分布偏移下的统计学习面临挑战。分布鲁棒学习旨在控制候选分布不确定性集合内的最坏统计性能，但如何恰当设定该集合仍具挑战性。为实现分布鲁棒性同时避免过度保守，本文提出一种具有形状约束的分布鲁棒学习方法，该方法融合了关于未知目标分布与其估计值差异方式的先验信息。具体而言，我们假设目标分布与其估计值之间的未知密度比关于某种偏序关系具有等渗性。在总体层面，我们给出了不涉及等渗约束的形状约束优化问题的解。在样本层面，我们为多种不同场景下目标的经验估计量提供了一致性结果。基于合成数据与真实数据案例的实证研究表明，所提出的形状约束方法具有更高的准确性。