不确定性量化的共形预测与Wasserstein分布鲁棒优化的统一视角

中文摘要 AI 辅助

有限数据的不确定性量化是机器学习、优化与自动化系统的核心，在有限样本和测试时分布偏移下，决策必须保持可靠。共形预测（CP）与分布鲁棒优化（DRO）提供了两种互补方法：CP构建依赖数据的预测集，在可交换性下具有无分布的有限样本有效性；DRO在经验分布周围的歧义集上优化最坏情况性能。我们通过将两者视为将有限校准数据转化为高概率使测试得分低于其的依赖数据分位数估计量的方式，建立了CP与DRO的统一概率视角。从该视角看，CP与DRO沿同一估计量族的两个坐标校正经验分位数：CP提高分位数水平，DRO通过歧义半径移动分位数值。两种方法对真实分布提供相同的校准条件保证，要求目标覆盖率在高概率下对校准样本成立。然而它们的构造不同：CP使用闭式、无分布的水平校正，DRO使用值空间校正，其认证半径依赖未知分布的性质，且额外保证在歧义集上的均匀覆盖率。这种区别出现在得分分布的尾部：由于CP依赖校准样本的稀疏上尾顺序统计量，当样本在目标分位数附近密集时，其水平膨胀几乎不移动估计量，而当样本稀疏时则会过冲；而选择得当的DRO半径在值空间进行校正，可避免这种过冲。

英文摘要

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

发表机构

• Contextual Robotics Institute, University of California San Diego（加州大学圣迭戈分校语境机器人研究所）
• University of Southern California（南加州大学）
• California Institute of Technology（加州理工学院）
• ETH Zürich（苏黎世联邦理工学院）

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