A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification
Abstract
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.
Related
- Higher-Order Sensitivity Analysis of Wasserstein and Adapted Wasserstein Distributionally Robust Optimization Problems
- First-Order Methods for Wasserstein Distributionally Robust Constrained Optimization
- Distributionally robust chance-constrained standard quadratic optimization with Wasserstein ambiguity
- Integrating Machine Learning and Uncertainty Quantification for Real-Time Decision-Making in Digital Twins: A System-Centric Review and Perspective
- Continuous-Time Wasserstein Distributionally Robust Control: Limits, Duality, and Stability
BibTeX
@misc{long2026unified,
title = {{A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification}},
author = {Kehan Long and Yiqi Zhao and Pol Mestres and Lars Lindemann and Nikolay Atanasov and Jorge Cortés},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29789},
eprint = {2608.29789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29789},
}
RIS
TY - UNPB
TI - A Unified Perspective on Conformal Prediction and Wasserstein Distributionally Robust Optimization for Uncertainty Quantification
AU - Kehan Long
AU - Yiqi Zhao
AU - Pol Mestres
AU - Lars Lindemann
AU - Nikolay Atanasov
AU - Jorge Cortés
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29789
UR - https://arxiv.org/abs/2608.29789
AB - 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.
ER -