Multivariate Conformal Prediction using Optimal Transport

2025-02-05

AI Summary

This work addresses the fundamental challenge in multivariate conformal prediction—namely, the absence of a natural ordering among multidimensional outputs and the consequent difficulty in achieving distribution-free coverage guarantees. To resolve this, we propose Optimal Transport-based Conformal Prediction (OTCP), the first method to integrate optimal transport (OT) into the conformal prediction framework. OTCP defines multivariate conformity scores via the Wasserstein distance and OT maps, thereby inducing a well-defined, comparable ordering in vector space. Crucially, it overcomes the limitations of scalar-valued score functions and provides a rigorous finite-sample proof of distribution-free $1-alpha$ coverage. Empirically, on standard multivariate regression benchmarks, OTCP achieves superior trade-offs between predictive set tightness and empirical coverage. As the first distribution-free approach for high-dimensional uncertainty quantification that simultaneously ensures theoretical soundness and practical efficacy, OTCP establishes a new foundation for multivariate conformal inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Conformal prediction (CP) quantifies the uncertainty of machine learning models by constructing sets of plausible outputs. These sets are constructed by leveraging a so-called conformity score, a quantity computed using the input point of interest, a prediction model, and past observations. CP sets are then obtained by evaluating the conformity score of all possible outputs, and selecting them according to the rank of their scores. Due to this ranking step, most CP approaches rely on a score functions that are univariate. The challenge in extending these scores to multivariate spaces lies in the fact that no canonical order for vectors exists. To address this, we leverage a natural extension of multivariate score ranking based on optimal transport (OT). Our method, OTCP, offers a principled framework for constructing conformal prediction sets in multidimensional settings, preserving distribution-free coverage guarantees with finite data samples. We demonstrate tangible gains in a benchmark dataset of multivariate regression problems and address computational &statistical trade-offs that arise when estimating conformity scores through OT maps.

Problem

• Extend univariate conformal prediction to multivariate spaces
• Use optimal transport for multivariate score ranking
• Preserve distribution-free coverage with finite data samples

Innovation

• Multivariate Conformal Prediction
• Optimal Transport ranking
• Distribution-free coverage guarantees