Beyond Uncertainty Sets: Leveraging Optimal Transport to Extend Conformal Predictive Distribution to Multivariate Settings

2025-11-19

AI Summary

This paper addresses the lack of finite-sample coverage guarantees for uncertainty quantification in multi-output settings. We propose the first Multivariate Conformal Prediction Distribution (MCPD) with rigorous finite-sample calibration. Methodologically, we innovatively integrate optimal transport theory with the conformal prediction framework: vector ranks and quantile regions are defined via transport maps; piecewise-constant assignment coupled with multivariate weighted quantile computation yields an analytically tractable predictive distribution. Our contributions are threefold: (1) We extend conformal prediction beyond univariate scores or scalar outputs to general multivariate outputs; (2) All derived uncertainty regions—regardless of shape or construction—automatically satisfy the user-specified marginal coverage probability; (3) We provide two practical implementations—conservative deterministic and exact randomized—thereby generalizing the Dempster–Hill procedure to multivariate settings for the first time.

Technology Category

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

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. A candidate output is included in the prediction set if its non-conformity score is not considered extreme relative to the scores observed on a set of calibration examples. However, this procedure is only straightforward when scores are scalar-valued, which has limited CP to real-valued scores or ad-hoc reductions to one dimension. The problem of ordering vectors has been studied via optimal transport (OT), which provides a principled method for defining vector-ranks and multivariate quantile regions, though typically with only asymptotic coverage guarantees. We restore finite-sample, distribution-free coverage by conformalizing the vector-valued OT quantile region. Here, a candidate's rank is defined via a transport map computed for the calibration scores augmented with that candidate's score. This defines a continuum of OT problems for which we prove that the resulting optimal assignment is piecewise-constant across a fixed polyhedral partition of the score space. This allows us to characterize the entire prediction set tractably, and provides the machinery to address a deeper limitation of prediction sets: that they only indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration, i.e., they define a valid multivariate distribution where any derived uncertainty region automatically has guaranteed coverage. We present both conservative and exact randomized versions, the latter resulting in a multivariate generalization of the classical Dempster-Hill procedure.

Problem

• Extends conformal prediction to multivariate settings using optimal transport
• Constructs valid multivariate predictive distributions with finite-sample guarantees
• Solves vector ordering problem to define multivariate ranks and quantile regions

Innovation

• Extends conformal prediction using optimal transport
• Constructs multivariate predictive distributions with finite-sample guarantees
• Provides both conservative and exact randomized distribution versions