Multivariate Conformal Prediction via Conformalized Gaussian Scoring

2025-07-28

AI Summary

Existing conformal prediction methods struggle to achieve approximate conditional coverage without strong regularity assumptions—particularly in modeling heteroscedasticity. Method: We propose a novel Gaussian-score-based conformal prediction framework that reformulates the CDF-based nonconformity score as a closed-form Mahalanobis distance, eliminating reliance on Monte Carlo sampling. The framework supports invertible output-space transformations, missing-value imputation, and local information updates, integrating conditional density estimation with an analytically tractable Gaussian scoring model. Contribution/Results: Our approach enables efficient, differentiable, and scalable construction of multivariate prediction sets. Experiments demonstrate substantial improvements in conditional coverage approximation accuracy for multivariate settings, alongside reduced computational overhead. By jointly addressing heteroscedasticity and structural flexibility, the method establishes a new paradigm for reliable uncertainty quantification under heterogeneous noise.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

While achieving exact conditional coverage in conformal prediction is unattainable without making strong, untestable regularity assumptions, the promise of conformal prediction hinges on finding approximations to conditional guarantees that are realizable in practice. A promising direction for obtaining conditional dependence for conformal sets--in particular capturing heteroskedasticity--is through estimating the conditional density $mathbb{P}_{Y|X}$ and conformalizing its level sets. Previous work in this vein has focused on nonconformity scores based on the empirical cumulative distribution function (CDF). Such scores are, however, computationally costly, typically requiring expensive sampling methods. To avoid the need for sampling, we observe that the CDF-based score reduces to a Mahalanobis distance in the case of Gaussian scores, yielding a closed-form expression that can be directly conformalized. Moreover, the use of a Gaussian-based score opens the door to a number of extensions of the basic conformal method; in particular, we show how to construct conformal sets with missing output values, refine conformal sets as partial information about $Y$ becomes available, and construct conformal sets on transformations of the output space. Finally, empirical results indicate that our approach produces conformal sets that more closely approximate conditional coverage in multivariate settings compared to alternative methods.

Problem

• Approximating conditional coverage in conformal prediction
• Avoiding costly sampling in nonconformity score computation
• Extending conformal methods for multivariate settings

Innovation

• Uses Gaussian scores for closed-form conformalization
• Extends conformal method to missing output values
• Refines conformal sets with partial information updates