Multivariate Standardized Residuals for Conformal Prediction

TL;DR

This paper extends conformal prediction to multivariate outputs by normalizing residuals with learned local covariance, improving conditional coverage and enabling practical extensions like handling missing data.

Contribution

It introduces a multivariate residual normalization method using Mahalanobis distance, providing a computationally efficient way to improve conditional coverage in conformal prediction.

Findings

Standardized residuals improve conditional coverage in heteroskedastic settings.

Mahalanobis distance captures inter-output correlations effectively.

The method enables handling missing data and transformations in conformal sets.

Abstract

While split conformal prediction guarantees marginal coverage, approaching the stronger property of conditional coverage is essential for reliable uncertainty quantification. Naive conformal scores, however, suffer from poor conditional coverage in heteroskedastic settings. In univariate regression, this is commonly addressed by normalizing non-conformity scores using an estimated local score variance. In this work, we propose a natural extension of this normalization to the multivariate setting, effectively whitening the residuals to decouple output correlations and standardize local variance. Furthermore, we derive a sufficient condition characterizing a broad class of distributions for which standardized residuals yield asymptotic conditional coverage. We demonstrate that using the Mahalanobis distance induced by a learned local covariance as a non-conformity score provides a closed-form, computationally efficient mechanism for capturing inter-output correlations and heteroskedasticity, avoiding the expensive sampling required by previous methods based on cumulative distribution functions. This structure unlocks several practical extensions, including the handling of missing output values, the refinement of conformal sets when partial information is revealed, and the construction of valid conformal sets for transformations of the output. Finally, we provide extensive empirical evidence on both synthetic and real-world datasets showing that our approach yields conformal sets that improve upon the conditional coverage of existing multivariate baselines.