Exact and Approximate Conformal Inference for Multi-Output Regression

Chancellor Johnstone, Eugene Ndiaye
Proceedings of the Fourteenth Symposium on Conformal and Probabilistic Prediction with Applications, PMLR 266:153-172, 2025.

Abstract

It is common in machine learning to estimate a response $y$ given covariate information $x$. However, these predictions alone do not quantify any uncertainty associated with said predictions. One way to overcome this deficiency is with conformal inference methods, which construct a set containing the unobserved response with a prescribed probability. Unfortunately, even with a one-dimensional response, conformal inference is computationally expensive despite recent encouraging advances. In this paper, we explore multi-output regression, delivering exact derivations of conformal inference p-values when the predictive model can be described as a linear function of $y$. Additionally, we introduce a multivariate extension of rootCP as well unionCP as efficient ways of approximating the conformal prediction region for a wide array of multi-output predictors, both linear and nonlinear, while preserving computational advantages. We also provide both theoretical and empirical evidence of the effectiveness of our methods using both real-world and simulated data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v266-johnstone25a, title = {Exact and Approximate Conformal Inference for Multi-Output Regression}, author = {Johnstone, Chancellor and Ndiaye, Eugene}, booktitle = {Proceedings of the Fourteenth Symposium on Conformal and Probabilistic Prediction with Applications}, pages = {153--172}, year = {2025}, editor = {Nguyen, Khuong An and Luo, Zhiyuan and Papadopoulos, Harris and Löfström, Tuwe and Carlsson, Lars and Boström, Henrik}, volume = {266}, series = {Proceedings of Machine Learning Research}, month = {10--12 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v266/main/assets/johnstone25a/johnstone25a.pdf}, url = {https://proceedings.mlr.press/v266/johnstone25a.html}, abstract = {It is common in machine learning to estimate a response $y$ given covariate information $x$. However, these predictions alone do not quantify any uncertainty associated with said predictions. One way to overcome this deficiency is with conformal inference methods, which construct a set containing the unobserved response with a prescribed probability. Unfortunately, even with a one-dimensional response, conformal inference is computationally expensive despite recent encouraging advances. In this paper, we explore multi-output regression, delivering exact derivations of conformal inference p-values when the predictive model can be described as a linear function of $y$. Additionally, we introduce a multivariate extension of rootCP as well unionCP as efficient ways of approximating the conformal prediction region for a wide array of multi-output predictors, both linear and nonlinear, while preserving computational advantages. We also provide both theoretical and empirical evidence of the effectiveness of our methods using both real-world and simulated data.} }
Endnote
%0 Conference Paper %T Exact and Approximate Conformal Inference for Multi-Output Regression %A Chancellor Johnstone %A Eugene Ndiaye %B Proceedings of the Fourteenth Symposium on Conformal and Probabilistic Prediction with Applications %C Proceedings of Machine Learning Research %D 2025 %E Khuong An Nguyen %E Zhiyuan Luo %E Harris Papadopoulos %E Tuwe Löfström %E Lars Carlsson %E Henrik Boström %F pmlr-v266-johnstone25a %I PMLR %P 153--172 %U https://proceedings.mlr.press/v266/johnstone25a.html %V 266 %X It is common in machine learning to estimate a response $y$ given covariate information $x$. However, these predictions alone do not quantify any uncertainty associated with said predictions. One way to overcome this deficiency is with conformal inference methods, which construct a set containing the unobserved response with a prescribed probability. Unfortunately, even with a one-dimensional response, conformal inference is computationally expensive despite recent encouraging advances. In this paper, we explore multi-output regression, delivering exact derivations of conformal inference p-values when the predictive model can be described as a linear function of $y$. Additionally, we introduce a multivariate extension of rootCP as well unionCP as efficient ways of approximating the conformal prediction region for a wide array of multi-output predictors, both linear and nonlinear, while preserving computational advantages. We also provide both theoretical and empirical evidence of the effectiveness of our methods using both real-world and simulated data.
APA
Johnstone, C. & Ndiaye, E.. (2025). Exact and Approximate Conformal Inference for Multi-Output Regression. Proceedings of the Fourteenth Symposium on Conformal and Probabilistic Prediction with Applications, in Proceedings of Machine Learning Research 266:153-172 Available from https://proceedings.mlr.press/v266/johnstone25a.html.

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