Scaling-Score Conformal Prediction for Multi-Target Regression

Sylvain Rousseau, Soundouss Messoudi
Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, PMLR 329:312-342, 2026.

Abstract

Multi-target regression requires a model to simultaneously predict several related outputs. Conformal prediction provides distribution-free, finite-sample marginal coverage guarantees, but extending these to joint multi-dimensional regions in a model-agnostic, sample-efficient manner remains challenging: max-aggregation ignores scale differences, copula-based methods are only asymptotically valid, rectangular methods typically split the calibration set, and quantile- or density-based methods require training a specialised model beyond a plain point predictor. We propose the scaling-score conformal method, which is model-agnostic (requires only component-wise absolute residuals), uses a single calibration set, and yields four nested output types: an outer rectangle (SCO) with valid joint coverage, the exact set R$\alpha$, a staircase (SC2) over-approximation of R$\alpha$, and an inner rectangle (SCI). A single hyperparameter $\gamma$ $\in$ (0,1) controls the base-rectangle quantile level independently of $\alpha$. We prove downward-closedness and a rectangular sandwich bound and derive a closed-form outer rectangle. Experiments on 29 real-world datasets confirm valid joint coverage; SC2 with $\gamma$ = 1 - $\alpha$ consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.

Cite this Paper

BibTeX
@InProceedings{pmlr-v329-rousseau26a, title = {Scaling-Score Conformal Prediction for Multi-Target Regression}, author = {Rousseau, Sylvain and Messoudi, Soundouss}, booktitle = {Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications}, pages = {312--342}, year = {2026}, editor = {Ahlberg, Ernst and Johansson, Ulf and Boström, Henrik and Carlevaro, Alberto and Hallberg Szabadváry, Johan and Carlsson, Lars}, volume = {329}, series = {Proceedings of Machine Learning Research}, month = {02--04 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v329/main/assets/rousseau26a/rousseau26a.pdf}, url = {https://proceedings.mlr.press/v329/rousseau26a.html}, abstract = {Multi-target regression requires a model to simultaneously predict several related outputs. Conformal prediction provides distribution-free, finite-sample marginal coverage guarantees, but extending these to joint multi-dimensional regions in a model-agnostic, sample-efficient manner remains challenging: max-aggregation ignores scale differences, copula-based methods are only asymptotically valid, rectangular methods typically split the calibration set, and quantile- or density-based methods require training a specialised model beyond a plain point predictor. We propose the scaling-score conformal method, which is model-agnostic (requires only component-wise absolute residuals), uses a single calibration set, and yields four nested output types: an outer rectangle (SCO) with valid joint coverage, the exact set R$\alpha$, a staircase (SC2) over-approximation of R$\alpha$, and an inner rectangle (SCI). A single hyperparameter $\gamma$ $\in$ (0,1) controls the base-rectangle quantile level independently of $\alpha$. We prove downward-closedness and a rectangular sandwich bound and derive a closed-form outer rectangle. Experiments on 29 real-world datasets confirm valid joint coverage; SC2 with $\gamma$ = 1 - $\alpha$ consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.} }
Endnote
%0 Conference Paper %T Scaling-Score Conformal Prediction for Multi-Target Regression %A Sylvain Rousseau %A Soundouss Messoudi %B Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications %C Proceedings of Machine Learning Research %D 2026 %E Ernst Ahlberg %E Ulf Johansson %E Henrik Boström %E Alberto Carlevaro %E Johan Hallberg Szabadváry %E Lars Carlsson %F pmlr-v329-rousseau26a %I PMLR %P 312--342 %U https://proceedings.mlr.press/v329/rousseau26a.html %V 329 %X Multi-target regression requires a model to simultaneously predict several related outputs. Conformal prediction provides distribution-free, finite-sample marginal coverage guarantees, but extending these to joint multi-dimensional regions in a model-agnostic, sample-efficient manner remains challenging: max-aggregation ignores scale differences, copula-based methods are only asymptotically valid, rectangular methods typically split the calibration set, and quantile- or density-based methods require training a specialised model beyond a plain point predictor. We propose the scaling-score conformal method, which is model-agnostic (requires only component-wise absolute residuals), uses a single calibration set, and yields four nested output types: an outer rectangle (SCO) with valid joint coverage, the exact set R$\alpha$, a staircase (SC2) over-approximation of R$\alpha$, and an inner rectangle (SCI). A single hyperparameter $\gamma$ $\in$ (0,1) controls the base-rectangle quantile level independently of $\alpha$. We prove downward-closedness and a rectangular sandwich bound and derive a closed-form outer rectangle. Experiments on 29 real-world datasets confirm valid joint coverage; SC2 with $\gamma$ = 1 - $\alpha$ consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.
APA
Rousseau, S. & Messoudi, S.. (2026). Scaling-Score Conformal Prediction for Multi-Target Regression. Proceedings of the Fifteenth Symposium on Conformal and Probabilistic Prediction with Applications, in Proceedings of Machine Learning Research 329:312-342 Available from https://proceedings.mlr.press/v329/rousseau26a.html.

Related Material