A Unified Comparative Study with Generalized Conformity Scores for Multi-output Conformal Regression - research findings

Aug 5, 2026

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Conformal prediction builds distribution-free prediction regions with finite-sample coverage guarantees, yet multi-output extensions remain difficult due to output dependencies and high computation. The study presents a unified comparison of nine multi-output conformal methods under a shared framework, clarifying key properties and relationships. It introduces two new multi-output conformity-score classes that generalize univariate counterparts, ensuring asymptotic conditional coverage while keeping exact finite-sample marginal coverage. Experiments across 13 tabular datasets evaluate all explored methods using a consistent implementation.

LanguageEnglish

CategoryResearch & Report

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Pages42

Reading Time1 h 46 min

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Abstract

Introduction

Motivation and challenges

Contributions

Conformity-score classes

Empirical evaluation

A Unified Comparative Study with Generalized Conformity Scores for Multi-Output Conformal Regression Victor Dheur 1 Matteo Fontana 2 Yorick Estievenart 1 Naomi Desobry 1 Souhaib Ben Taieb 1 3 Abstract Conformal prediction provides a powerful framework for constructing distribution-free prediction regions with finite-sample coverage guarantees. While extensively studied in univariate settings, its extension to multi-output problems presents additional challenges, including complex output dependencies and high computational costs, and remains relatively underexplored. In this work, we present a unified comparative study of nine conformal methods with different multivariate base models for constructing multivariate prediction regions within the same framework. This study highlights their key properties while also exploring the connections between them. Additionally, we introduce two novel classes of conformity scores for multi-output regression that generalize their univariate counterparts. These scores ensure asymptotic conditional coverage while maintaining exact finite-sample marginal coverage. One class is compatible with any generative model, offering broad applicability, while the other is computationally efficient, leveraging the properties of invertible generative models. Finally, we conduct a comprehensive empirical evaluation across 13 tabular datasets, comparing all the multi-output conformal methods explored in this work. To ensure a fair and consistent comparison, all methods are implemented within a unified code base 1. 1Department of Computer Science, University of Mons, Mons, Belgium 2Department of Computer Science, Royal Holloway, University of London, Egham, United Kingdom 3Department of Statistics and Data Science, Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates. Correspondence to: Victor Dheur <[victor.dheur@umons.ac.be](victor.dheur@umons.ac.be)> . Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025 . Copyright 2025 by the author(s) . 1 [https://github. com/Vekteur/](https://github. com/Vekteur/)[ ](https://github. com/Vekteur/)multi-output-conformal-regression Y2 3 2 1 0 ~~ ~~ 1 ~~ ~~ 2 ~~ ~~ 3 PCP C-PCP DR-CP C-HDR M-CP L-CP Figure 1: Examples of bivariate prediction regions with an 80% coverage level for a toy example. 1. Introduction Quantifying uncertainty in model predictions is crucial in many real-world applications, often involving prediction problems with multiple output variables and complex statistical dependencies. For example, in medical diagnostics, the progression of a disease can be studied by analysing multiple health indicators that exhibit nonlinear dependencies, such as blood pressure and cholesterol levels of a patient (Rajkomar et al., 2018) . Although modern probabilistic AI models can model complex relationships between variables, they may produce unreliable or overly confident predictions (Nalisnick et al., 2018) . Conformal prediction (CP) offers a robust framework for improving model reliability by generating distribution-free prediction regions with a finite-sample coverage guarantee (Vovk et al., 1999) . Although substantial research has focused on univariate prediction problems (Romano et al., 2019; Sesia and Romano, 2021; Rossellini et al., 2024), multivariate settings have received less attention. Among existing work, Zhou et al. (2024) achieves marginal coverage by combining univariate prediction regions, but fails to capture dependencies between variables. Other methods, such as density-based approaches (Izbicki et al., 2022) or sample-based techniques (Wang et al., 2023b; Plassier et al., 2025), suffer from high computational costs. An alterna- tive method (Sadinle et al., 2019) optimises the size of the region, but does not achieve asymptotic conditional coverage. For a toy bivariate example, Figure 1 illustrates the diversity of p

FAQs

What problem does the document address in conformal prediction?

It addresses extending conformal prediction to multi-output regression, where dependencies across outputs and computational costs make existing approaches harder to apply reliably.

What are the main contributions of the study?

The work compares nine multi-output conformal methods within a unified framework and proposes two new classes of conformity scores for multi-output regression.

How do the proposed conformity scores affect coverage guarantees?

They ensure asymptotic conditional coverage while maintaining exact finite-sample marginal coverage, with one class applicable to any generative model and another computationally efficient for invertible generative models.

Aug 5, 2026

Upload by Aurelia