Multi-Variable Conformal Prediction: Optimizing Prediction Sets using Multiple Calibration Variables
Abstract
Split conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is limited to adjusting a single threshold parameter. Hence, the shape of the prediction sets must be fixed beforehand, typically by fitting it on a reserved part of the calibration data. We introduce multi-variable conformal prediction (MCP), a framework that jointly optimizes and calibrates multiple prediction-set parameters in a single optimization problem, eliminating calibration data split without sacrificing finite-sample coverage guarantees. Building on scenario theory, a principled framework for certifying data-driven decisions, we propose two tractable variants: RemMCP, grounded in constrained optimization with constraint removal, which generalizes split conformal prediction; and RelMCP, based on penalized optimization with constraint relaxation, which also supports non-convex problems, albeit at a higher computational cost. In experiments with ellipsoidal and multi-modal prediction sets, RemMCP and a heuristic, adaptive version of RelMCP consistently meet the target coverage, with lower variance across calibration runs than baselines with calibration data split, and in most settings with smaller or comparable prediction sets.
Related
- When Does Trimming Help Conformal Prediction? A Retained-Law Diagnostic under Calibration Contamination
- When does conformal calibration need censoring weights? Cause-of-failure prediction sets under competing risks
- Sketched Calibration for Conformal Prediction under Covariate Shift
- Particle-Based Conformal Prediction for Contact-Aware Uncertainty Calibration in Stratified Configuration Spaces
- Abstention Spends Its Budget on Local Label Variance More Than on Novelty: Auditing Conformal Calibration and Selective Prediction under Molecular Scaffold Shift
BibTeX
@misc{lutzow2026multivariable,
title = {{Multi-Variable Conformal Prediction: Optimizing Prediction Sets using Multiple Calibration Variables}},
author = {Laura Lützow and Simone Garatti and Marco C. Campi and Lars Lindemann and Matthias Althoff},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.12341},
eprint = {2605.12341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12341},
}
RIS
TY - UNPB
TI - Multi-Variable Conformal Prediction: Optimizing Prediction Sets using Multiple Calibration Variables
AU - Laura Lützow
AU - Simone Garatti
AU - Marco C. Campi
AU - Lars Lindemann
AU - Matthias Althoff
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2605.12341
UR - https://arxiv.org/abs/2605.12341
AB - Split conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is limited to adjusting a single threshold parameter. Hence, the shape of the prediction sets must be fixed beforehand, typically by fitting it on a reserved part of the calibration data. We introduce multi-variable conformal prediction (MCP), a framework that jointly optimizes and calibrates multiple prediction-set parameters in a single optimization problem, eliminating calibration data split without sacrificing finite-sample coverage guarantees. Building on scenario theory, a principled framework for certifying data-driven decisions, we propose two tractable variants: RemMCP, grounded in constrained optimization with constraint removal, which generalizes split conformal prediction; and RelMCP, based on penalized optimization with constraint relaxation, which also supports non-convex problems, albeit at a higher computational cost. In experiments with ellipsoidal and multi-modal prediction sets, RemMCP and a heuristic, adaptive version of RelMCP consistently meet the target coverage, with lower variance across calibration runs than baselines with calibration data split, and in most settings with smaller or comparable prediction sets.
ER -