# Multi-Variable Conformal Prediction: Optimizing Prediction Sets without Data Splitting

arXiv: https://arxiv.org/abs/2605.12341 · HTML: https://mlsanity.com/paper/2605.12341/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 45 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 18 of 18)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Copula-based conformal prediction for Multi-Target Regression](https://mlsanity.com/paper/2101.12002/) (2021)
- [Calibrated Multiple-Output Quantile Regression with Representation Learning](https://mlsanity.com/paper/2110.00816/) (2021)
- [Conformal Uncertainty Sets for Robust Optimization](https://mlsanity.com/paper/2105.14957/) (2021)
- [Conformal prediction for multi-dimensional time series by ellipsoidal sets](https://mlsanity.com/paper/2403.03850/) (2024)
- [Conformal Prediction Regions for Time Series using Linear Complementarity Programming](https://mlsanity.com/paper/2304.01075/) (2023)
- [Minimum Volume Conformal Sets for Multivariate Regression](https://mlsanity.com/paper/2503.19068/) (2025)
- [Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction](https://mlsanity.com/paper/2312.08604/) (2023)
- [On the exact feasibility of convex scenario programs with discarded constraints](https://mlsanity.com/paper/2003.13433/) (2020)
- [Relational Conformal Prediction for Correlated Time Series](https://mlsanity.com/paper/2502.09443/) (2025)
- [Conformal Structured Prediction](https://mlsanity.com/paper/2410.06296/) (2024)
- [Guaranteed prediction sets for functional surrogate models](https://mlsanity.com/paper/2501.18426/) (2025)
- [Volume-Sorted Prediction Set: Efficient Conformal Prediction for Multi-Target Regression](https://mlsanity.com/paper/2503.02205/) (2025)
- [Guarantees for Real Robotic Systems: Unifying Formal Controller Synthesis and Reachset-Conformant Identification](https://mlsanity.com/paper/2103.01626/) (2021)
- [Reachset-Conformant System Identification](https://mlsanity.com/paper/2407.11692/) (2024)
- [Bridging conformal prediction and scenario optimization](https://mlsanity.com/paper/2503.23561/) (2025)
- [Zono-Conformal Prediction: Zonotope-Based Uncertainty Quantification for Regression and Classification Tasks](https://mlsanity.com/paper/2508.11025/) (2025)
- [CONTRA: Conformal Prediction Region via Normalizing Flow Transformation](https://mlsanity.com/paper/2605.08561/) (2026)

## Abstract (arXiv)

Conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is structurally constrained to a scalar score function and a single threshold variable - forcing shapes of prediction sets to be fixed before calibration, typically through data splitting. We introduce multi-variable conformal prediction (MCP), a framework that extends conformal prediction to vector-valued score functions with multiple simultaneous calibration variables. Building on scenario theory as a principled framework for certifying data-driven decisions, MCP unifies prediction set design and calibration into a single optimization problem, eliminating data splitting without sacrificing coverage guarantees. We propose two computationally efficient variants: RemMCP, grounded in constrained optimization with constraint removal, which admits a clean generalization of split conformal prediction; and RelMCP, based on iterative optimization with constraint relaxation, which supports non-convex score functions at the cost of possibly greater conservatism. Through numerical experiments on ellipsoidal and multi-modal prediction sets, we demonstrate that RemMCP and RelMCP consistently meet the target coverage with prediction set sizes smaller than or comparable to those of baselines with data split, while considerably reducing variance across calibration runs - a direct consequence of using all available data for shape optimization and calibration simultaneously.