# Scaling-Score Conformal Prediction for Multi-Target Regression

arXiv: https://arxiv.org/abs/2609.17091 · HTML: https://mlsanity.com/paper/2609.17091/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 19 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [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)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Distributional Random Forests: Heterogeneity Adjustment and Multivariate Distributional Regression](https://mlsanity.com/paper/2005.14458/) (2020)
- [Nonparametric Multiple-Output Center-Outward Quantile Regression](https://mlsanity.com/paper/2204.11756/) (2022)
- [Conformal Multi-Target Hyperrectangles](https://mlsanity.com/paper/2406.04498/) (2024)
- [Interpretable Multivariate Conformal Prediction with Fast Transductive Standardization](https://mlsanity.com/paper/2512.15383/) (2025)

## Abstract (arXiv)

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 α , a staircase (SC 2 ) over approximation of R α , and an inner rectangle (SCI). A single hyperparameter γ ∈ (0, 1) controls the base-rectangle quantile level independently of α. We prove downward-closedness and a rectangular sandwich bound and derive a closed-form outer rectangle. Experiments on 29 realworld datasets confirm valid joint coverage; SC 2 with γ = 1-α consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.