Scaling-Score Conformal Prediction for Multi-Target Regression
TL;DR
The scaling-score conformal method is proposed, which is model-agnostic (requires only component-wise absolute residuals), uses a single calibration set, and yields four nested output types: an outer rectangle with valid joint coverage, the exact set R $\alpha$ , a staircase over approximation of R $\alpha$ , and an inner rectangle (SCI).
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 (SC 2 ) 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 realworld datasets confirm valid joint coverage; SC 2 with $\gamma$ = 1-$\alpha$ consistently achieves competitive volume relative to baselines, with the advantage growing with output dimension d.
