Scaling-Score Conformal Prediction for Multi-Target Regression

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.