SelPI trains prediction, interval and abstention together, and width thresholding is shown to reject the right inputs on 18 tabular datasets
Synopsis
High-stakes regression decisions need a point estimate, a risk interval and an option to abstain, yet interval methods and selective prediction each supply only part of that triad. SelPI trains interval, value and two abstention heads jointly under one loss, with a Fréchet penalty tying the two gates' budgets to the joint acceptance rate they deliver. Across 18 tabular benchmarks, interval width predicts large point errors well but is anti-correlated with interval misses, so abstaining on wide intervals discards inputs the interval got right; with the base predictor held fixed, the learned gate beats width thresholding on a proper interval score on all 18 tabular datasets, and the advantage grows with task heteroscedasticity.
Interpretation
SelPI lets one regression model return an interval, an in-interval point estimate and an answer-or-abstain decision in a single forward pass, with abstention decided by two gates aimed at interval misses and value-position error. Prediction-interval methods answer every query, and selective-prediction methods learn abstention only for point predictions, with no risk statement on the accepted set. Across 18 tabular benchmarks, interval width ranks large point errors well but is anti-correlated with interval misses, so a reject-if-wide rule systematically discards inputs whose intervals were correct.
Each gate is trained on the loss whose failures it predicts and carries its own acceptance budget, and an input is answered only if both gates admit it. A single gate must encode two failure modes in one scalar, with the balance set implicitly by loss weights rather than by the decision-maker. A Fréchet penalty fires when the gates' joint acceptance rate falls below the floor their budgets imply, tying the separate budgets to the delivered joint rate; with the base predictor held fixed, the learned gate beats width thresholding on a proper interval score on all 18 tabular datasets and closes a median of the gap to an oracle selector.
Perspective
The demonstrated scope is 18 tabular regression datasets plus IPIV's two image tasks, where bone age has a single 200-image test split and is treated as illustrative while IMDB-WIKI uses five-fold cross-validation as the reliable image benchmark; the accepted set carries no finite-sample coverage guarantee, so an abstention decision should be read as a learned signal of predictive difficulty rather than a certification of safety.
The advantage grows with residual heteroscedasticity and vanishes on near-deterministic tasks, and on IMDB-WIKI the single gate is better at three of five comparable acceptance rates; the shift analysis picks the covariate tail with knowledge of the error, so it cannot claim the advantage grows with shift severity and does not cover the image tasks.