Authors: Zhanli Wu, Fabrizio Leisen, Miguel-Angel Luque-Fernandez, F. Javier Rubio
Abstract
The Super Learner (SL) is a widely used ensemble method that combines point predictions from a library of learners based on their predictive performance. Interval predictions are of considerable practical interest because they allow uncertainty in predictions produced by an individual learner or an ensemble to be quantified. Several methods have been proposed for constructing interval predictions based on the SL, however, these approaches are typically justified using asymptotic arguments or rely on computationally intensive procedures such as the bootstrap. Conformal prediction (CP) is a machine learning framework for constructing prediction intervals with finite-sample and asymptotic coverage guarantees under mild conditions. We propose coupling CP with the SL through a natural construction that mirrors the original SL framework, using individual learner weights and combining learner-specific conformity scores via a weighted majority vote. We characterize the properties of the resulting SL-based prediction intervals for continuous outcomes. We cover settings under exchangeability, potential violations of exchangeability, and data-generating mechanisms exhibiting heteroscedasticity, sparsity, and other forms of distributional heterogeneity. A comprehensive simulation study shows that the conformalized SL achieves valid finite-sample coverage with competitive performance relative to the true data-generating mechanism. A central contribution of this work is an application to predicting creatinine levels using socio-demographic, biometric, and laboratory measurements. This example demonstrates the benefits of an ensemble with carefully selected learners designed to capture key aspects of complex regression functions, including non-linear effects, interactions, sparsity, heteroscedasticity, and robustness to outliers.
A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.
Daniel Bensimon, Sean Xiang Yu, Eric D. Kolaczyk +1
Conformal Prediction (CP) provides robust uncertainty guarantees for predictive models, but is typically applied post hoc, which misaligns model training with the conformal goal of producing efficient (i.e., narrow) intervals. We propose SPACR (Single-Pass Adaptive Conformal Regressor), a novel method for directly training uncertainty-aware regressors within a differentiable loss. SPACR jointly optimizes accuracy, efficiency, and validity without batch-splitting or a predefined confidence level during training. As a result, a single SPACR model yields valid prediction intervals at multiple confidence levels during inference, avoiding the costly retraining required by methods like Directly Optimized Inductive Conformal Regression (DOICR). Experiments on diverse tabular and image datasets show that SPACR consistently gives tighter intervals and better coverage-efficiency trade-offs compared to standard CP and DOICR, while significantly reducing computational costs relative to retraining-dependent baselines.
Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an O(hβ) localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.