Adaptive Population-Risk Reconstruction for Learning under Selective Labels

Selective labels arise when outcomes are observed only for a nonrandom subset of the population, causing empirical risk minimization (ERM) to optimize prediction for the selected sample rather than the deployment population. We propose Adaptive Population-Risk Reconstruction (APR), a framework that exploits bridge observations: labeled cases that are unexpectedly selected yet resemble regions where outcomes are largely unobserved. APR combines selection surprise with affinity to the unlabeled population to construct continuous Target-relevance scores, and uses these scores to reconstruct separate Core and Target components of deployment-population risk. A global reliability term limits reconstruction when effective bridge information is weak, while an observation-specific gate restricts adaptation to inputs for which the reconstructed predictor is locally relevant. Bridge discovery is completed and frozen before outcome-model training. In progressive bridge-information simulations, APR recovers 94% of the AUC gap between ERM and a full-information oracle. Across six selective-label benchmark settings, APR achieves the highest mean Target-population AUC and the lowest mean log-loss among the compared methods. In a real-covariate semi-synthetic benchmark based on UCI Bank Marketing, APR also achieves the best mean Target- and full-population AUC, log-loss, and Brier score. Additional stress tests show robustness to bridge contamination, identification noise, moderate transportability violations, and moderate misspecification of the Target-population share, while performance degrades as bridge identification or outcome transportability approaches its limiting failure regime.