LEC: Linear Expectation Constraints for Selection-Conditioned Risk Control in Selective Prediction and Routing Systems

Foundation models often generate unreliable answers, while heuristic uncertainty estimators fail to fully distinguish correct from incorrect outputs, causing users to accept erroneous answers without any statistical guarantee. We address this problem through selection-conditioned risk control, aiming to ensure that an accepted prediction has an error probability no larger than a user-specified risk level. To this end, we propose LEC, a principled framework that reframes selective prediction as a decision problem governed by a linear expectation constraint over selection and error indicators. This formulation directly controls the ratio between the expected number of accepted errors and the expected number of accepted predictions, which corresponds to the marginal error probability conditioned on selection. Under exchangeability, we derive a finite-sample sufficient condition that relies only on a held-out calibration set, enabling the computation of a risk-constrained, retention-maximizing threshold. Furthermore, we extend LEC to two-model routing systems: if the primary model's uncertainty exceeds its calibrated threshold, the input is delegated to a subsequent model, while maintaining system-level selectionconditioned error control. Experiments on both closed-ended and open-ended question answering (QA) and vision question answering (VQA) demonstrate that LEC maintains the prescribed risk level in accepted predictions and substantially improves sample retention compared to baselines. Figure 1. Illustration of selective prediction in single-model and two-model routing systems. By calibrating when to accept, escalate, or abstain, LEC provides system-level selection-conditioned error control. Code is available here.