LEC: Linear Expectation Constraints for False-Discovery Control in Selective Prediction and Routing Systems

2025-12-01

AI Summary

Large language models (LLMs) frequently produce unreliable outputs, and existing uncertainty estimation methods lack statistical guarantees for reliably identifying incorrect answers. To address this, we propose a false discovery rate (FDR) control framework that formulates selective prediction as a decision problem with linear expectation constraints, deriving a sufficient condition for finite-sample FDR control via calibration sets. We further design an uncertainty-aware dual-model routing mechanism that intelligently allocates tasks across models while maintaining unified FDR guarantees. Our method performs offline calibration under the exchangeability assumption, balancing theoretical rigor with practical applicability. Experiments on multiple question-answering benchmarks demonstrate significant improvements in both FDR control accuracy and effective sample retention rate. Moreover, under strict FDR control, our approach substantially increases the number of correctly accepted answers—thereby enhancing reliability without sacrificing coverage.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Large language models (LLMs) often generate unreliable answers, while heuristic uncertainty methods fail to fully distinguish correct from incorrect predictions, causing users to accept erroneous answers without statistical guarantees. We address this issue through the lens of false discovery rate (FDR) control, ensuring that among all accepted predictions, the proportion of errors does not exceed a target risk level. To achieve this in a principled way, we propose LEC, which reinterprets selective prediction as a constrained decision problem by enforcing a Linear Expectation Constraint over selection and error indicators. Then, we establish a finite-sample sufficient condition, which relies only on a held-out set of exchangeable calibration samples, to compute an FDR-constrained, coverage-maximizing threshold. Furthermore, we extend LEC to a two-model routing mechanism: given a prompt, if the current model's uncertainty exceeds its calibrated threshold, we delegate it to a stronger model, while maintaining a unified FDR guarantee. Evaluations on closed-ended and open-ended question-answering (QA) datasets show that LEC achieves tighter FDR control and substantially improves sample retention over prior methods. Moreover, the two-model routing mechanism achieves lower risk levels while accepting more correct samples than each individual model.

Problem

• Control false discovery rate in LLM predictions
• Ensure error proportion below target risk level
• Extend FDR control to two-model routing systems

Innovation

• Linear Expectation Constraint for FDR control
• Finite-sample threshold from exchangeable calibration data
• Two-model routing with unified FDR guarantee