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

arXiv: https://arxiv.org/abs/2512.01556 · HTML: https://mlsanity.com/paper/2512.01556/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 54 references that are in the map.
- First posted on arXiv 1 Dec 2025.

## Builds on (most cited first, 30 of 41)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [Improved Baselines with Visual Instruction Tuning](https://mlsanity.com/paper/2310.03744/) (2023)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [InternVL: Scaling up Vision Foundation Models and Aligning for Generic Visual-Linguistic Tasks](https://mlsanity.com/paper/2312.14238/) (2023)
- [DeBERTa: Decoding-enhanced BERT with Disentangled Attention](https://mlsanity.com/paper/2006.03654/) (2020)
- [Learn to Explain: Multimodal Reasoning via Thought Chains for Science Question Answering](https://mlsanity.com/paper/2209.09513/) (2022)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [CommonsenseQA: A Question Answering Challenge Targeting Commonsense Knowledge](https://mlsanity.com/paper/1811.00937/) (2018)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models](https://mlsanity.com/paper/2305.19187/) (2023)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [OpenChat: Advancing Open-source Language Models with Mixed-Quality Data](https://mlsanity.com/paper/2309.11235/) (2023)
- [Conformal Language Modeling](https://mlsanity.com/paper/2306.10193/) (2023)
- [Shifting Attention to Relevance: Towards the Predictive Uncertainty Quantification of Free-Form Large Language Models](https://mlsanity.com/paper/2307.01379/) (2023)
- [A Survey on Uncertainty Quantification of Large Language Models: Taxonomy, Open Research Challenges, and Future Directions](https://mlsanity.com/paper/2412.05563/) (2024)
- [Trust or Escalate: LLM Judges with Provable Guarantees for Human Agreement](https://mlsanity.com/paper/2407.18370/) (2024)
- [Uncertainty Quantification and Confidence Calibration in Large Language Models: A Survey](https://mlsanity.com/paper/2503.15850/) (2025)
- [VL-Uncertainty: Detecting Hallucination in Large Vision-Language Model via Uncertainty Estimation](https://mlsanity.com/paper/2411.11919/) (2024)
- [Selection by Prediction with Conformal p-values](https://mlsanity.com/paper/2210.01408/) (2022)
- [ConU: Conformal Uncertainty in Large Language Models with Correctness Coverage Guarantees](https://mlsanity.com/paper/2407.00499/) (2024)
- [PAC Confidence Sets for Deep Neural Networks via Calibrated Prediction](https://mlsanity.com/paper/2001.00106/) (2019)
- [MM-Vet v2: A Challenging Benchmark to Evaluate Large Multimodal Models for Integrated Capabilities](https://mlsanity.com/paper/2408.00765/) (2024)
- [Conformal Alignment: Knowing When to Trust Foundation Models with Guarantees](https://mlsanity.com/paper/2405.10301/) (2024)

## Built on by (most cited first, 7 of 7)

- [Proactive Routing to Interpretable Surrogates with Distribution-Free Safety Guarantees](https://mlsanity.com/paper/2603.14623/) (2026)
- [MiRD: Reliable Set-Valued Prediction for Open-Ended Question Answering via Miscoverage Risk Decomposition](https://mlsanity.com/paper/2605.27091/) (2026)
- [Benchmarking candidate coverage and rejection policy transfer in typed decision models](https://mlsanity.com/paper/2610.03387/) (2026)
- [Witness Evidence Portfolios: Single-Prefill Risk Detection for Closed Multimodal Answers](https://mlsanity.com/paper/2607.27667/) (2026)
- [Certify or Refuse: A Cross-Model Map for Selective Risk Control with Coverage Floors under Covariate Shift](https://mlsanity.com/paper/2608.10893/) (2026)
- [Asymptotic Risk Calibration for Selective Question Answering](https://mlsanity.com/paper/2608.12008/) (2026)
- [SafeRestore: Detector-Relative Risk Certificates for Selective Industrial Image Restoration](https://mlsanity.com/paper/2609.03475/) (2026)

## Abstract (arXiv)

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 selection-conditioned 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.