# Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty

arXiv: https://arxiv.org/abs/2507.16806 · HTML: https://mlsanity.com/paper/2507.16806/

## Key facts

- Cited by 117 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 39 of its 56 references that are in the map.
- First posted on arXiv 22 Jul 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 39)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [HotpotQA: A Dataset for Diverse, Explainable Multi-hop Question Answering](https://mlsanity.com/paper/1809.09600/) (2018)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [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)
- [Can LLMs Express Their Uncertainty? An Empirical Evaluation of Confidence Elicitation in LLMs](https://mlsanity.com/paper/2306.13063/) (2023)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Teaching Models to Express Their Uncertainty in Words](https://mlsanity.com/paper/2205.14334/) (2022)
- [The Internal State of an LLM Knows When It's Lying](https://mlsanity.com/paper/2304.13734/) (2023)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Reasoning Models Don't Always Say What They Think](https://mlsanity.com/paper/2505.05410/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Measuring short-form factuality in large language models](https://mlsanity.com/paper/2411.04368/) (2024)
- [2 OLMo 2 Furious](https://mlsanity.com/paper/2501.00656/) (2024)
- [LLMs Know More Than They Show: On the Intrinsic Representation of LLM Hallucinations](https://mlsanity.com/paper/2410.02707/) (2024)
- [Reducing conversational agents' overconfidence through linguistic calibration](https://mlsanity.com/paper/2012.14983/) (2020)
- [Scalable Best-of-N Selection for Large Language Models via Self-Certainty](https://mlsanity.com/paper/2502.18581/) (2025)
- [Fact-Checking the Output of Large Language Models via Token-Level Uncertainty Quantification](https://mlsanity.com/paper/2403.04696/) (2024)
- [AbstentionBench: Reasoning LLMs Fail on Unanswerable Questions](https://mlsanity.com/paper/2506.09038/) (2025)
- [Language Model Cascades: Token-level uncertainty and beyond](https://mlsanity.com/paper/2404.10136/) (2024)
- [Confidence Improves Self-Consistency in LLMs](https://mlsanity.com/paper/2502.06233/) (2025)
- [Taming Overconfidence in LLMs: Reward Calibration in RLHF](https://mlsanity.com/paper/2410.09724/) (2024)
- [Big-Math: A Large-Scale, High-Quality Math Dataset for Reinforcement Learning in Language Models](https://mlsanity.com/paper/2502.17387/) (2025)
- [SaySelf: Teaching LLMs to Express Confidence with Self-Reflective Rationales](https://mlsanity.com/paper/2405.20974/) (2024)
- [On the Weaknesses of Reinforcement Learning for Neural Machine Translation](https://mlsanity.com/paper/1907.01752/) (2019)
- [Reasoning Models Better Express Their Confidence](https://mlsanity.com/paper/2505.14489/) (2025)

## Built on by (most cited first, 30 of 117)

- [Why Language Models Hallucinate](https://mlsanity.com/paper/2509.04664/) (2025)
- [Time Is a Feature: Exploiting Temporal Dynamics in Diffusion Language Models](https://mlsanity.com/paper/2508.09138/) (2025)
- [TruthRL: Incentivizing Truthful LLMs via Reinforcement Learning](https://mlsanity.com/paper/2509.25760/) (2025)
- [BrokenMath: A Benchmark for Sycophancy in Theorem Proving with LLMs](https://mlsanity.com/paper/2510.04721/) (2025)
- [Scaling Open-Ended Reasoning to Predict the Future](https://mlsanity.com/paper/2512.25070/) (2025)
- [Trace Length is a Simple Uncertainty Signal in Reasoning Models](https://mlsanity.com/paper/2510.10409/) (2025)
- [Mitigating LLM Hallucination via Behaviorally Calibrated Reinforcement Learning](https://mlsanity.com/paper/2512.19920/) (2025)
- [The Confidence Dichotomy: Analyzing and Mitigating Miscalibration in Tool-Use Agents](https://mlsanity.com/paper/2601.07264/) (2026)
- [Decoupling Reasoning and Confidence: Resurrecting Calibration in Reinforcement Learning from Verifiable Rewards](https://mlsanity.com/paper/2603.09117/) (2026)
- [Process-Supervised Reinforcement Learning for Interactive Multimodal Tool-Use Agents](https://mlsanity.com/paper/2509.14480/) (2025)
- [Calibrating LLM Judges: Linear Probes for Fast and Reliable Uncertainty Estimation](https://mlsanity.com/paper/2512.22245/) (2025)
- [Agentic Uncertainty Reveals Agentic Overconfidence](https://mlsanity.com/paper/2602.06948/) (2026)
- [Understanding Reasoning in LLMs through Strategic Information Allocation under Uncertainty](https://mlsanity.com/paper/2603.15500/) (2026)
- [The Illusion of Certainty: Decoupling Capability and Calibration in On-Policy Distillation](https://mlsanity.com/paper/2604.16830/) (2026)
- [C2GSPG: Confidence-calibrated Group Sequence Policy Gradient towards Self-aware Reasoning](https://mlsanity.com/paper/2509.23129/) (2025)
- [Are LLM Decisions Faithful to Verbal Confidence?](https://mlsanity.com/paper/2601.07767/) (2026)
- [VL-Calibration: Decoupled Confidence Calibration for Large Vision-Language Models Reasoning](https://mlsanity.com/paper/2604.09529/) (2026)
- [Just Ask Jev: Reinforcement Learning for Calibrated Decisions as a Zero-Shot Detector of AI Alignment Failures](https://mlsanity.com/paper/2609.29429/) (2026)
- [Knowledge Graphs are Implicit Reward Models: Path-Derived Signals Enable Compositional Reasoning](https://mlsanity.com/paper/2601.15160/) (2026)
- [On Calibration of Large Language Models: From Response To Capability](https://mlsanity.com/paper/2602.13540/) (2026)
- [FutureSim: Replaying World Events to Evaluate Adaptive Agents](https://mlsanity.com/paper/2605.15188/) (2026)
- [Zero-Overhead Introspection for Adaptive Test-Time Compute](https://mlsanity.com/paper/2512.01457/) (2025)
- [NOVA: NOise-aware Verbal Confidence CAlibration for Robust Large Language Models in RAG Systems](https://mlsanity.com/paper/2601.11004/) (2026)
- [Reaching Beyond the Mode: RL for Distributional Reasoning in Language Models](https://mlsanity.com/paper/2603.24844/) (2026)
- [Generalized Correctness Models: Learning Calibrated and Model-Agnostic Correctness Predictors from Historical Patterns](https://mlsanity.com/paper/2509.24988/) (2025)
- [Honesty over Accuracy: Trustworthy Language Models through Reinforced Hesitation](https://mlsanity.com/paper/2511.11500/) (2025)
- [Calibration-Aware Policy Optimization for Reasoning LLMs](https://mlsanity.com/paper/2604.12632/) (2026)
- [Scaling Large Reasoning Models beyond Human Supervision: A Path toward Superintelligence](https://mlsanity.com/paper/2608.31075/) (2026)
- [Self-Filtered Distillation with LLMs-generated Trust Indicators for Reliable Patent Classification](https://mlsanity.com/paper/2510.05431/) (2025)
- [CritiCal: Can Critique Help LLM Uncertainty or Confidence Calibration?](https://mlsanity.com/paper/2510.24505/) (2025)

## Abstract (arXiv)

When language models (LMs) are trained via reinforcement learning (RL) to generate natural language "reasoning chains", their performance improves on a variety of difficult question answering tasks. Today, almost all successful applications of RL for reasoning use binary reward functions that evaluate the correctness of LM outputs. Because such reward functions do not penalize guessing or low-confidence outputs, they often have the unintended side-effect of degrading calibration and increasing the rate at which LMs generate incorrect responses (or "hallucinate") in other problem domains. This paper describes RLCR (Reinforcement Learning with Calibration Rewards), an approach to training reasoning models that jointly improves accuracy and calibrated confidence estimation. During RLCR, LMs generate both predictions and numerical confidence estimates after reasoning. They are trained to optimize a reward function that augments a binary correctness score with a Brier score -- a scoring rule for confidence estimates that incentivizes calibrated prediction. We first prove that this reward function (or any reward function that uses a bounded, proper scoring rule) yields models whose predictions are both accurate and well-calibrated. We next show that across diverse datasets, RLCR substantially improves calibration with no loss in accuracy, on both in-domain and out-of-domain evaluations -- outperforming both ordinary RL training and classifiers trained to assign post-hoc confidence scores. While ordinary RL hurts calibration, RLCR improves it. Finally, we demonstrate that verbalized confidence can be leveraged at test time to improve accuracy and calibration via confidence-weighted scaling methods. Our results show that explicitly optimizing for calibration can produce more generally reliable reasoning models. Code, models, and further info is available at https://rl-calibration.github.io/.