mlsanityData to 14 Oct 2026

LLM Critics Help Catch Bugs in Mathematics: Towards a Better Mathematical Verifier with Natural Language Feedback

arXiv 2406.14024 ↗First posted 20 Jun 2024cs.CLcited by 41

On the map: LLM reasoning → Mathematical reasoning

Abstract (arXiv)

In recent progress, mathematical verifiers have achieved success in mathematical reasoning tasks by validating the correctness of solutions generated by policy models. However, existing verifiers are trained with binary classification labels, which are not informative enough for the model to accurately assess the solutions. To mitigate the aforementioned insufficiency of binary labels, we introduce step-wise natural language feedback as rationale labels, that is, the correctness of each step and the detailed explanations. In this paper, we propose Math-Minos, a natural language feedback-enhanced verifier by constructing automatically generated training data and a two-stage training paradigm for effective training and efficient inference. Our experiments reveal that a small set of natural language feedback can significantly boost the performance of the verifier in both verification and reinforcement learning. We have released the code and data for further exploration.

Key facts. Cited by 41 papers in the map (data through 14 Oct 2026). Placed in Mathematical reasoning, one of 17 directions in LLM reasoning. Builds on 22 of its 26 references that are in the map. First posted on arXiv 20 Jun 2024.

Builds on

22 of 26 references.

Closest ancestors · 6

Same direction, newest first

  1. 2024InternLM-Math: Open Math Large Language Models Toward Verifiable Reasoning9 Feb
  2. 2024DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models5 Feb
  3. 2023OVM, Outcome-supervised Value Models for Planning in Mathematical Reasoning16 Nov
  4. 2023Llemma: An Open Language Model For Mathematics16 Oct
  5. 2023WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct18 Aug
  6. 2022Solving Math Word Problems via Cooperative Reasoning induced Language Models28 Oct

Other references · 16

  1. 2022Training language models to follow instructions with human feedbackcited by 15,971
  2. 2017Proximal Policy Optimization Algorithmscited by 15,215
  3. 2023LLaMA: Open and Efficient Foundation Language Modelscited by 14,132
  4. 2021Training Verifiers to Solve Math Word Problemscited by 10,109
  5. 2023Direct Preference Optimization: Your Language Model is Secretly a Reward Modelcited by 8,065
  6. 2022Self-Consistency Improves Chain of Thought Reasoning in Language Modelscited by 5,708
  7. 2021Measuring Mathematical Problem Solving With the MATH Datasetcited by 5,371
  8. 2023Let's Verify Step by Stepcited by 3,789
  9. 2023Code Llama: Open Foundation Models for Codecited by 2,139
  10. 2023Large Language Models Cannot Self-Correct Reasoning Yetcited by 1,017
  11. 2023Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotationscited by 928
  12. 2022Complexity-Based Prompting for Multi-Step Reasoningcited by 388
4 more
  1. 2022Making Large Language Models Better Reasoners with Step-Aware Verifiercited by 233
  2. 2023RestGPT: Connecting Large Language Models with Real-World RESTful APIscited by 135
  3. 2023Making Large Language Models Better Reasoners with Alignmentcited by 63
  4. 2023When Do Program-of-Thoughts Work for Reasoning?cited by 19

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 4; 2025: 30; 2026: 7.

41 papers.

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  1. 2025The Lessons of Developing Process Reward Models in Mathematical Reasoningcited by 415
  2. 2025Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Modelscited by 384
  3. 2024HuatuoGPT-o1, Towards Medical Complex Reasoning with LLMscited by 265
  4. 2025A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?cited by 223
  5. 2025RM-R1: Reward Modeling as Reasoningcited by 154
  6. 2025PRMBench: A Fine-grained and Challenging Benchmark for Process-Level Reward Modelscited by 99
  7. 2025Unlocking Multimodal Mathematical Reasoning via Process Reward Modelcited by 65
  8. 2025Evaluating Step-by-step Reasoning Traces: A Surveycited by 63
  9. 2025A Survey on Large Language Models for Mathematical Reasoningcited by 52
  10. 2025GRPO-CARE: Consistency-Aware Reinforcement Learning for Multimodal Reasoningcited by 47
  11. 2024Improving LLM Reasoning through Scaling Inference Computation with Collaborative Verificationcited by 40
  12. 2025Enhancing LLM Reasoning with Iterative DPO: A Comprehensive Empirical Investigationcited by 37
29 more
  1. 2024Critic-CoT: Boosting the reasoning abilities of large language model via Chain-of-thoughts Criticcited by 28
  2. 2025AdaptiveStep: Automatically Dividing Reasoning Step through Model Confidencecited by 24
  3. 2025MathAgent: Leveraging a Mixture-of-Math-Agent Framework for Real-World Multimodal Mathematical Error Detectioncited by 24
  4. 2025Retrieval-Augmented Process Reward Model for Generalizable Mathematical Reasoningcited by 23
  5. 2024Towards a Unified View of Preference Learning for Large Language Models: A Surveycited by 17
  6. 2025DeepCritic: Deliberate Critique with Large Language Modelscited by 17
  7. 2025Dynamic and Generalizable Process Reward Modelingcited by 16
  8. 2025GTR: Guided Thought Reinforcement Prevents Thought Collapse in RL-based VLM Agent Trainingcited by 14
  9. 2025Sailing by the Stars: A Survey on Reward Models and Learning Strategies for Learning from Rewardscited by 14
  10. 2025A Survey on Feedback-based Multi-step Reasoning for Large Language Models on Mathematicscited by 11
  11. 2025FreePRM: Training Process Reward Models Without Ground Truth Process Labelscited by 11
  12. 2025LaSeR: Reinforcement Learning with Last-Token Self-Rewardingcited by 9
  13. 2025Graph of Verification: Structured Verification of LLM Reasoning with Directed Acyclic Graphscited by 8
  14. 2025Enhancing Large Language Model Reasoning with Reward Models: An Analytical Surveycited by 8
  15. 2026No More Stale Feedback: Co-Evolving Critics for Open-World Agent Learningcited by 8
  16. 2026Reward Modeling for Reinforcement Learning-Based LLM Reasoning: Design, Challenges, and Evaluationcited by 7
  17. 2025Demystifying Multilingual Chain-of-Thought in Process Reward Modelingcited by 6
  18. 2025MathDebugger: Detecting and Diagnosing Errors in Synthetic Mathematical Datacited by 6
  19. 2025SCAN: Self-Denoising Monte Carlo Annotation for Robust Process Reward Learningcited by 6
  20. 2025Generative Adversarial Reasoner: Enhancing LLM Reasoning with Adversarial Reinforcement Learningcited by 6
  21. 2025LogicReward: Incentivizing LLM Reasoning via Step-Wise Logical Supervisioncited by 6
  22. 2026Steer, Don't Solve: Training Small Critic Models for Large Code Agentscited by 5
  23. 2026Pseudo-Formalization for Automatic Proof Verificationcited by 4
  24. 2025GTR-Turbo: Merged Checkpoint is Secretly a Free Teacher for Agentic VLM Trainingcited by 3
  25. 2025MathEDU: Feedback Generation on Problem-Solving Processes for Mathematical Learning Supportcited by 2
  26. 2025RPM-MCTS: Knowledge-Retrieval as Process Reward Model with Monte Carlo Tree Search for Code Generationcited by 1
  27. 2026When Tokenization is Secretly Output Supervisioncited by 0
  28. 2026Decision Shifts, Lost Label Functionality, and an Inconclusive Grounding Audit in Correctness-Gated Multi-Teacher Distillationcited by 0
  29. 2026One-Sided Verification Under Limited Expert Reviewcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Step-KTO: Optimizing Mathematical Reasoning through Stepwise Binary Feedback
  2. 2025DeepTheorem: Advancing LLM Reasoning for Theorem Proving Through Natural Language and Reinforcement Learning
  3. 2025DeepSeekMath-V2: Towards Self-Verifiable Mathematical Reasoning
  4. 2025Scaling Generative Verifiers For Natural Language Mathematical Proof Verification And Selection
  5. 2025VerifyBench: A Systematic Benchmark for Evaluating Reasoning Verifiers Across Domains
  6. 2025Teaching LLMs for Step-Level Automatic Math Correction via Reinforcement Learning
  7. 2025No Free Lunch: Rethinking Internal Feedback for LLM Reasoning
  8. 2025From Accuracy to Robustness: A Study of Rule- and Model-based Verifiers in Mathematical Reasoning