# From Accuracy to Robustness: A Study of Rule- and Model-based Verifiers in Mathematical Reasoning

arXiv: https://arxiv.org/abs/2505.22203 · HTML: https://mlsanity.com/paper/2505.22203/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 24 of its 32 references that are in the map.
- First posted on arXiv 28 May 2025.

## Placed in

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

## Builds on (most cited first, 24 of 24)

- [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)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [Monitoring Reasoning Models for Misbehavior and the Risks of Promoting Obfuscation](https://mlsanity.com/paper/2503.11926/) (2025)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [Crossing the Reward Bridge: Expanding RL with Verifiable Rewards Across Diverse Domains](https://mlsanity.com/paper/2503.23829/) (2025)
- [General-Reasoner: Advancing LLM Reasoning Across All Domains](https://mlsanity.com/paper/2505.14652/) (2025)
- [Optimizing Test-Time Compute via Meta Reinforcement Fine-Tuning](https://mlsanity.com/paper/2503.07572/) (2025)
- [xVerify: Efficient Answer Verifier for Reasoning Model Evaluations](https://mlsanity.com/paper/2504.10481/) (2025)

## Built on by (most cited first, 22 of 22)

- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [VerifyBench: A Systematic Benchmark for Evaluating Reasoning Verifiers Across Domains](https://mlsanity.com/paper/2507.09884/) (2025)
- [An Imperfect Verifier is Good Enough: Learning with Noisy Rewards](https://mlsanity.com/paper/2604.07666/) (2026)
- [Generate, Filter, Control, Replay: A Comprehensive Survey of Rollout Strategies for LLM Reinforcement Learning](https://mlsanity.com/paper/2605.02913/) (2026)
- [Delay, Plateau, or Collapse: Evaluating the Impact of Systematic Verification Error on RLVR](https://mlsanity.com/paper/2605.02909/) (2026)
- [Gaming the Judge: Unfaithful Chain-of-Thought Can Undermine Agent Evaluation](https://mlsanity.com/paper/2601.14691/) (2026)
- [Scaling Large Reasoning Models beyond Human Supervision: A Path toward Superintelligence](https://mlsanity.com/paper/2608.31075/) (2026)
- [Aligning Multilingual Reasoning with Verifiable Semantics from a High-Resource Expert Model](https://mlsanity.com/paper/2509.25543/) (2025)
- [Where the Verifier Fails: A Category-Level Audit of Reward Signals in RLVR](https://mlsanity.com/paper/2609.01354/) (2026)
- [OmniVerifier-M1: Multimodal Meta-Verifier with Explicit Structured Recalibration](https://mlsanity.com/paper/2605.28805/) (2026)
- [Judging Is Not Enumerating: Silent Omissions in LLM-Authored Acceptable Sets](https://mlsanity.com/paper/2608.01000/) (2026)
- [Are Verifier Errors Independent Within a GRPO Group? Evidence from Qwen2.5 Rollouts](https://mlsanity.com/paper/2609.06386/) (2026)
- [Rethinking Multiple-Choice Questions for RLVR: Unlocking Potential via Distractor Design](https://mlsanity.com/paper/2603.12826/) (2026)
- [When Errors Can Be Beneficial: A Categorization of Imperfect Rewards for Policy Gradient](https://mlsanity.com/paper/2604.25872/) (2026)
- [A Primer in Post-Training Reasoning Data: What We Know About How It Works](https://mlsanity.com/paper/2606.02113/) (2026)
- [Step-Tagging: Toward controlling the generation of Language Reasoning Models through step monitoring](https://mlsanity.com/paper/2512.14332/) (2025)
- [TRACES: Tagging Reasoning Steps for Adaptive Cost-Efficient Early-Stopping](https://mlsanity.com/paper/2604.21057/) (2026)
- [AutoVerifier: Residual-Guided Non-Parametric Optimization for Reference-Based Answer Verification](https://mlsanity.com/paper/2608.25637/) (2026)
- [Learned Reporting Preferences in RLVR Can Conflict with the Current Request](https://mlsanity.com/paper/2609.36587/) (2026)
- [When a Data Artifact Isn't a Shortcut: Causal Auditing of Synthetic RLVR Corpora](https://mlsanity.com/paper/2610.00202/) (2026)
- [Who Sets the Pace of Judge Hacking? Judge Capability versus the Policy-Judge Gap](https://mlsanity.com/paper/2610.15778/) (2026)
- [Checking the Checkers: A Synthesis of Verifier Noise and Robustness in RLVR](https://mlsanity.com/paper/2610.17824/) (2026)

## Abstract (arXiv)

Trustworthy verifiers are essential for the success of reinforcement learning with verifiable reward (RLVR), which is the core methodology behind various large reasoning models such as DeepSeek-R1. In complex domains like mathematical reasoning, rule-based verifiers have been widely adopted in previous works to train strong reasoning models. However, the reliability of these verifiers and their impact on the RL training process remain poorly understood. In this work, we take mathematical reasoning as a case study and conduct a comprehensive analysis of various verifiers in both static evaluation and RL training scenarios. We show that widely used rule-based verifiers fail to recognize equivalent answers in different formats, leading to substantial false negatives that increasingly hinder RL performance as the policy model gets stronger. Model-based verifiers substantially improve static accuracy but are highly susceptible to reward hacking during RL, where they misclassify certain patterns in responses as correct, particularly after fine-tuning. Our findings underscore the challenges inherent to both rule- and model-based verifiers and provide insights toward developing more accurate and robust reward systems for reinforcement learning.