# Verifier-Free RL for LLMs via Intrinsic Gradient-Norm Reward

arXiv: https://arxiv.org/abs/2605.09920 · HTML: https://mlsanity.com/paper/2605.09920/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 12 of 12)

- [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)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [CRUXEval: A Benchmark for Code Reasoning, Understanding and Execution](https://mlsanity.com/paper/2401.03065/) (2024)
- [The Unreasonable Effectiveness of Entropy Minimization in LLM Reasoning](https://mlsanity.com/paper/2505.15134/) (2025)
- [Right Question is Already Half the Answer: Fully Unsupervised LLM Reasoning Incentivization](https://mlsanity.com/paper/2504.05812/) (2025)
- [MiMo: Unlocking the Reasoning Potential of Language Model -- From Pretraining to Posttraining](https://mlsanity.com/paper/2505.07608/) (2025)
- [Can Large Reasoning Models Self-Train?](https://mlsanity.com/paper/2505.21444/) (2025)
- [Step-KTO: Optimizing Mathematical Reasoning through Stepwise Binary Feedback](https://mlsanity.com/paper/2501.10799/) (2025)

## Built on by (most cited first, 3 of 3)

- [Multilingual Verifier Bias in RLVR: Benchmark, Rollout Diagnosis, and the Cross-Lingual Selection Bottleneck](https://mlsanity.com/paper/2608.20362/) (2026)
- [Unsupervised Post-Training of Foundation Models: A Survey](https://mlsanity.com/paper/2608.24982/) (2026)
- [Test-time Calibration Learning for Large Language Model Reasoning](https://mlsanity.com/paper/2610.02695/) (2026)

## Abstract (arXiv)

While Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a promising post-training paradigm for Large Language Models (LLMs), its dependency on the gold label or domain-specific verifiers limits its scalability to new tasks and domains. In this work, we propose Verifier-free Intrinsic Gradient-Norm Reward (VIGOR), a simple reward that uses only the policy model itself. Given a prompt, VIGOR samples a group of completions and assigns higher within-group rewards to outputs that induce smaller ℓ2 norms of the teacher-forced negative log-likelihood gradients under the current parameters. Intuitively, lower gradient norms suggest the completion aligns better with the current policy, serving as an intrinsic preference signal for policy optimization. To make this intrinsic signal practical for RL, we correct the systematic length bias of averaged token-level gradients with a √T scaling, and apply group-wise rank shaping to stabilize reward scales across prompts. Across mathematical reasoning benchmarks, VIGOR outperforms the state-of-the-art Reinforcement Learning from Internal Feedback (RLIF) baseline, and it also exhibits cross-domain transfer to code benchmarks when trained only on math data. For instance, on Qwen2.5-7B-Base post-trained on MATH, VIGOR improves the average math accuracy by +3.31% and the average code accuracy by +1.91% over this baseline, while exhibiting more stable training dynamics. The code is available at https://github.com/ZJUSCL/VIGOR.