# Save the Good Prefix: Precise Error Penalization via Process-Supervised RL to Enhance LLM Reasoning

arXiv: https://arxiv.org/abs/2601.18984 · HTML: https://mlsanity.com/paper/2601.18984/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 48 of its 51 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Placed in

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

## Builds on (most cited first, 30 of 48)

- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [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)
- [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)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [MathArena: Evaluating LLMs on Uncontaminated Math Competitions](https://mlsanity.com/paper/2505.23281/) (2025)
- [ProcessBench: Identifying Process Errors in Mathematical Reasoning](https://mlsanity.com/paper/2412.06559/) (2024)
- [ReFT: Reasoning with Reinforced Fine-Tuning](https://mlsanity.com/paper/2401.08967/) (2024)
- [R-Zero: Self-Evolving Reasoning LLM from Zero Data](https://mlsanity.com/paper/2508.05004/) (2025)
- [Light-R1: Curriculum SFT, DPO and RL for Long COT from Scratch and Beyond](https://mlsanity.com/paper/2503.10460/) (2025)
- [Free Process Rewards without Process Labels](https://mlsanity.com/paper/2412.01981/) (2024)
- [The Surprising Effectiveness of Negative Reinforcement in LLM Reasoning](https://mlsanity.com/paper/2506.01347/) (2025)
- [Stream of Search (SoS): Learning to Search in Language](https://mlsanity.com/paper/2404.03683/) (2024)
- [Rule Based Rewards for Language Model Safety](https://mlsanity.com/paper/2411.01111/) (2024)
- [PRMBench: A Fine-grained and Challenging Benchmark for Process-Level Reward Models](https://mlsanity.com/paper/2501.03124/) (2025)
- [Self-Rewarding Vision-Language Model via Reasoning Decomposition](https://mlsanity.com/paper/2508.19652/) (2025)
- [Efficient Reinforcement Finetuning via Adaptive Curriculum Learning](https://mlsanity.com/paper/2504.05520/) (2025)
- [On Designing Effective RL Reward at Training Time for LLM Reasoning](https://mlsanity.com/paper/2410.15115/) (2024)
- [Outcome-based Exploration for LLM Reasoning](https://mlsanity.com/paper/2509.06941/) (2025)
- [Process Reward Model with Q-Value Rankings](https://mlsanity.com/paper/2410.11287/) (2024)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [One Token to Fool LLM-as-a-Judge](https://mlsanity.com/paper/2507.08794/) (2025)
- [UFT: Unifying Supervised and Reinforcement Fine-Tuning](https://mlsanity.com/paper/2505.16984/) (2025)
- [Stop Summation: Min-Form Credit Assignment Is All Process Reward Model Needs for Reasoning](https://mlsanity.com/paper/2504.15275/) (2025)
- [VisPlay: Self-Evolving Vision-Language Models from Images](https://mlsanity.com/paper/2511.15661/) (2025)

## Built on by (most cited first, 11 of 11)

- [Too Correct to Learn: Reinforcement Learning on Saturated Reasoning Data](https://mlsanity.com/paper/2604.18493/) (2026)
- [Reinforcing Multimodal Reasoning Against Visual Degradation](https://mlsanity.com/paper/2605.09262/) (2026)
- [Know When to Stop: Segment-Level Credit Assignment for Reducing Overthinking](https://mlsanity.com/paper/2607.00482/) (2026)
- [HPFA: Hypergraph-Based Paired Failure Attribution for LLM Reasoning](https://mlsanity.com/paper/2608.02026/) (2026)
- [Internalizing Outcome Supervision into Process Supervision: A New Paradigm for Reinforcement Learning for Reasoning](https://mlsanity.com/paper/2605.05226/) (2026)
- [Process Rewards with Learned Reliability](https://mlsanity.com/paper/2605.15529/) (2026)
- [ConSteer-RL: Steering Reasoning Capabilities in Large Language Models via Confidence-Aware Reinforcement Learning](https://mlsanity.com/paper/2606.08088/) (2026)
- [When Do Models Admit They Are Wrong? Failure Disclosure Is Unstable Under Reinforcement Learning](https://mlsanity.com/paper/2609.33220/) (2026)
- [Targeting Pivotal Decisions for Credit Assignment in Agentic Reinforcement Learning](https://mlsanity.com/paper/2609.36178/) (2026)
- [Align Then Reason: A Multimodal Lip-Sync Judge for Dubbing](https://mlsanity.com/paper/2610.00825/) (2026)
- [Tropical Reinforcement Learning](https://mlsanity.com/paper/2610.02478/) (2026)

## Abstract (arXiv)

Reinforcement learning (RL) has emerged as a powerful framework for improving the reasoning capabilities of large language models (LLMs). However, most existing RL approaches rely on sparse outcome rewards, which fail to credit correct intermediate steps in partially successful solutions. Process reward models (PRMs) offer fine-grained step-level supervision, but their scores are often noisy and difficult to evaluate. As a result, recent PRM benchmarks focus on a more objective capability: detecting the first incorrect step in a reasoning path. However, this evaluation target is misaligned with how PRMs are typically used in RL, where their step-wise scores are treated as raw rewards to maximize. To bridge this gap, we propose Verifiable Prefix Policy Optimization (VPPO), which uses PRMs only to localize the first error during RL. Given an incorrect rollout, VPPO partitions the trajectory into a verified correct prefix and an erroneous suffix based on the first error, rewarding the former while applying targeted penalties only after the detected mistake. This design yields stable, interpretable learning signals and improves credit assignment. Across multiple reasoning benchmarks, VPPO consistently outperforms sparse-reward RL and prior PRM-guided baselines on both Pass@1 and Pass@K.