# Entropy-Regularized Process Reward Model

arXiv: https://arxiv.org/abs/2412.11006 · HTML: https://mlsanity.com/paper/2412.11006/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 32 of its 35 references that are in the map.
- First posted on arXiv 15 Dec 2024.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 30 of 32)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [Least-to-Most Prompting Enables Complex Reasoning in Large Language Models](https://mlsanity.com/paper/2205.10625/) (2022)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment](https://mlsanity.com/paper/2304.06767/) (2023)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [Iterative Preference Learning from Human Feedback: Bridging Theory and Practice for RLHF under KL-Constraint](https://mlsanity.com/paper/2312.11456/) (2023)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [Step-DPO: Step-wise Preference Optimization for Long-chain Reasoning of LLMs](https://mlsanity.com/paper/2406.18629/) (2024)
- [RLHF Workflow: From Reward Modeling to Online RLHF](https://mlsanity.com/paper/2405.07863/) (2024)
- [Iterative Reasoning Preference Optimization](https://mlsanity.com/paper/2404.19733/) (2024)
- [Monte Carlo Tree Search Boosts Reasoning via Iterative Preference Learning](https://mlsanity.com/paper/2405.00451/) (2024)
- [OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data](https://mlsanity.com/paper/2410.01560/) (2024)
- [Advancing LLM Reasoning Generalists with Preference Trees](https://mlsanity.com/paper/2404.02078/) (2024)
- [AlphaMath Almost Zero: Process Supervision without Process](https://mlsanity.com/paper/2405.03553/) (2024)
- [Accessing GPT-4 level Mathematical Olympiad Solutions via Monte Carlo Tree Self-refine with LLaMa-3 8B](https://mlsanity.com/paper/2406.07394/) (2024)
- [OpenMathInstruct-1: A 1.8 Million Math Instruction Tuning Dataset](https://mlsanity.com/paper/2402.10176/) (2024)
- [DPO Meets PPO: Reinforced Token Optimization for RLHF](https://mlsanity.com/paper/2404.18922/) (2024)
- [Some things are more CRINGE than others: Iterative Preference Optimization with the Pairwise Cringe Loss](https://mlsanity.com/paper/2312.16682/) (2023)
- [DART-Math: Difficulty-Aware Rejection Tuning for Mathematical Problem-Solving](https://mlsanity.com/paper/2407.13690/) (2024)
- [Solving Math Word Problems via Cooperative Reasoning induced Language Models](https://mlsanity.com/paper/2210.16257/) (2022)
- [Step-level Value Preference Optimization for Mathematical Reasoning](https://mlsanity.com/paper/2406.10858/) (2024)
- [Building Math Agents with Multi-Turn Iterative Preference Learning](https://mlsanity.com/paper/2409.02392/) (2024)

## Built on by (most cited first, 30 of 38)

- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [Can 1B LLM Surpass 405B LLM? Rethinking Compute-Optimal Test-Time Scaling](https://mlsanity.com/paper/2502.06703/) (2025)
- [GenPRM: Scaling Test-Time Compute of Process Reward Models via Generative Reasoning](https://mlsanity.com/paper/2504.00891/) (2025)
- [Offline Reinforcement Learning for LLM Multi-Step Reasoning](https://mlsanity.com/paper/2412.16145/) (2024)
- [On Entropy Control in LLM-RL Algorithms](https://mlsanity.com/paper/2509.03493/) (2025)
- [A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models](https://mlsanity.com/paper/2510.08049/) (2025)
- [StepWiser: Stepwise Generative Judges for Wiser Reasoning](https://mlsanity.com/paper/2508.19229/) (2025)
- [Med-PRM: Medical Reasoning Models with Stepwise, Guideline-verified Process Rewards](https://mlsanity.com/paper/2506.11474/) (2025)
- [ARES: Multimodal Adaptive Reasoning via Difficulty-Aware Token-Level Entropy Shaping](https://mlsanity.com/paper/2510.08457/) (2025)
- [d-TreeRPO: Towards More Reliable Policy Optimization for Diffusion Language Models](https://mlsanity.com/paper/2512.09675/) (2025)
- [Qsharp: Provably Optimal Distributional RL for LLM Post-Training](https://mlsanity.com/paper/2502.20548/) (2025)
- [Multi-Turn On-Policy Distillation with Prefix Replay](https://mlsanity.com/paper/2607.04763/) (2026)
- [Value-Guided Search for Efficient Chain-of-Thought Reasoning](https://mlsanity.com/paper/2505.17373/) (2025)
- [EntroPIC: Towards Stable Long-Term Training of LLMs via Entropy Stabilization with Proportional-Integral Control](https://mlsanity.com/paper/2511.15248/) (2025)
- [A Survey on Feedback-based Multi-step Reasoning for Large Language Models on Mathematics](https://mlsanity.com/paper/2502.14333/) (2025)
- [VulnLLM-R: Specialized Reasoning LLM with Agent Scaffold for Vulnerability Detection](https://mlsanity.com/paper/2512.07533/) (2025)
- [More Bang for the Buck: Process Reward Modeling with Entropy-Driven Uncertainty](https://mlsanity.com/paper/2503.22233/) (2025)
- [Hybrid Reward Normalization for Process-supervised Non-verifiable Agentic Tasks](https://mlsanity.com/paper/2509.25598/) (2025)
- [Advancing Reasoning in Diffusion Language Models with Denoising Process Rewards](https://mlsanity.com/paper/2510.01544/) (2025)
- [Enhancing Large Language Model Reasoning with Reward Models: An Analytical Survey](https://mlsanity.com/paper/2510.01925/) (2025)
- [Fractured Chain-of-Thought Reasoning](https://mlsanity.com/paper/2505.12992/) (2025)
- [Simultaneous Multi-objective Alignment Across Verifiable and Non-verifiable Rewards](https://mlsanity.com/paper/2510.01167/) (2025)
- [From Mathematical Reasoning to Code: Generalization of Process Reward Models in Test-Time Scaling](https://mlsanity.com/paper/2506.00027/) (2025)
- [PaLMR: Towards Faithful Visual Reasoning via Multimodal Process Alignment](https://mlsanity.com/paper/2603.06652/) (2026)
- [PAIR: Prefix-Aware Internal Reward Model for Multi-Turn Agent Optimization](https://mlsanity.com/paper/2605.17877/) (2026)
- [VL-GenRM: Enhancing Vision-Language Verification via Vision Experts and Iterative Training](https://mlsanity.com/paper/2506.13888/) (2025)
- [Grounding the Score: Explicit Visual Premise Verification for Reliable Vision-Language Process Reward Models](https://mlsanity.com/paper/2603.16253/) (2026)
- [SDR: Set-Distance Rewards for Radiology Report Generation](https://mlsanity.com/paper/2606.00440/) (2026)
- [What Makes Reasoning Invalid: Echo Reflection Mitigation for Large Language Models](https://mlsanity.com/paper/2511.06380/) (2025)
- [PROMISE: Process Reward Models Unlock Test-Time Scaling Laws in Generative Recommendations](https://mlsanity.com/paper/2601.04674/) (2026)

## Abstract (arXiv)

Large language models (LLMs) have shown promise in performing complex multi-step reasoning, yet they continue to struggle with mathematical reasoning, often making systematic errors. A promising solution is reinforcement learning (RL) guided by reward models, particularly those focusing on process rewards, which score each intermediate step rather than solely evaluating the final outcome. This approach is more effective at guiding policy models towards correct reasoning trajectories. In this work, we propose an entropy-regularized process reward model (ER-PRM) that integrates KL-regularized Markov Decision Processes (MDP) to balance policy optimization with the need to prevent the policy from shifting too far from its initial distribution. We derive a novel reward construction method based on the theoretical results. Our theoretical analysis shows that we could derive the optimal reward model from the initial policy sampling. Our empirical experiments on the MATH and GSM8K benchmarks demonstrate that ER-PRM consistently outperforms existing process reward models, achieving 1% improvement on GSM8K and 2-3% improvement on MATH under best-of-N evaluation, and more than 1% improvement under RLHF. These results highlight the efficacy of entropy-regularization in enhancing LLMs' reasoning capabilities.