# Kalman Filter Enhanced GRPO for Reinforcement Learning-Based Language Model Reasoning

arXiv: https://arxiv.org/abs/2505.07527 · HTML: https://mlsanity.com/paper/2505.07527/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 12 May 2025.

## Placed in

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

## Builds on (most cited first, 14 of 14)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Soft Actor-Critic Algorithms and Applications](https://mlsanity.com/paper/1812.05905/) (2018)
- [Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models](https://mlsanity.com/paper/2401.01335/) (2024)
- [Reinforced Self-Training (ReST) for Language Modeling](https://mlsanity.com/paper/2308.08998/) (2023)
- [KTO: Model Alignment as Prospect Theoretic Optimization](https://mlsanity.com/paper/2402.01306/) (2024)

## Built on by (most cited first, 14 of 14)

- [MAPO: Mixed Advantage Policy Optimization](https://mlsanity.com/paper/2509.18849/) (2025)
- [λ-GRPO: Unifying the GRPO Frameworks with Learnable Token Preferences](https://mlsanity.com/paper/2510.06870/) (2025)
- [On-Policy Supervised Fine-Tuning for Efficient Reasoning](https://mlsanity.com/paper/2602.13407/) (2026)
- [Kernelized Advantage Estimation: From Nonparametric Statistics to LLM Reasoning](https://mlsanity.com/paper/2604.28005/) (2026)
- [EQA-RM: A Generative Embodied Reward Model with Test-time Scaling](https://mlsanity.com/paper/2506.10389/) (2025)
- [Online Causal Kalman Filtering for Stable and Effective Policy Optimization](https://mlsanity.com/paper/2602.10609/) (2026)
- [BASIS: Batchwise Advantage Estimation from Single-Rollout Information Sharing for LLM Reasoning](https://mlsanity.com/paper/2605.27293/) (2026)
- [GAPO: Robust Advantage Estimation for Real-World Code LLMs](https://mlsanity.com/paper/2510.21830/) (2025)
- [MHPO: Modulated Hazard-aware Policy Optimization for Stable Reinforcement Learning](https://mlsanity.com/paper/2603.16929/) (2026)
- [K-Score: Kalman Filter as a Principled Alternative to Reward Normalization in Reinforcement Learning](https://mlsanity.com/paper/2604.23056/) (2026)
- [CRPO: Character-centric Group Relative Policy Optimization for Role-aware Reasoning in Role-playing Agents](https://mlsanity.com/paper/2605.25511/) (2026)
- [BV-Blend: Uncertainty-Weighted Historical Baselines for Stable Critic-Free RL with Verifiable Rewards](https://mlsanity.com/paper/2606.28707/) (2026)
- [Revisiting On-policy Adversarial Black-Box Distillation: Calibrating Groupwise Reward Geometry for Effective Advantage Construction](https://mlsanity.com/paper/2609.39757/) (2026)
- [When Does External Guidance Help LLM Reasoning? A Bias-Variance Theory of Guidance-Augmented GRPO](https://mlsanity.com/paper/2610.06861/) (2026)

## Abstract (arXiv)

The advantage function is a central concept in RL that helps reduce variance in policy gradient estimates. For language modeling, Group Relative Policy Optimization (GRPO) was proposed to use the within-group sample mean as a baseline for advantage normalization. This estimator can be sensitive to small group size and rollout-level stochasticity, which may lead to suboptimal advantage estimates in some settings. In this paper, we propose Kalman Filter Enhanced Group Relative Policy Optimization (KRPO), a lightweight variant that treats per-group rewards as noisy observations of a latent prompt-level reward baseline and uses a 1D Kalman filter to estimate both the baseline and its uncertainty. KRPO introduces no additional learned parameters and can be integrated into GRPO with minimal computational overhead. On mathematical reasoning benchmarks, KRPO consistently improves training reward curves and final accuracy over GRPO. These results suggest that adaptive advantage estimation is a promising direction for critic-free reinforcement learning in language model reasoning. The code is available at https://github.com/billhhh/KRPO_LLMs_RL.