# One Ring to Rule Them All: Unifying Group-Based RL via Dynamic Power-Mean Geometry

arXiv: https://arxiv.org/abs/2601.22521 · HTML: https://mlsanity.com/paper/2601.22521/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 29 of its 35 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Placed in

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

## Builds on (most cited first, 29 of 29)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [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)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Learning to Reason without External Rewards](https://mlsanity.com/paper/2505.19590/) (2025)
- [Advancing LLM Reasoning Generalists with Preference Trees](https://mlsanity.com/paper/2404.02078/) (2024)
- [GPG: A Simple and Strong Reinforcement Learning Baseline for Model Reasoning](https://mlsanity.com/paper/2504.02546/) (2025)
- [A Minimalist Approach to LLM Reasoning: from Rejection Sampling to Reinforce](https://mlsanity.com/paper/2504.11343/) (2025)
- [Right Question is Already Half the Answer: Fully Unsupervised LLM Reasoning Incentivization](https://mlsanity.com/paper/2504.05812/) (2025)
- [Geometric-Mean Policy Optimization](https://mlsanity.com/paper/2507.20673/) (2025)
- [CPPO: Accelerating the Training of Group Relative Policy Optimization-Based Reasoning Models](https://mlsanity.com/paper/2503.22342/) (2025)
- [SEED-GRPO: Semantic Entropy Enhanced GRPO for Uncertainty-Aware Policy Optimization](https://mlsanity.com/paper/2505.12346/) (2025)
- [S-GRPO: Early Exit via Reinforcement Learning in Reasoning Models](https://mlsanity.com/paper/2505.07686/) (2025)
- [GRPO-LEAD: A Difficulty-Aware Reinforcement Learning Approach for Concise Mathematical Reasoning in Language Models](https://mlsanity.com/paper/2504.09696/) (2025)
- [Sample More to Think Less: Group Filtered Policy Optimization for Concise Reasoning](https://mlsanity.com/paper/2508.09726/) (2025)
- [SRPO: A Cross-Domain Implementation of Large-Scale Reinforcement Learning on LLM](https://mlsanity.com/paper/2504.14286/) (2025)
- [RePO: Replay-Enhanced Policy Optimization](https://mlsanity.com/paper/2506.09340/) (2025)
- [DLER: Doing Length pEnalty Right - Incentivizing More Intelligence per Token via Reinforcement Learning](https://mlsanity.com/paper/2510.15110/) (2025)
- [On-Policy RL with Optimal Reward Baseline](https://mlsanity.com/paper/2505.23585/) (2025)
- [BNPO: Beta Normalization Policy Optimization](https://mlsanity.com/paper/2506.02864/) (2025)
- [Stable Reinforcement Learning for Efficient Reasoning](https://mlsanity.com/paper/2505.18086/) (2025)

## Built on by (most cited first, 1 of 1)

- [Holder Policy Optimisation](https://mlsanity.com/paper/2605.12058/) (2026)

## Abstract (arXiv)

Group-based reinforcement learning has evolved from the arithmetic mean of GRPO to the geometric mean of GMPO. While GMPO improves stability by constraining a conservative objective, it shares a fundamental limitation with GRPO: reliance on a fixed aggregation geometry that ignores the evolving and heterogeneous nature of each trajectory. In this work, we unify these approaches under Power-Mean Policy Optimization (PMPO), a generalized framework that parameterizes the aggregation geometry via the power-mean geometry exponent p. Within this framework, GRPO and GMPO are recovered as special cases. Theoretically, we demonstrate that adjusting p modulates the concentration of gradient updates, effectively reweighting tokens based on their advantage contribution. To determine p adaptively, we introduce a Clip-aware Effective Sample Size (ESS) mechanism. Specifically, we propose a deterministic rule that maps a trajectory clipping fraction to a target ESS. Then, we solve for the specific p to align the trajectory induced ESS with this target one. This allows PMPO to dynamically transition between the aggressive arithmetic mean for reliable trajectories and the conservative geometric mean for unstable ones. Experiments on multiple mathematical reasoning benchmarks demonstrate that PMPO outperforms strong baselines.