# GRPO, Dr. GRPO, and DAPO Are Three Operations on One Number: The Group-Standard-Deviation Identity

arXiv: https://arxiv.org/abs/2607.00152 · HTML: https://mlsanity.com/paper/2607.00152/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 30 Jun 2026.

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

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (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)
- [Big-Math: A Large-Scale, High-Quality Math Dataset for Reinforcement Learning in Language Models](https://mlsanity.com/paper/2502.17387/) (2025)
- [It Takes Two: Your GRPO Is Secretly DPO](https://mlsanity.com/paper/2510.00977/) (2025)
- [Advantage Shaping as Surrogate Reward Maximization: Unifying Pass@K Policy Gradients](https://mlsanity.com/paper/2510.23049/) (2025)
- [GRPO is Secretly a Process Reward Model](https://mlsanity.com/paper/2509.21154/) (2025)
- [Machine Learning vs Deep Learning: The Generalization Problem](https://mlsanity.com/paper/2403.01621/) (2024)
- [Gradient Starvation in Binary-Reward GRPO: Why Group-Mean Centering Fails and Why the Simplest Fix Works](https://mlsanity.com/paper/2605.07689/) (2026)
- [When More Sampling Hurts: The Modal Ceiling and Correlation Ceiling of Test-Time Scaling](https://mlsanity.com/paper/2606.28661/) (2026)
- [Why GRPO Needs Normalization: A Local-Curvature Perspective on Adaptive Gradients](https://mlsanity.com/paper/2601.23135/) (2026)
- [Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression](https://mlsanity.com/paper/2606.23575/) (2026)
- [A First-Principles Derivation of LLM Policy Optimization: From Expected Reward to GRPO and Its Structural Extensions](https://mlsanity.com/paper/2606.16733/) (2026)

## Built on by (most cited first, 2 of 2)

- [The Dark Room in the Reward Channel: Dense Prediction Rewards Collapse GRPO-Trained LLM Agents -- and The Channel, Not the Content, Decides What Works](https://mlsanity.com/paper/2607.21273/) (2026)
- [Jina-OCR-v1: Efficient Document Parsing with Speculative Decoding and Dense Verifiable Rewards](https://mlsanity.com/paper/2609.03181/) (2026)

## Abstract (arXiv)

Three of the most popular methods for training language models to reason look like three different tricks. They are not. All three adjust a single number: standard deviation, reflecting how much a prompt's sampled answers disagree. When such a model is trained, it answers each problem many times, and an automatic checker marks every answer right or wrong. The standard deviation of those marks measures the disagreement: largest when the answers split evenly between right and wrong, and zero when they all agree. Group Relative Policy Optimization (GRPO) divides by this number, GRPO Done Right (Dr. GRPO) drops the division, and Decoupled Clip and Dynamic Sampling Policy Optimization (DAPO) discards the groups where it is zero. Each is presented as its own fix, yet this paper proves they are three settings of one dial. That dial is not cosmetic: for right-or-wrong rewards, the disagreement is exactly the size of the training update, the group-standard-deviation identity. A split group teaches the most, while a unanimous group teaches nothing and falls silent. The same result says which problems deserve the most weight and how many tries each one needs. This paper confirms the intuition on a large real difficulty dataset (Big-Math) and in a controlled training run. What looks like a harmless normalization step is the dial that decides where learning happens and how strongly.