GRPO, Dr. GRPO, and DAPO Are Three Operations on One Number: The Group-Standard-Deviation Identity
A new paper proves that three popular methods for training language models to reason—GRPO, Dr. GRPO, and DAPO—are all adjusting the same number: the standard deviation of answer correctness across multiple samples. The group-standard-deviation identity shows that disagreement directly determines the size of the training update, validated on the Big-Math dataset and in controlled experiments.
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[Submitted on 30 Jun 2026]
Title:GRPO, Dr. GRPO, and DAPO Are Three Operations on One Number: The Group-Standard-Deviation Identity
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Abstract: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.
Comments: 18 pages, 10 figures, 4 tables. Code and data: this https URL
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computation and Language (cs.CL); Machine Learning (stat.ML)
MSC classes: 68T05, 62F99
ACM classes: I.2.6; I.2.7; G.3
Cite as: arXiv:2607.00152 [cs.LG]
(or arXiv:2607.00152v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.00152
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Yong Yi Bay [view email] [v1] Tue, 30 Jun 2026 20:28:08 UTC (186 KB)
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