[2606.30789] Predictable GRPO: A Closed-Form Model of Training Dynamics

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Computer Science > Machine Learning

arXiv:2606.30789 (cs)

[Submitted on 29 Jun 2026]

Title:Predictable GRPO: A Closed-Form Model of Training Dynamics

Authors:Rajat Ghosh, Datta Nimmaturi, Aryan Singhal, Vaishnavi Bhargava, Henry Wong, Johnu George, Debojyoti Dutta<br>View a PDF of the paper titled Predictable GRPO: A Closed-Form Model of Training Dynamics, by Rajat Ghosh and 6 other authors

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Abstract:Group Relative Policy Optimization (GRPO) has become a standard tool for improving the reasoning ability of large language models, yet its training dynamics are still described empirically: reward trajectories are fit with low-parameter functional forms whose constants carry no mechanistic meaning, and hyperparameter choices remain a matter of trial and error. We develop a first-principles reduced-order model of these dynamics. The reduction has three consequences. First, it subsumes the empirical single-exponential saturation law as its overdamped limit, recasting the fitted plateau, timescale, and size exponent as the fixed point, inverse stiffness, and curvature-scaling exponent of the underlying potential, and adding, through the retained inertial term, the slow-start phase the single exponential cannot represent. Second, it yields predictions tied to independently measurable quantities rather than fitted ones: group-size invariance of the deterministic trajectory with a $1/G$ stationary fluctuation, a sharp stability threshold in the refresh interval, and an overdamped-to-oscillatory transition. Third, it furnishes diagnostics that separate failure modes a reward curve alone conflates -- reward hacking, advantage degeneracy, policy concentration, and dynamical instability. Across three models and two group sizes, the closed-form trajectory fits training reward to $R^2 \geq 0.91$ and the predicted group-size invariance holds on both the reward curve and out-of-distribution transfer to eight math benchmarks. The stability and oscillatory predictions are exercised in a controlled exact-reduction setting where the mean-field assumption holds exactly: a softmax-bandit reduction reproduces the predicted overdamped-to-oscillatory transition and locates the refresh-interval stability threshold at the independently measured stiffness, with a deep-network demonstration left to future work.

Subjects:

Machine Learning (cs.LG); Machine Learning (stat.ML)

Cite as:<br>arXiv:2606.30789 [cs.LG]

(or<br>arXiv:2606.30789v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.30789

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arXiv-issued DOI via DataCite (pending registration)

Submission history<br>From: Rajat Ghosh [view email]<br>[v1]<br>Mon, 29 Jun 2026 18:19:09 UTC (752 KB)

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