Predictable GRPO: A Closed-Form Model of Training Dynamics
This paper develops a first-principles reduced-order model of GRPO training dynamics, subsuming the empirical single-exponential saturation law as its overdamped limit and adding an inertial term to capture the slow-start phase. It yields predictions tied to independently measurable quantities such as group-size invariance, a sharp stability threshold in the refresh interval, and an overdamped-to-oscillatory transition. The closed-form trajectory fits training reward with R² ≥ 0.91 across three models and two group sizes, and the predicted group-size invariance holds on both the reward curve and out-of-distribution transfer to eight math benchmarks. Additionally, the model furnishes diagnostics that separate failure modes conflated by the reward curve alone, such as reward hacking, advantage degeneracy, policy concentration, and dynamical instability.
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[Submitted on 29 Jun 2026]
Title:Predictable GRPO: A Closed-Form Model of Training Dynamics
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: arXiv:2606.30789 [cs.LG]
(or arXiv:2606.30789v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.30789
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Rajat Ghosh [view email] [v1] Mon, 29 Jun 2026 18:19:09 UTC (752 KB)
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