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

arXiv: https://arxiv.org/abs/2606.30789 · HTML: https://mlsanity.com/paper/2606.30789/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 20 references that are in the map.
- First posted on arXiv 29 Jun 2026.

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

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [OpenMathInstruct-2: Accelerating AI for Math with Massive Open-Source Instruction Data](https://mlsanity.com/paper/2410.01560/) (2024)
- [GSM-Plus: A Comprehensive Benchmark for Evaluating the Robustness of LLMs as Mathematical Problem Solvers](https://mlsanity.com/paper/2402.19255/) (2024)
- [Reinforcement Learning with Verifiable Rewards: GRPO's Effective Loss, Dynamics, and Success Amplification](https://mlsanity.com/paper/2503.06639/) (2025)
- [What is the Alignment Objective of GRPO?](https://mlsanity.com/paper/2502.18548/) (2025)
- [Spurious Rewards Paradox: Mechanistically Understanding How RLVR Activates Memorization Shortcuts in LLMs](https://mlsanity.com/paper/2601.11061/) (2026)
- [Predictive Scaling Laws for Efficient GRPO Training of Large Reasoning Models](https://mlsanity.com/paper/2507.18014/) (2025)
- [New Skills or Sharper Primitives? A Probabilistic Perspective on the Emergence of Reasoning in RLVR](https://mlsanity.com/paper/2602.08281/) (2026)

## Abstract (arXiv)

We develop a first-principles reduced-order model of these dynamics. Under a single mean-field assumption that summarizes the policy by its expected reward, we reduce the GRPO update to a stochastically-forced damped oscillator whose mass, damping, and stiffness are fixed in closed form by the optimizer hyperparameters together with a single measured curvature scale -- momentum supplies the inertia, off-policy lag erodes the damping, and the group size enters, to leading order, as a noise temperature. 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 R2 ≥ 0.91 and the mean trajectory is group-size invariant to leading order -- on both the reward curve and out-of-distribution transfer to eight math benchmarks -- while the within-group reward spread retains a residual G-dependence that the leading-order temperature picture does not capture.