# Linear Dynamics in the RLVR Training of Large Language Models

arXiv: https://arxiv.org/abs/2601.04537 · HTML: https://mlsanity.com/paper/2601.04537/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 36 references that are in the map.
- First posted on arXiv 8 Jan 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 31)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [CodeRL: Mastering Code Generation through Pretrained Models and Deep Reinforcement Learning](https://mlsanity.com/paper/2207.01780/) (2022)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)
- [AIMO-2 Winning Solution: Building State-of-the-Art Mathematical Reasoning Models with OpenMathReasoning dataset](https://mlsanity.com/paper/2504.16891/) (2025)
- [The Unreasonable Effectiveness of Entropy Minimization in LLM Reasoning](https://mlsanity.com/paper/2505.15134/) (2025)
- [A Kernel-Based View of Language Model Fine-Tuning](https://mlsanity.com/paper/2210.05643/) (2022)
- [The Invisible Leash: Why RLVR May or May Not Escape Its Origin](https://mlsanity.com/paper/2507.14843/) (2025)
- [Reasoning with Sampling: Your Base Model is Smarter Than You Think](https://mlsanity.com/paper/2510.14901/) (2025)
- [Critical Tokens Matter: Token-Level Contrastive Estimation Enhances LLM's Reasoning Capability](https://mlsanity.com/paper/2411.19943/) (2024)
- [JustRL: Scaling a 1.5B LLM with a Simple RL Recipe](https://mlsanity.com/paper/2512.16649/) (2025)
- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)
- [Reinforcement Learning Finetunes Small Subnetworks in Large Language Models](https://mlsanity.com/paper/2505.11711/) (2025)
- [Model Extrapolation Expedites Alignment](https://mlsanity.com/paper/2404.16792/) (2024)
- [On Predictability of Reinforcement Learning Dynamics for Large Language Models](https://mlsanity.com/paper/2510.00553/) (2025)

## Built on by (most cited first, 4 of 4)

- [Low-rank Optimization Trajectories Modeling for LLM RLVR Acceleration](https://mlsanity.com/paper/2604.11446/) (2026)
- [Extrapolative Weight Averaging Reveals Correctness-Efficiency Frontiers in Code RL](https://mlsanity.com/paper/2605.28751/) (2026)
- [Gradient Extrapolation-Based Policy Optimization](https://mlsanity.com/paper/2605.06755/) (2026)
- [SFT Leaves a Reproducible Signature in Activation Geometry: Calibrating Post-Training Interpretability Against Run-to-Run Noise](https://mlsanity.com/paper/2610.17093/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) has driven significant performance gains in reasoning-oriented large language models (LLMs), yet its internal training dynamics remain largely a black box. In this work, we perform a comprehensive trajectory-level analysis of RLVR and uncover a striking regularity: across various model families, RL algorithms, and training configurations, RLVR consistently enters a robust linear regime, where both parameter weights and output log-probabilities, measured rigorously via teacher-forced evaluation, evolve in a highly linear manner (R2 > 0.7). Through controlled experiments and theoretical analysis, we demonstrate that this linearity is not a coincidence, but stems from the high-variance, noisy nature of RLVR training signals, which act as a low-pass filter to concentrate optimization along a stable, low-dimensional drift. Moreover, we show that this linear structure is not merely descriptive but powerfully predictive and actionable. Specifically, weight-space extrapolation matches the performance of standard RL optimization while achieving a 6.1x training speedup through periodic re-grounding. Meanwhile, output-space extrapolation serves as a lightweight intervention that effectively bypasses late-stage model collapse, consistently outperforming standard RL across mathematical and coding benchmarks, with an average performance improvement of 4.2%. Our code is available at https://github.com/Miaow-Lab/RLVR-Linearity.