Linear Dynamics in the RLVR Training of Large Language Models
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916034282258432 |
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| author | Wang, Tianle Liu, Jiayu Wu, Zhongyuan Jin, Shenghao Chen, Wei Xu, Hao Miao, Ning |
| author_facet | Wang, Tianle Liu, Jiayu Wu, Zhongyuan Jin, Shenghao Chen, Wei Xu, Hao Miao, Ning |
| contents | 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 ($R^2 > 0.7$). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04537 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear Dynamics in the RLVR Training of Large Language Models Wang, Tianle Liu, Jiayu Wu, Zhongyuan Jin, Shenghao Chen, Wei Xu, Hao Miao, Ning Machine Learning Computation and Language 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 ($R^2 > 0.7$). 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. |
| title | Linear Dynamics in the RLVR Training of Large Language Models |
| topic | Machine Learning Computation and Language |
| url | https://arxiv.org/abs/2601.04537 |