该理论解释了多域RL训练中性能下降的机制，并提出了有效的恢复方法，对从事LLM后训练和多任务学习的开发者有直接指导意义，建议关注其刷新策略。
该研究揭示了多域强化学习（RL）中一个关键问题：在数学推理、代码生成等单一领域训练会损害其他领域性能。现有解释（如灾难性遗忘或全局梯度冲突）不完整，因为即使全模型梯度几乎正交时，干扰仍会发生。研究发现，单域RL产生稀疏、小幅度的参数编辑，不同领域共享大量活跃计算路径，更新方向决定协同或冲突。基于局部扰动模型，作者证明后期训练主要通过二阶损伤项损害早期领域，该损伤集中在低维共享冲突子空间。通过短暂领域刷新（如代码→数学→问答→创意写作后重新训练数学），数学性能从57.66恢复至66.04，且其他领域性能保持良好，平均得分达66.39。此外，无训练的回滚方法也部分恢复了数学性能，提供了局部损伤的直接证据。
A Local Perturbation Theory for Cross-Domain Interference and Recovery in Multi-Domain RL
Reinforcement learning (RL) post-training improves large language models (LLMs) on individual domains such as mathematical reasoning, code generation, question answering, and creative writing (CW), but training on one domain often degrades performance on others. Existing explanations based on catastrophic forgetting or global gradient conflict are incomplete: substantial interference can occur even when full-model gradients are nearly orthogonal. We show that single-domain RL produces sparse, small-magnitude parameter edits with weak overlap among top-changed neurons, while different domains still share substantial active computation routes on which update directions determine whether they act synergistically or conflict. Guided by this observation, we prove under a local perturbation model of multi-domain RL that later-domain training harms an earlier domain mainly through a second-order damage term, which under the observed sparse route structure concentrates in a low-dimensional shared conflict subspace. Moreover, a short domain refresh contracts the harmful component on this subspace, enabling selective recovery with limited collateral damage. Consistent with the theory, a brief Re-Math refresh after Code $\rightarrow$ Math $\rightarrow$ QA $\rightarrow$ CW recovers Math from 57.66 to 66.04 while largely preserving performance on the other domains, yielding the best average score of 66.39. Beyond refresh, a training-free rollback on a sparse proxy conflict coordinate set for the Math-QA pair partially restores Math, providing direct proxy-level evidence for localized damage. These results provide a localized mechanistic account of interference and recovery in multi-domain RL.