# A Local Perturbation Theory for Cross-Domain Interference and Recovery in Multi-Domain RL

arXiv: https://arxiv.org/abs/2606.02398 · HTML: https://mlsanity.com/paper/2606.02398/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 33 of its 40 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Placed in

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

## Builds on (most cited first, 30 of 33)

- [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)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [TIES-Merging: Resolving Interference When Merging Models](https://mlsanity.com/paper/2306.01708/) (2023)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Merging Models with Fisher-Weighted Averaging](https://mlsanity.com/paper/2111.09832/) (2021)
- [Knowledge Neurons in Pretrained Transformers](https://mlsanity.com/paper/2104.08696/) (2021)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Continual Learning of Large Language Models: A Comprehensive Survey](https://mlsanity.com/paper/2404.16789/) (2024)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs](https://mlsanity.com/paper/2605.00674/) (2026)
- [Identifying and Controlling Important Neurons in Neural Machine Translation](https://mlsanity.com/paper/1811.01157/) (2018)
- [WritingBench: A Comprehensive Benchmark for Generative Writing](https://mlsanity.com/paper/2503.05244/) (2025)
- [Does Math Reasoning Improve General LLM Capabilities? Understanding Transferability of LLM Reasoning](https://mlsanity.com/paper/2507.00432/) (2025)
- [Retaining by Doing: The Role of On-Policy Data in Mitigating Forgetting](https://mlsanity.com/paper/2510.18874/) (2025)
- [Finding Skill Neurons in Pre-trained Transformer-based Language Models](https://mlsanity.com/paper/2211.07349/) (2022)
- [Spurious Forgetting in Continual Learning of Language Models](https://mlsanity.com/paper/2501.13453/) (2025)

## Built on by (most cited first, 3 of 3)

- [One Step, One Lead: Mitigating Higher-Order Interference in Multi-Domain Reinforcement Learning via Cross-Step Control](https://mlsanity.com/paper/2609.06469/) (2026)
- [Beyond Static Interpretability: Anticipating Post-SFT Mechanisms from Pre-SFT Parameters for Better Tuning](https://mlsanity.com/paper/2608.24482/) (2026)
- [A helps B while B hurts A: directed transfer in instruction-tuning mixture](https://mlsanity.com/paper/2609.39702/) (2026)

## Abstract (arXiv)

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.