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

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Summary

This paper proposes a local perturbation theory to explain cross-domain interference in multi-domain RL for LLMs, showing that interference is driven by a second-order damage term in a low-dimensional conflict subspace, and demonstrates that brief domain refresh or training-free rollback can selectively recover lost capabilities.

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.
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Source: https://huggingface.co/papers/2606.02398

Abstract

Multi-domain reinforcement learning in language models causes performance degradation through shared computational pathways, but targeted refresh and rollback techniques can selectively recover lost capabilities with minimal side effects.

Reinforcement learning(RL) post-training improveslarge 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 oncatastrophic forgettingor globalgradient conflictare incomplete: substantial interference can occur even when full-model gradients are nearly orthogonal. We show that single-domain RL produces sparse, small-magnitudeparameter editswith 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 alocal perturbation modelof multi-domain RL that later-domain training harms an earlier domain mainly through asecond-order damage term, which under the observed sparse route structure concentrates in a low-dimensional sharedconflict subspace. Moreover, a shortdomain refreshcontracts 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-freerollbackon 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.

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