Abstract Fine-tuning Large Language Models (LLMs) on some task-specific datasets has been a primary use of LLMs. However, it has been empirically observed that this approach to enhancing capability inevitably compromises safety, a phenomenon also known as the safety-capability trade-off in LLM fine-tuning. This paper presents a theoretical framework for understanding the interplay between safety and capability in two primary safety-aware LLM fine-tuning strategies, providing new insights into th