The Influence Spectrum: The Power of Layer-Wise Importance in Large Language Models

The black-box nature of Large Language Models (LLMs) poses significant challenges to their transparency and controllability. Understanding which layers are most relevant to a given task provides a practical basis for interpreting LLM behavior and enabling effective task adaptation. We propose the influence spectrum framework to characterize task-specific layer importance in LLMs. At its core, the Parameter-space Influence Function (PIF) measures local loss sensitivity to parameter perturbations through first-order gradient information and second-order directional curvature. Using Hessian-vector products (HVPs), PIF condenses massive second-order operations into just two sequential gradient evaluations. Layer-masking experiments validate the diagnostic value of the resulting influence spectra, while extensive evaluations across diverse open-source LLMs and benchmarks demonstrate the utility of our framework in Parameter-Efficient Fine-Tuning (PEFT), continual learning, dynamic model defense, and knowledge editing.