LoRA (Low-Rank Adaptation) is a widely used model fine-tuning method. In fine-tuning, the law among model performance, model parameters, and data complexity has been a focal issue in the field. Existing methods often leverage external metrics (such as cross-entropy or perplexity) to evaluate model performance. In the fine-tuning process for large models, two types of knowledge are typically involved: the frozen, general knowledge acquired by the model during pre-training and the new knowledge learned through the LoRA module from the current data. Generally, the less LoRA's learned knowledge relies on the large model, the more it captures the specific knowledge of new data, thereby enhancing its adaptability to new tasks. However, external metrics do not readily capture the dependency relationship between these two types of knowledge. Therefore, we designed an internal metric based on the Mutual Information Upper Bound (MIUB) theory to investigate the scaling law of large-model LoRA fine-tuning. In our experiments, we validated this approach on benchmark datasets, using the Llama3-8B and Phi3-3B models. The results show that the proposed MIUB metric aligns more accurately and stably with the scaling law of LoRA fine-tuning compared to cross-entropy and perplexity.

本文研究了模型性能、参数和数据复杂性之间的关系，特别是在大型模型的LoRA微调过程中。提出了一种基于互信息上界的内部度量，能够更准确地捕捉微调过程中新旧知识之间的依赖关系。实验结果表明，该指标在评估LoRA微调的缩放法则方面优于传统的外部度量方法。

基于互信息上界的LoRA缩放法则