The Expressive Power of Low-Rank Adaptation
It addresses the lack of theoretical understanding for LoRA, a widely used parameter-efficient fine-tuning method in AI, providing foundational insights for researchers and practitioners.
This paper theoretically analyzes the expressive power of Low-Rank Adaptation (LoRA), proving that for fully connected neural networks, LoRA can adapt any model to accurately represent any smaller target model if the LoRA-rank meets a specific threshold, and for Transformers, it can adapt to a target model of the same size with rank-(embedding size/2) adapters.
Low-Rank Adaptation (LoRA), a parameter-efficient fine-tuning method that leverages low-rank adaptation of weight matrices, has emerged as a prevalent technique for fine-tuning pre-trained models such as large language models and diffusion models. Despite its huge success in practice, the theoretical underpinnings of LoRA have largely remained unexplored. This paper takes the first step to bridge this gap by theoretically analyzing the expressive power of LoRA. We prove that, for fully connected neural networks, LoRA can adapt any model $f$ to accurately represent any smaller target model $\overline{f}$ if LoRA-rank $\geq(\text{width of }f) \times \frac{\text{depth of }\overline{f}}{\text{depth of }f}$. We also quantify the approximation error when LoRA-rank is lower than the threshold. For Transformer networks, we show any model can be adapted to a target model of the same size with rank-$(\frac{\text{embedding size}}{2})$ LoRA adapters.