The Expressive Power of Low-Rank Adaptation

TL;DR

This paper provides a theoretical analysis of Low-Rank Adaptation (LoRA), demonstrating its expressive power in fine-tuning neural networks and establishing rank thresholds for accurate model adaptation.

Contribution

It offers the first theoretical insights into LoRA's expressive capabilities, including rank conditions for approximating target models in neural networks and Transformers.

Findings

LoRA can adapt any fully connected network to a smaller target model with sufficient rank.

Quantifies approximation errors when LoRA rank is below the threshold.

Any Transformer model can be adapted with rank proportional to embedding size.

Abstract

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.