Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapters
It addresses the reliability of single fine-tuning runs for practitioners using LoRA, offering foundational theoretical insights.
This paper tackles the problem of understanding the generalization behavior of asymmetric Low-Rank Adaptation (LoRA) in foundation models, providing theoretical bounds that show a sample complexity of O(√r/√N) with high probability and a matching lower bound of O(1/√N).
Low-Rank Adaptation (LoRA) has emerged as a widely adopted parameter-efficient fine-tuning (PEFT) technique for foundation models. Recent work has highlighted an inherent asymmetry in the initialization of LoRA's low-rank factors, which has been present since its inception and was presumably derived experimentally. This paper focuses on providing a comprehensive theoretical characterization of asymmetric LoRA with frozen random factors. First, while existing research provides upper-bound generalization guarantees based on averages over multiple experiments, the behaviour of a single fine-tuning run with specific random factors remains an open question. We address this by investigating the concentration of the typical LoRA generalization gap around its mean. Our main upper bound reveals a sample complexity of $\tilde{\mathcal{O}}\left(\frac{\sqrt{r}}{\sqrt{N}}\right)$ with high probability for rank $r$ LoRAs trained on $N$ samples. Additionally, we also determine the fundamental limits in terms of sample efficiency, establishing a matching lower bound of $\mathcal{O}\left(\frac{1}{\sqrt{N}}\right)$. By more closely reflecting the practical scenario of a single fine-tuning run, our findings offer crucial insights into the reliability and practicality of asymmetric LoRA.