Title: Initialization using Update Approximation is a Silver Bullet for Extremely Efficient Low-Rank Fine-Tuning
Primary area: transfer learning, meta learning, and lifelong learning
Keywords: LoRA, Low-rank adaptation, PEFT, Parameter-efficient fine-tuning, Fine-tuning

Abstract: Low-rank adapters have become standard for efficiently fine-tuning large language models, but they often fall short of achieving the performance of full fine-tuning. We propose a method, **LoRA** **S**ilver **B**ullet or **LoRA-SB**, that approximates full fine-tuning within low-rank subspaces using a carefully designed initialization strategy. We theoretically demonstrate that the architecture of LoRA-XS, which inserts a learnable $r \times r$ matrix between $B$ and $A$ while keeping other matrices fixed, provides the precise conditions needed for this approximation. We leverage its constrained update space to achieve optimal scaling for high-rank gradient updates while removing the need for hyperparameter tuning. We prove that our initialization offers an optimal low-rank approximation of the initial gradient and preserves update directions throughout training. Extensive experiments across mathematical reasoning, commonsense reasoning, and language understanding tasks demonstrate that our approach exceeds the performance of LoRA (and baselines) while using **27-90** times fewer learnable parameters, and comprehensively outperforms LoRA-XS. Our findings establish that it is possible to simulate full fine-tuning in low-rank subspaces, and achieve significant parameter efficiency gains without sacrificing performance. Anonymous code is available at: https://anonymous.4open.science/r/lora-sb-anonymous-5BEE.