Parameter-Efficient Fine-Tuning with Column Space Projection

Fine-tuning large language models (LLMs) with minimal computational overhead is essential for efficiently adapting them to downstream tasks under resource constraints. Parameter-efficient fine-tuning (PEFT) methods, such as Low-Rank Adaptation (LoRA), facilitate this by updating only a small subset of parameters. However, recent studies show that LoRA diverges from full fine-tuning (Full FT) in its learning behavior, particularly in terms of spectral properties. Motivated by these findings, we propose PiCa, the first theoretically grounded PEFT method based on the spectral properties of fine-tuned weights. PiCa projects gradients onto the low-rank column subspace of pre-trained weights and exhibits learning patterns more closely aligned with Full FT. Furthermore, we show that combining PiCa with weight sharing drastically reduces the number of trainable parameters without compromising performance, enabling to achieve superior performance than LoRA using 13x fewer trainable parameters. Extensive experiments demonstrate PiCa achieves the state-of-the-art performance compared to existing PEFT methods.