Curvature-Guided LoRA: Matching Full Fine-Tuning in Function Space

Parameter-efficient fine-tuning methods such as LoRA enable efficient adaptation of large pretrained models, but often lag behind full fine-tuning in both convergence speed and final performance. Recent approaches aim to reduce this gap by aligning LoRA parameter updates with those of full fine-tuning, but such parameter-space alignment only indirectly controls model predictions. Instead, we adopt a function-space perspective and formulate the \emph{prediction alignment problem}, whose objective is to match the outputs of LoRA fine-tuning to those of full fine-tuning. We show that this objective naturally leads to a curvature-aware, second-order formulation, where optimal low-rank updates correspond to a Newton-like, curvature-whitened gradient. Based on this insight, we propose Curvature-Guided LoRA (CG-LoRA), an algorithm that selects adaptation directions using local curvature information. Our method is computationally efficient and avoids explicit second-order matrix construction. Experiments on standard natural language understanding benchmarks demonstrate improved performance and faster convergence compared to existing LoRA variants.