A Riemannian Geometry for Low-rank Adaptation

Low-rank adaptation (LoRA) is widely used as a parameter-efficient fine-tuning technique for pre-trained deep neural networks, which approximates the weight update via full fine-tuning by a low-rank matrix BA^\top. This parameterization leads to the equivalence relation (B, A) \sim (BG^{-1}, AG^\top) for any invertible matrix G because BA^\top = BG^{-1}(AG^\top)^\top and thus both pairs yield the same loss value. This relation induces a quotient manifold where matrices (BG^{-1}, AG^\top) for all G are identified, eliminating redundant directions along which the loss value remains unchanged. To respect the geometry of this manifold, the original search space is endowed with a Riemannian metric that is invariant under the equivalence relation. Such a metric induces preconditioning at each gradient step and ensures that each weight update via LoRA changes the loss value, leading to efficient optimization. In this paper, we propose a new Riemannian metric that is specifically tailored to LoRA to close the gap to full fine-tuning at the weight level. We theoretically show that LoRA with our preconditioning induced by this metric satisfies the following two properties at each iteration: (i) The weight update follows the direction closest to the gradient of full fine-tuning within the subspace of first-order weight changes allowed by the LoRA parameterization. (ii) The updated weight matrix is closer in Frobenius norm to that of full fine-tuning than the updated weight matrices of LoRA with conventional preconditioning and without preconditioning. These theoretical insights suggest that our preconditioning makes LoRA better approximate full fine-tuning, thereby leading to more efficient optimization. Experiments show the effectiveness and efficiency of our preconditioning for LoRA on fine-tuning tasks with language and vision domains.