aiqbench analysis
This paper introduces a novel Riemannian metric specifically designed for Low-rank adaptation (LoRA) to address redundancy in the parameter search space. By enforcing geometric invariance, the proposed method ensures that gradient updates meaningfully change the loss value and better approximate full fine-tuning performance. Theoretical analysis and experiments across language and vision domains demonstrate improved optimization efficiency without requiring additional computational overhead.
A Riemannian Geometry for Low-rank Adaptation
arXiv:2610.08049v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG)
arXiv:2610.08049 [cs.LG]
(or arXiv:2610.08049v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.08049 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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