LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently

Yuanhe Zhang, Fanghui Liu, Yudong Chen
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:75513-75574, 2025.

Abstract

This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA) (Hu et al., 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately—applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-zhang25ax, title = {{L}o{RA}-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently}, author = {Zhang, Yuanhe and Liu, Fanghui and Chen, Yudong}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {75513--75574}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/zhang25ax/zhang25ax.pdf}, url = {https://proceedings.mlr.press/v267/zhang25ax.html}, abstract = {This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA) (Hu et al., 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately—applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.} }
Endnote
%0 Conference Paper %T LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently %A Yuanhe Zhang %A Fanghui Liu %A Yudong Chen %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-zhang25ax %I PMLR %P 75513--75574 %U https://proceedings.mlr.press/v267/zhang25ax.html %V 267 %X This paper explores how theory can guide and enhance practical algorithms, using Low-Rank Adaptation (LoRA) (Hu et al., 2022) in large language models as a case study. We rigorously prove that, under gradient descent, LoRA adapters align with specific singular subspaces of the one-step full fine-tuning gradient. This result suggests that, by properly initializing the adapters using the one-step full gradient, subspace alignment can be achieved immediately—applicable to both linear and nonlinear models. Building on our theory, we propose a theory-driven algorithm, LoRA-One, where the linear convergence (as well as generalization) is built and incorporating preconditioners theoretically helps mitigate the effects of ill-conditioning. Besides, our theory reveals connections between LoRA-One and other gradient-alignment-based methods, helping to clarify misconceptions in the design of such algorithms. LoRA-One achieves significant empirical improvements over LoRA and its variants across benchmarks in natural language understanding, mathematical reasoning, and code generation. Code is available at: https://github.com/YuanheZ/LoRA-One.
APA
Zhang, Y., Liu, F. & Chen, Y.. (2025). LoRA-One: One-Step Full Gradient Could Suffice for Fine-Tuning Large Language Models, Provably and Efficiently. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:75513-75574 Available from https://proceedings.mlr.press/v267/zhang25ax.html.

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