In this work we study the enhancement of Low Rank Adaptation (LoRA) fine-tuning procedure by introducing a Riemannian preconditioner in its optimization step. Specifically, we introduce an $r\times r$ preconditioner in each gradient step where $r$ is the LoRA rank. This preconditioner requires a small change to existing optimizer code and creates virtually minuscule storage and runtime overhead. Our experimental results with both large language models and text-to-image diffusion models show that with our preconditioner, the convergence and reliability of SGD and AdamW can be significantly enhanced. Moreover, the training process becomes much more robust to hyperparameter choices such as learning rate. Theoretically, we show that fine-tuning a two-layer ReLU network in the convex paramaterization with our preconditioner has convergence rate independent of condition number of the data matrix. This new Riemannian preconditioner, previously explored in classic low-rank matrix recovery, is introduced to deep learning tasks for the first time in our work. We release our code at https://github.com/pilancilab/Riemannian_Preconditioned_LoRA.

通过引入Riemannian预处理器，研究Low Rank Adaptation（LoRA）微调过程的增强，实验结果表明，使用我们的预处理器可以显著提高SGD和AdamW的收敛性和稳定性。

利用黎曼预条件的LoRA方法对基础模型进行微调