Low-rank adaptation (LoRA) is the standard way to fine-tune large models, yet when its two factors are trained independently, the update ignores the geometry of the low-rank weight change it induces. We introduce LoRA-TSD, an optimizer that treats every LoRA step as a tangent vector of the fixed-rank matrix manifold and takes the spectral-norm steepest-descent step of Muon inside that tangent space, mapping the result back to the factors through a retraction native to the LoRA parametrization.
The paper introduces ISO-LoRA, an optimizer that improves rank utilization in Low‑Rank Adaptation (LoRA) by coupling factor updates through spectral descent on the induced tangent perturbation in weight space. Experiments on GPT‑2 adaptation show that standard optimizers like AdamW concentrate updates in a few singular directions, whereas ISO-LoRA distributes energy more evenly, leading to higher effective rank and better downstream performance across 0.1B‑7B models. The authors provide theoretical guarantees under a stylized spiked‑gradient model and demonstrate that ISO-LoRA consistently outperforms factor‑wise optimizers, especially at moderate‑to‑large LoRA ranks.
By Zihan Zhu, Zhehang Du, Xuyang Chen, Tim Tsz-Kit Lau, Jiayuan Wu, X. Y. Han, Qi Long, Weijie Su
LoRA-TSD introduces a new optimizer for low‑rank adaptation (LoRA) that treats each update as a tangent vector on the fixed‑rank matrix manifold and applies a Muon‑style spectral‑norm steepest‑descent step within that tangent space. The method avoids costly full‑matrix operations and offers a retraction that is up to 2.8× cheaper than previous manifold approaches. The authors prove that their surrogate recovers LoRA‑Pro, identify the Riemannian gradient as the natural stationarity measure, and provide the first global convergence guarantees for both LoRA‑Pro and LoRA‑TSD, achieving superior performance across multiple benchmarks with Llama and Qwen models.
By Dmitrii Andriianov, Andrey Veprikov, Aleksandr Beznosikov
arXiv:2607. 13246v1 Announce Type: cross Abstract: Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training.
By Ali Parviz, Gal Mishne, Alex Cloninger
arXiv:2607. 17620v1 Announce Type: new Abstract: Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices.
By Nikhil Ghosh, Tetiana Parshakova, Robert M. Gower
arXiv:2607. 26247v1 Announce Type: new Abstract: Low-rank adaptation (LoRA) fine-tunes large pretrained models at a fraction of the cost of full fine-tuning, but its performance depends strongly on how the adapters are initialized.
By Dianze Liu, Farshid Ghezelbash