LoRA-TSD: Tangent-Space Spectral Descent for LoRA via Muon-Style Updates

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 step avoids expensive operations on full weight matrices, and its retraction is up to 2.8× cheaper than the truncated-SVD retraction used by prior manifold methods. We prove that the Frobenius-norm version of our surrogate recovers LoRA-Pro, and we identify the tangent-projected gradient, the Riemannian gradient of the manifold, as the stationarity measure natural to LoRA training and computable from the factor gradients alone. Under this measure we give the first global convergence guarantees for both LoRA-Pro and LoRA-TSD, with rates that drive the factor-gradient norms to zero. Across six commonsense and natural-language-inference benchmarks with Llama-3.2-1B, Llama-3.1-8B and Qwen3-32B, LoRA-TSD outperforms every competing LoRA optimizer and stays robust to the adapter rank. Code is available at https://github.com/brain-lab-research/LoRA-TSD.