arXiv:2607. 25299v1 Announce Type: cross Abstract: Optimization over the Stiefel manifold plays a significant role in various machine learning tasks.
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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.
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 a geometry-aware Bayesian fine‑tuning method that uses Stein variational gradient descent on the Stiefel manifold. By transporting low‑rank adapter matrices along this manifold, the approach preserves orthogonality constraints and yields multiple inference solutions, enabling uncertainty quantification. Experiments demonstrate improved model calibration and higher prediction accuracy compared to Euclidean‑space SVGD and related methods.
arXiv:2603. 29824v2 Announce Type: replace Abstract: Parameter-efficient fine-tuning methods such as LoRA enable efficient adaptation of large pretrained models, but often lag behind full fine-tuning in both convergence speed and final performance.
arXiv:2510. 01878v2 Announce Type: replace Abstract: Low-rank gradient optimization for large language models is currently divided into two categories: structured methods that rigorously identify subspaces, and randomized approaches employed primarily for computational efficiency.
arXiv:2606. 16454v1 Announce Type: cross Abstract: Low-Rank Adaptation (LoRA) enables efficient adaptation of large pre-trained models to downstream tasks by parameterizing weight updates with low-rank matrices.