RiemannLoRA: A Unified Riemannian Framework for Ambiguity-Free LoRA
  Optimization
Authors: name: Vladimir Bogachev, name: Vladimir Aletov, name: Alexander Molozhavenko, name: Denis Bobkov, name: Vera Soboleva, name: Aibek Alanov, name: Maxim Rakhuba

Low-Rank Adaptation (LoRA) has become a widely adopted standard for
parameter-efficient fine-tuning of large language models (LLMs), significantly
reducing memory and computational demands. However, challenges remain,
including finding optimal initialization strategies or mitigating
overparametrization in low-rank matrix factorization. In this work, we propose
a novel approach that addresses both of the challenges simultaneously within a
unified framework. Our method treats a set of fixed-rank LoRA matrices as a
smooth manifold. Considering adapters as elements on this manifold removes
overparametrization, while determining the direction of the fastest loss
decrease along the manifold provides initialization. Special care is taken to
obtain numerically stable and computationally efficient implementation of our
method, using best practices from numerical linear algebra and Riemannian
optimization. Experimental results on LLM and diffusion model architectures
demonstrate that RiemannLoRA consistently improves both convergence speed and
final performance over standard LoRA and its state-of-the-art modifications.