On the Optimization Landscape of Low Rank Adaptation Methods for Large Language Models

Training Large Language Models (LLMs) poses significant memory challenges, making low-rank adaptation methods an attractive solution. Previously, Low-Rank Adaptation (LoRA) addressed this by adding a trainable low-rank matrix to the frozen pre-trained weights in each layer, reducing the number of trainable parameters and optimizer states. GaLore, which compresses the gradient matrix instead of the weight matrix, has demonstrated superior performance to LoRA with faster convergence and reduced memory consumption. Despite their empirical success, the performance of these methods has not been fully understood or explained theoretically. In this paper, we analyze the optimization landscapes of LoRA, GaLore, and full-rank methods, revealing that GaLore benefits from fewer spurious local minima and a larger region that satisfies the PL * condition, a variant of Polyak-Łojasiewicz (PL) condition, leading to faster convergence. Our analysis leads to a novel method, GaRare, which further improves GaLore by using gradient random projection to reduce computational overhead. Practically, GaRare achieves strong performance in both pre-training and fine-tuning tasks, offering a more efficient approach to large-scale model adaptation. Code is available at https://github.com/liuxhym/GaRare.git . * Work done during Xu-Hui Liu's visit at King's College London.