From Low Rank Gradient Subspace Stabilization to Low-Rank Weights: Observations, Theories, and Applications

• URL: http://arxiv.org/abs/2407.11239v2
• Date: Sat, 07 Jun 2025 06:59:36 GMT
• Title: From Low Rank Gradient Subspace Stabilization to Low-Rank Weights: Observations, Theories, and Applications
• Authors: Ajay Jaiswal, Yifan Wang, Lu Yin, Shiwei Liu, Runjin Chen, Jiawei Zhao, Ananth Grama, Yuandong Tian, Zhangyang Wang,
• Abstract summary: We study the non-uniform low-rank properties of weight matrices in Large Language Models.<n>We present Weight Low-Rank Projection (WeLore) that unifies weight compression and memory-efficient fine-tuning into one.
• Score: 85.17672240603011
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Large Language Models' (LLMs) weight matrices can often be expressed in low-rank form with potential to relax memory and compute resource requirements. Unlike prior efforts that focus on developing novel matrix decompositions, in this work we study the non-uniform low-rank properties of weight matrices in LLMs through the lens of stabilizing gradient subspace. First, we provide a theoretical framework to understand the stabilization of gradient subspaces through Hessian analysis. Second, we empirically establish an important relationship between gradient dynamics and low-rank expressiveness of weight matrices. Our findings reveal that different LLM components exhibit varying levels of converged low-rank structures, necessitating variable rank reduction across them to minimize drop in performance due to compression. Drawing on this result, we present Weight Low-Rank Projection(WeLore) that unifies weight compression and memory-efficient fine-tuning into one, in a data-agnostic and one-shot manner. When used as a compression technique, WeLore categorizes weight matrices into Low-rank Components (LRCs) and Non-Low-rank Components (N-LRCs) and suitably encodes them for minimum performance loss. Our gradient dynamics perspective illustrates that LRCs tend to have better fine-tuning capabilities and their standalone fine-tuning can closely mimic and sometimes outperform the training loss trajectory and performance of full fine-tuning with notable memory and compute footprint reduction. Codes are available at https://github.com/VITA-Group/WeLore.

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