Optimal Brain Decomposition for Accurate LLM Low-Rank Approximation

TL;DR

This paper introduces OBD-LLM, a novel second-order Hessian-based method for optimal low-rank decomposition of language model weights, significantly improving accuracy over previous SVD-based approaches.

Contribution

It presents a closed-form, loss-aware decomposition method utilizing Kronecker-factorization of the Hessian, considering both input and output layer information.

Findings

Achieves 20-40% better results than previous methods.

Utilizes a bi-directional whitening process for weight decomposition.

Provides a theoretical framework for optimal low-rank approximation in LLMs.

Abstract

Low-rank decomposition has emerged as an important problem in Large Language Model (LLM) fine-tuning and inference. Through Singular Value Decomposition (SVD), the weight matrix can be factorized into low-rank spaces optimally. Previously, a common practice was to decompose the weight in the activation-whitened space, and then achieve satisfying results. In this work, we propose Optimal Brain Decomposition LLM (OBD-LLM), which studies the decomposition problem in the model space by utilizing second-order Hessian information. Through a rigorous Kronecker-factorization of the Hessian, we show that the decomposition needs to consider both input and output information of the layer, and achieves much better decomposition results compared to input only method. Our loss-aware decomposition method involves a bi-directional whitening on the weight matrix. As a result, OBD-LLM is a closed-form solution for the optimal decomposition of weights in the language model. Remarkably, we achieve ~20-40\% better results than previous state-of-the-art decomposition methods, the SVD-LLM.