COALA: Numerically Stable and Efficient Framework for Context-Aware Low-Rank Approximation

Uliana Parkina, Maxim Rakhuba

Keywords: LLM, model compression, low-rank approximation, matrix decompositions, regularization

TL;DR: We introduce an inversion-free regularized framework using stable decompositions to overcome numerical instabilities in context-aware low-rank approximation for neural networks.

Abstract:

Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks.
In this type of approximation, a norm is weighted by a matrix of input activations, significantly improving metrics over the unweighted case.
Nevertheless, existing methods for neural networks suffer from numerical instabilities due to their reliance on classical formulas involving explicit Gram matrix computation and their subsequent inversion.
We demonstrate that this can degrade the approximation quality or cause numerically singular matrices.

To address these limitations, we propose a novel _inversion-free regularized framework_ that is based entirely on stable decompositions and overcomes the numerical pitfalls of prior art.
Our method can handle all possible challenging scenarios: (1) when calibration matrices exceed GPU memory capacity, (2) when input activation matrices are nearly singular, and even (3) when insufficient data prevents unique approximation.
For the latter, we prove that our solution converges to a desired approximation and derive explicit error bounds.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 22925