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

TL;DR

This paper introduces COALA, a numerically stable and efficient framework for context-aware low-rank approximation in neural networks, overcoming previous numerical issues and handling challenging scenarios with provable convergence.

Contribution

The paper presents a novel inversion-free regularized framework for context-aware low-rank approximation that is numerically stable and effective in challenging scenarios.

Findings

Addresses numerical instability in existing methods

Handles large calibration matrices and near-singular inputs

Provides convergence guarantees and explicit error bounds

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 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.