Compressing Large Language Models using Low Rank and Low Precision Decomposition

TL;DR

This paper presents CALDERA, a novel post-training compression method for large language models that combines low-rank and low-precision decomposition to significantly reduce model size while maintaining performance.

Contribution

CALDERA introduces a new low-rank, low-precision decomposition technique for LLM compression, with theoretical error bounds and improved performance over existing methods.

Findings

Outperforms existing compression techniques at less than 2.5 bits per parameter.

Effectively compresses LLaMa-2 and LLaMa-3 models with minimal performance loss.

Provides theoretical bounds on approximation error and tradeoffs between compression ratio and accuracy.

Abstract

The prohibitive sizes of Large Language Models (LLMs) today make it difficult to deploy them on memory-constrained edge devices. This work introduces $\rm CALDERA$ -- a new post-training LLM compression algorithm that harnesses the inherent low-rank structure of a weight matrix $\mathbf{W}$ by approximating it via a low-rank, low-precision decomposition as $\mathbf{W} \approx \mathbf{Q} + \mathbf{L}\mathbf{R}$. Here, $\mathbf{L}$ and $\mathbf{R}$ are low rank factors, and the entries of $\mathbf{Q}$, $\mathbf{L}$ and $\mathbf{R}$ are quantized. The model is compressed by substituting each layer with its $\mathbf{Q} + \mathbf{L}\mathbf{R}$ decomposition, and the zero-shot performance of the compressed model is evaluated. Additionally, $\mathbf{L}$ and $\mathbf{R}$ are readily amenable to low-rank adaptation, consequently enhancing the zero-shot performance. $\rm CALDERA$ obtains this decomposition by formulating it as an optimization problem $\min_{\mathbf{Q},\mathbf{L},\mathbf{R}}\lVert(\mathbf{Q} + \mathbf{L}\mathbf{R} - \mathbf{W})\mathbf{X}^\top\rVert_{\rm F}^2$, where $\mathbf{X}$ is the calibration data, and $\mathbf{Q}, \mathbf{L}, \mathbf{R}$ are constrained to be representable using low-precision formats. Theoretical upper bounds on the approximation error of $\rm CALDERA$ are established using a rank-constrained regression framework, and the tradeoff between compression ratio and model performance is studied by analyzing the impact of target rank and quantization bit budget. Results illustrate that compressing LlaMa-$2$ $7$B/$13B$/$70$B and LlaMa-$3$ $8$B models using $\rm CALDERA$ outperforms existing post-training LLM compression techniques in the regime of less than $2.5$ bits per parameter. The implementation is available at: https://github.com/pilancilab/caldera.