Optimal Formats for Weight Quantisation

TL;DR

This paper introduces a systematic framework for designing weight quantisation formats that leverage variable-length coding, leading to improved model compression and efficiency in deep learning models.

Contribution

It connects quantisation format design with classical theory, develops non-linear quantisation curves, and derives optimal bit-width allocation across model layers.

Findings

Variable-length formats outperform fixed-length formats.

Optimal bit-width allocation saves up to 0.25 bits per parameter.

Formats exploiting variable-length encoding improve model efficiency.

Abstract

Weight quantisation is an essential technique for enabling efficient training and deployment of modern deep learning models. However, the recipe book of quantisation formats is large and formats are often chosen empirically. In this paper, we propose a framework for systematic design and analysis of quantisation formats. By connecting the question of format design with the classical quantisation theory, we show that the strong practical performance of popular formats comes from their ability to represent values using variable-length codes. We frame the problem as minimising the KL divergence between original and quantised model outputs under a model size constraint, which can be approximated by minimising the squared quantisation error, a well-studied problem where entropy-constrained quantisers with variable-length codes are optimal. We develop non-linear quantisation curves for block-scaled data across multiple distribution families and observe that these formats, along with sparse outlier formats, consistently outperform fixed-length formats, indicating that they also exploit variable-length encoding. Finally, by using the relationship between the Fisher information and KL divergence, we derive the optimal allocation of bit-widths to individual parameter tensors across the model's layers, saving up to 0.25 bits per parameter when applied to large language models.