Entropy Law: The Story Behind Data Compression and LLM Performance

Data is the cornerstone of large language models (LLMs), but not all data is useful for model learning. Carefully selected data can better elicit the capabilities of LLMs with much less computational overhead. Most methods concentrate on evaluating the quality of individual samples in data selection, while the combinatorial effects among samples are neglected. Even if each sample is of perfect quality, their combinations may be suboptimal in teaching LLMs due to their intrinsic homogeneity or contradiction. In this paper, we aim to uncover the underlying relationships between LLM performance and data selection. Inspired by the information compression nature of LLMs, we uncover an ``entropy law'' that connects LLM performance with data compression ratio and first-epoch training loss, which reflect the information redundancy of a dataset and the mastery of inherent knowledge encoded in this dataset, respectively. Through both theoretical deduction and empirical evaluation, we find that model performance is negatively correlated to the compression ratio of training data, which usually yields a lower training loss. Based on the findings of the entropy law, we propose a quite efficient and universal data selection method named \textbf{ZIP} for training LLMs, which aim to prioritize data subsets exhibiting a low compression ratio. Based on a multi-stage algorithm that selects diverse data in a greedy manner, we can obtain a good data subset with satisfactory diversity. Extensive experiments have been conducted to validate the entropy law and the superiority of ZIP across different LLM backbones and alignment stages. We also present an interesting application of entropy law that can detect potential performance risks at the beginning of model training.