# Entropy Law: The Story Behind Data Compression and LLM Performance

arXiv: https://arxiv.org/abs/2407.06645 · HTML: https://mlsanity.com/paper/2407.06645/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 37 references that are in the map.
- First posted on arXiv 9 Jul 2024.

## Builds on (most cited first, 25 of 25)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Self-Instruct: Aligning Language Models with Self-Generated Instructions](https://mlsanity.com/paper/2212.10560/) (2022)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [LIMA: Less Is More for Alignment](https://mlsanity.com/paper/2305.11206/) (2023)
- [Super-NaturalInstructions: Generalization via Declarative Instructions on 1600+ NLP Tasks](https://mlsanity.com/paper/2204.07705/) (2022)
- [Enhancing Chat Language Models by Scaling High-quality Instructional Conversations](https://mlsanity.com/paper/2305.14233/) (2023)
- [OpenAssistant Conversations -- Democratizing Large Language Model Alignment](https://mlsanity.com/paper/2304.07327/) (2023)
- [ChemCrow: Augmenting large-language models with chemistry tools](https://mlsanity.com/paper/2304.05376/) (2023)
- [How Far Can Camels Go? Exploring the State of Instruction Tuning on Open Resources](https://mlsanity.com/paper/2306.04751/) (2023)
- [What Makes Good Data for Alignment? A Comprehensive Study of Automatic Data Selection in Instruction Tuning](https://mlsanity.com/paper/2312.15685/) (2023)
- [Data Selection for Language Models via Importance Resampling](https://mlsanity.com/paper/2302.03169/) (2023)
- [A Survey on Data Selection for Language Models](https://mlsanity.com/paper/2402.16827/) (2024)
- [Camels in a Changing Climate: Enhancing LM Adaptation with Tulu 2](https://mlsanity.com/paper/2311.10702/) (2023)
- [QuRating: Selecting High-Quality Data for Training Language Models](https://mlsanity.com/paper/2402.09739/) (2024)
- [Superfiltering: Weak-to-Strong Data Filtering for Fast Instruction-Tuning](https://mlsanity.com/paper/2402.00530/) (2024)
- [#InsTag: Instruction Tagging for Analyzing Supervised Fine-tuning of Large Language Models](https://mlsanity.com/paper/2308.07074/) (2023)
- [MoDS: Model-oriented Data Selection for Instruction Tuning](https://mlsanity.com/paper/2311.15653/) (2023)
- [Rethinking Data Selection for Supervised Fine-Tuning](https://mlsanity.com/paper/2402.06094/) (2024)
- [Data Management For Training Large Language Models: A Survey](https://mlsanity.com/paper/2312.01700/) (2023)
- [gzip Predicts Data-dependent Scaling Laws](https://mlsanity.com/paper/2405.16684/) (2024)

## Built on by (most cited first, 30 of 44)

- [Pangu Embedded: An Efficient Dual-system LLM Reasoner with Metacognition](https://mlsanity.com/paper/2505.22375/) (2025)
- [VideoLLM-MoD: Efficient Video-Language Streaming with Mixture-of-Depths Vision Computation](https://mlsanity.com/paper/2408.16730/) (2024)
- [Beyond Random Sampling: Efficient Language Model Pretraining via Curriculum Learning](https://mlsanity.com/paper/2506.11300/) (2025)
- [Rethinking Data Selection at Scale: Random Selection is Almost All You Need](https://mlsanity.com/paper/2410.09335/) (2024)
- [The Rise of Small Language Models in Healthcare: A Comprehensive Survey](https://mlsanity.com/paper/2504.17119/) (2025)
- [Pangu Pro MoE: Mixture of Grouped Experts for Efficient Sparsity](https://mlsanity.com/paper/2505.21411/) (2025)
- [Alignment and Safety in Large Language Models: Safety Mechanisms, Training Paradigms, and Emerging Challenges](https://mlsanity.com/paper/2507.19672/) (2025)
- [Pangu Ultra: Pushing the Limits of Dense Large Language Models on Ascend NPUs](https://mlsanity.com/paper/2504.07866/) (2025)
- [EntroPIC: Towards Stable Long-Term Training of LLMs via Entropy Stabilization with Proportional-Integral Control](https://mlsanity.com/paper/2511.15248/) (2025)
- [MIG: Automatic Data Selection for Instruction Tuning by Maximizing Information Gain in Semantic Space](https://mlsanity.com/paper/2504.13835/) (2025)
- [Breaking Determinism: Fuzzy Modeling of Sequential Recommendation Using Discrete State Space Diffusion Model](https://mlsanity.com/paper/2410.23994/) (2024)
- [SelfAug: Mitigating Catastrophic Forgetting in Retrieval-Augmented Generation via Distribution Self-Alignment](https://mlsanity.com/paper/2509.03934/) (2025)
- [ALinFiK: Learning to Approximate Linearized Future Influence Kernel for Scalable Third-Party LLM Data Valuation](https://mlsanity.com/paper/2503.01052/) (2025)
- [Difficulty-Based Preference Data Selection by DPO Implicit Reward Gap](https://mlsanity.com/paper/2508.04149/) (2025)
- [Bridging User Dynamics: Transforming Sequential Recommendations with Schr"odinger Bridge and Diffusion Models](https://mlsanity.com/paper/2409.10522/) (2024)
- [LLM Safety Alignment is Divergence Estimation in Disguise](https://mlsanity.com/paper/2502.00657/) (2025)
- [Layerwise Recall and the Geometry of Interwoven Knowledge in LLMs](https://mlsanity.com/paper/2502.10871/) (2025)
- [Revealing the Power of Post-Training for Small Language Models via Knowledge Distillation](https://mlsanity.com/paper/2509.26497/) (2025)
- [DisenGCD: A Meta Multigraph-assisted Disentangled Graph Learning Framework for Cognitive Diagnosis](https://mlsanity.com/paper/2410.17564/) (2024)
- [RAG-IGBench: Innovative Evaluation for RAG-based Interleaved Generation in Open-domain Question Answering](https://mlsanity.com/paper/2512.05119/) (2025)
- [SPARD: Self-Paced Curriculum for RL Alignment via Integrating Reward Dynamics and Data Utility](https://mlsanity.com/paper/2604.07837/) (2026)
- [Data Selection for Multi-turn Dialogue Instruction Tuning](https://mlsanity.com/paper/2604.07892/) (2026)
- [Denoising Pre-Training and Customized Prompt Learning for Efficient Multi-Behavior Sequential Recommendation](https://mlsanity.com/paper/2408.11372/) (2024)
- [Finding the Minimal Parameter Budget for Implicit Reasoning: A Data Complexity Driven Scaling Law for Language Models](https://mlsanity.com/paper/2504.03635/) (2025)
- [Revisiting scalable sequential recommendation with Multi-Embedding Approach and Mixture-of-Experts](https://mlsanity.com/paper/2510.25285/) (2025)
- [Instruction Data Selection via Answer Divergence](https://mlsanity.com/paper/2604.10448/) (2026)
- [Concept than Document: Context Compression via AMR-based Conceptual Entropy](https://mlsanity.com/paper/2511.18832/) (2025)
- [EDCO: Dynamic Curriculum Orchestration for Domain-specific Large Language Model Fine-tuning](https://mlsanity.com/paper/2601.03725/) (2026)
- [MDAP: A Multi-view Disentangled and Adaptive Preference Learning Framework for Cross-Domain Recommendation](https://mlsanity.com/paper/2410.05877/) (2024)
- [Call for Rigor in Reporting Quality of Instruction Tuning Data](https://mlsanity.com/paper/2503.04807/) (2025)

## Abstract (arXiv)

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 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.