Predictable Scale: Part I, Step Law -- Optimal Hyperparameter Scaling Law in Large Language Model Pretraining

• URL: http://arxiv.org/abs/2503.04715v6
• Date: Wed, 16 Jul 2025 07:46:43 GMT
• Title: Predictable Scale: Part I, Step Law -- Optimal Hyperparameter Scaling Law in Large Language Model Pretraining
• Authors: Houyi Li, Wenzhen Zheng, Qiufeng Wang, Hanshan Zhang, Zili Wang, Shijie Xuyang, Yuantao Fan, Zhenyu Ding, Haoying Wang, Ning Ding, Shuigeng Zhou, Xiangyu Zhang, Daxin Jiang,
• Abstract summary: In this study, we conduct an unprecedented empirical investigationtext- training over 3,700 Large Language Models (LLMs) from scratch across 100 trillion tokens.<n>We empirically observe that, under fixed model size ($N$) and dataset size ($D$), the hyperparameter landscape exhibits convexity with a broad optimum.<n>Building on this insight, we formally define and empirically validate the Step Law: The optimal learning rate follows a power-law relationship with $N$ and $D$, while the optimal batch size is primarily influenced by $D$ and remains largely invariant to $N$.
• Score: 59.369484219304866
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The impressive capabilities of Large Language Models (LLMs) across diverse tasks are now well\text{-}established, yet their effective deployment necessitates careful hyperparameter optimization. Although existing methods have explored the influence of hyperparameters on model performance, a principled and generalizable framework across model architectures and data recipes remains absent. In this study, we conduct an unprecedented empirical investigation\text{-} training over 3,700 LLMs from scratch across 100 trillion tokens, consuming nearly one million NVIDIA H800 GPU hours to establish a universal Scaling Law for hyperparameter optimization in LLM Pre-training, called \textbf{Step Law}. We empirically observe that, under fixed model size ($N$) and dataset size ($D$), the hyperparameter landscape exhibits convexity with a broad optimum, substantially reducing the complexity of hyperparameter search. Building on this insight, we formally define and empirically validate the Step Law: The optimal learning rate follows a power-law relationship with $N$ and $D$, while the optimal batch size is primarily influenced by $D$ and remains largely invariant to $N$.Notably, our estimated optima deviate from the global best performance found via exhaustive search by merely \textbf{0.094\%} on the test set. To our best known, Step Law is the \textbf{first} that unifies different model shapes and structures, such as Mixture-of-Experts models and dense transformers, as well as establishes optimal hyperparameter scaling laws across diverse data recipes. We contribute a universal, plug-and-play optimal hyperparameter tool for the community, which is expected to advance efficient LLM training at scale. All experimental code, data and checkpoints are publicly available at \href{https://github.com/step-law/steplaw}{https://github.com/step-law/steplaw}.

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