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

The impressive capabilities of Large Language Models (LLMs) across diverse tasks are now well 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 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 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 0.094\% on the test set. To our best known, Step Law is the 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 https://github.com/step-law/steplaw