统一稀疏与密集大模型的缩放规律，助力资源最优配置

Despite recent advancements of large language models (LLMs), optimally predicting the model size for LLM pretraining or allocating optimal resources still remains a challenge. Several efforts have addressed the challenge by proposing different empirical scaling laws, but almost all of them are architecture-specific (dense or sparse). In this work we revisit existing empirical scaling laws and propose a generalized scaling law to provide a unified framework that is applicable to both dense and sparse large language models. We evaluate and compare our proposed scaling law with existing scaling laws and demonstrate that our proposed scaling law captures the scaling behavior of existing scaling laws. Further, we show an IsoFLOP comparison between our proposed scaling law and the state-of-the-art scaling law to illustrate the effectiveness of our proposed scaling law for Mixture-of-Expert (MoE)-based very large LLMs like DeepSeek-V3. Our proposed scaling law can be used to estimate the best model hyperparameters (Model size, Tokens and Compute) for a given sparsity or to identify the optimal sparsity for the given model hyperparameters.