A Minimalist Optimizer Design for LLM Pretraining

Training large language models (LLMs) typically relies on adaptive optimizers such as Adam, which require significant memory to maintain first- and second-moment matrices, which are known as optimizer states. While recent works such as GaLore, Fira and APOLLO have proposed state-compressed variants to reduce memory consumption, a fundamental question remains: What is the minimal amount of optimizer state that is truly necessary to retain state-of-the-art performance in LLM pretraining? In this work, we systematically investigate this question using a bottom-up approach. We find that two (memory- and compute-efficient) optimization techniques are particularly effective: (1) column-wise gradient normalization significantly boosts the performance of plain SGD without requiring momentum; and (2) adding first-order momentum only to the output layer -- where gradient variance is highest -- yields performance competitive with fully adaptive methods such as Muon. Based on these insights, we propose SCALE (Stochastic Column-normAlized Last-layer momEntum), a new optimizer that combines column-normalized SGD with last-layer momentum, where column normalization refers to normalizing the gradient along the output dimension. Across multiple LLaMA models (60M–1B), SCALE matches or exceeds the performance of Adam while using only 35–45% of the total memory. It also consistently outperforms memory-efficient optimizers such as GaLore, Fira and APOLLO, making it a strong candidate for large-scale pretraining under memory constraints.