用动量自身做归一化，让大模型训练更高效

We introduce AdamS, a simple yet effective alternative to Adam for large language model (LLM) pretraining and post-training. By leveraging a novel denominator, i.e., the root of weighted sum of squares of the momentum and the current gradient, AdamS eliminates the need for second-moment estimates. Hence, AdamS is efficient, matching the memory and compute footprint of SGD with momentum while delivering superior optimization performance. Moreover, AdamS is easy to adopt: it can directly inherit hyperparameters of AdamW, and is entirely model-agnostic, integrating seamlessly into existing pipelines without modifications to optimizer APIs or architectures. The motivation behind AdamS stems from the observed $(L_0, L_1)$ smoothness properties in transformer objectives, where local smoothness is governed by gradient magnitudes that can be further approximated by momentum magnitudes. We establish rigorous theoretical convergence guarantees and provide practical guidelines for hyperparameter selection. Empirically, AdamS demonstrates strong performance in various tasks, including pre-training runs on GPT-2 and Llama2 (up to 13B parameters) and reinforcement learning in post-training regimes. With its efficiency, simplicity, and theoretical grounding, AdamS stands as a compelling alternative to existing optimizers.