# AdamS: Momentum Itself Can Be A Normalizer for LLM Pretraining and Post-training

arXiv: https://arxiv.org/abs/2505.16363 · HTML: https://mlsanity.com/paper/2505.16363/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 44 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Scaling Language Models: Methods, Analysis & Insights from Training Gopher](https://mlsanity.com/paper/2112.11446/) (2021)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Large Scale Private Learning via Low-rank Reparametrization](https://mlsanity.com/paper/2106.09352/) (2021)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [A Theory on Adam Instability in Large-Scale Machine Learning](https://mlsanity.com/paper/2304.09871/) (2023)
- [Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD](https://mlsanity.com/paper/2302.06570/) (2023)
- [CAME: Confidence-guided Adaptive Memory Efficient Optimization](https://mlsanity.com/paper/2307.02047/) (2023)
- [AdaLomo: Low-memory Optimization with Adaptive Learning Rate](https://mlsanity.com/paper/2310.10195/) (2023)
- [Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case](https://mlsanity.com/paper/2307.11782/) (2023)

## Built on by (most cited first, 6 of 6)

- [Stochastic Approximation with Block Coordinate Optimal Stepsizes](https://mlsanity.com/paper/2507.08963/) (2025)
- [Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations](https://mlsanity.com/paper/2604.12968/) (2026)
- [FOAM: Blocked State Folding for Memory-Efficient LLM Training](https://mlsanity.com/paper/2512.07112/) (2025)
- [PowerStep: Memory-Efficient Adaptive Optimization via ℓp-Norm Steepest Descent](https://mlsanity.com/paper/2605.10335/) (2026)
- [SCAPE: Accurate and Efficient LLM Training with Extreme Sparse Communication](https://mlsanity.com/paper/2607.01678/) (2026)
- [When Does Momentum Help in Reinforcement Learning with Verifiable Rewards?](https://mlsanity.com/paper/2610.20413/) (2026)

## Abstract (arXiv)

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 (L0, L1) 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.