# SWAN: SGD with Normalization and Whitening Enables Stateless LLM Training

arXiv: https://arxiv.org/abs/2412.13148 · HTML: https://mlsanity.com/paper/2412.13148/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 60 of its 76 references that are in the map.
- First posted on arXiv 17 Dec 2024.

## Builds on (most cited first, 30 of 60)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Reducing Activation Recomputation in Large Transformer Models](https://mlsanity.com/paper/2205.05198/) (2022)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis](https://mlsanity.com/paper/1806.03884/) (2018)
- [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)
- [Full Parameter Fine-tuning for Large Language Models with Limited Resources](https://mlsanity.com/paper/2306.09782/) (2023)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)

## Built on by (most cited first, 20 of 20)

- [Memory-Efficient LLM Pretraining via Minimalist Optimizer Design](https://mlsanity.com/paper/2506.16659/) (2025)
- [Mano: Restriking Manifold Optimization for LLM Training](https://mlsanity.com/paper/2601.23000/) (2026)
- [Powering Up Zeroth-Order Training via Subspace Gradient Orthogonalization](https://mlsanity.com/paper/2602.17155/) (2026)
- [On the Width Scaling of Neural Optimizers Under Matrix Operator Norms I: Row/Column Normalization and Hyperparameter Transfer](https://mlsanity.com/paper/2603.09952/) (2026)
- [Gradient Multi-Normalization for Stateless and Scalable LLM Training](https://mlsanity.com/paper/2502.06742/) (2025)
- [Reparameterized LLM Training via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2506.08001/) (2025)
- [FedMuon: Federated Learning with Bias-corrected LMO-based Optimization](https://mlsanity.com/paper/2509.26337/) (2025)
- [Good Arguments Against the People Pleasers: How Reasoning Mitigates (Yet Masks) LLM Sycophancy](https://mlsanity.com/paper/2603.16643/) (2026)
- [Stabilizing Native Low-Rank LLM Pretraining](https://mlsanity.com/paper/2602.12429/) (2026)
- [Taming Momentum: Rethinking Optimizer States Through Low-Rank Approximation](https://mlsanity.com/paper/2602.24283/) (2026)
- [GWT: Scalable Optimizer State Compression for Large Language Model Training](https://mlsanity.com/paper/2501.07237/) (2025)
- [Subspace Control: Turning Constrained Model Steering into Controllable Spectral Optimization](https://mlsanity.com/paper/2604.04231/) (2026)
- [SAGE: Sign-Adaptive Gradient for Memory-Efficient LLM Optimization](https://mlsanity.com/paper/2604.07663/) (2026)
- [Demystifying Manifold Constraints in LLM Pre-training](https://mlsanity.com/paper/2605.04418/) (2026)
- [Budget-aware Auto Optimizer Configurator](https://mlsanity.com/paper/2605.04711/) (2026)
- [Muse: Representation Geometry of Muon Beyond Normalized Momentum](https://mlsanity.com/paper/2607.14536/) (2026)
- [PowerStep: Memory-Efficient Adaptive Optimization via ℓp-Norm Steepest Descent](https://mlsanity.com/paper/2605.10335/) (2026)
- [Blog: Survey of Optimizers](https://mlsanity.com/paper/2608.28557/) (2026)
- [Normalize-Then-Precondition: A Hierarchical Approach to Marginal Scale and Interaction Geometry for LLM Training](https://mlsanity.com/paper/2609.36692/) (2026)
- [The Row Normalization Puzzle in Muon](https://mlsanity.com/paper/2609.39114/) (2026)

## Abstract (arXiv)

Adaptive optimizers such as Adam (Kingma & Ba, 2015) have been central to the success of large language models. However, they often require to maintain optimizer states throughout training, which can result in memory requirements several times greater than the model footprint. This overhead imposes constraints on scalability and computational efficiency. Stochastic Gradient Descent (SGD), in contrast, is a stateless optimizer, as it does not track state variables during training. Consequently, it achieves optimal memory efficiency. However, its capability in LLM training is limited (Zhao et al., 2024b). In this work, we show that pre-processing SGD in a stateless manner can achieve the same performance as the Adam optimizer for LLM training, while drastically reducing the memory cost. Specifically, we propose to pre-process the instantaneous stochastic gradients using normalization and whitening. We show that normalization stabilizes gradient distributions, and whitening counteracts the local curvature of the loss landscape. This results in SWAN (SGD with Whitening And Normalization), a stochastic optimizer that eliminates the need to store any optimizer states. Empirically, SWAN has the same memory footprint as SGD, achieving ≈ 50% reduction on total end-to-end memory compared to Adam. In language modeling tasks, SWAN demonstrates comparable or even better performance than Adam: when pre-training the LLaMA model with 350M and 1.3B parameters, SWAN achieves a 2x speedup by reaching the same evaluation perplexity using half as many tokens.