# SAGE: Sign-Adaptive Gradient for Memory-Efficient LLM Optimization

arXiv: https://arxiv.org/abs/2604.07663 · HTML: https://mlsanity.com/paper/2604.07663/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 30 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [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)
- [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)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [One weird trick for parallelizing convolutional neural networks](https://mlsanity.com/paper/1404.5997/) (2014)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [Is C4 Dataset Optimal for Pruning? An Investigation of Calibration Data for LLM Pruning](https://mlsanity.com/paper/2410.07461/) (2024)
- [SWAN: SGD with Normalization and Whitening Enables Stateless LLM Training](https://mlsanity.com/paper/2412.13148/) (2024)

## Built on by (most cited first, 1 of 1)

- [MESH: Memory-Efficient Sinkhorn Optimization for Mixture-of-Experts Training](https://mlsanity.com/paper/2608.04407/) (2026)

## Abstract (arXiv)

The AdamW optimizer, while standard for LLM pretraining, is a critical memory bottleneck, consuming optimizer states equivalent to twice the model's size. Although light-state optimizers like SinkGD attempt to address this issue, we identify the embedding layer dilemma: these methods fail to handle the sparse, high-variance gradients inherent to embeddings, forcing a hybrid design that reverts to AdamW and partially negates the memory gains. We propose SAGE (Sign Adaptive GradiEnt), a novel optimizer that resolves this dilemma by replacing AdamW in this hybrid structure. SAGE combines a Lion-style update direction with a new, memory-efficient O(d) adaptive scale. This scale acts as a "safe damper," provably bounded by 1.0, which tames high-variance dimensions more effectively than existing methods. This superior stability allows SAGE to achieve better convergence. On Llama models up to 1.3B parameters, our SAGE-based hybrid achieves new state-of-the-art perplexity, outperforming all baselines, including SinkGD hybrid, while significantly reducing optimizer state memory.