# PowerStep: Memory-Efficient Adaptive Optimization via ℓp-Norm Steepest Descent

arXiv: https://arxiv.org/abs/2605.10335 · HTML: https://mlsanity.com/paper/2605.10335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 57 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [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)
- [8-bit Optimizers via Block-wise Quantization](https://mlsanity.com/paper/2110.02861/) (2021)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [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)
- [Convergence of Adam Under Relaxed Assumptions](https://mlsanity.com/paper/2304.13972/) (2023)
- [Memory Efficient Optimizers with 4-bit States](https://mlsanity.com/paper/2309.01507/) (2023)
- [Toward Understanding Why Adam Converges Faster Than SGD for Transformers](https://mlsanity.com/paper/2306.00204/) (2023)
- [Deconstructing What Makes a Good Optimizer for Language Models](https://mlsanity.com/paper/2407.07972/) (2024)
- [Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity](https://mlsanity.com/paper/2310.17998/) (2023)

## Abstract (arXiv)

Adaptive optimizers such as Adam are standard for training Transformers, but storing gradient first and second moments incurs substantial memory overhead. We introduce PowerStep, a memory-efficient optimizer that achieves coordinate-wise adaptivity without storing second-moment statistics. Motivated by ℓp-norm steepest descent, PowerStep applies a signed-power transform directly to one momentum buffer. We establish a finite-horizon stationarity bound for exact, unregularized updates, with an O(1/√T) term and a noise-dependent residual. Experiments on Transformers from 124M to 235B parameters show competitive validation quality while halving textttfp32 optimizer-state memory relative to AdamW. Combined with uniform textttint8 quantization, PowerStep remains numerically stable and reduces optimizer-state memory by ~8× compared to textttfp32 AdamW. PowerStep thus provides a simple, memory-efficient alternative for large-scale training.