PowerStep: Memory-Efficient Adaptive Optimization via ${\ell }_p$-Norm Steepest Descent

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 ${\ell }_p$-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 $O(1/\sqrt{T})$ term and a noise-dependent residual. Experiments on Transformers from 124M to 235B parameters show competitive validation quality while halving $\text{fp32}$ optimizer-state memory. Combined with uniform $\text{int8}$ quantization, PowerStep remains numerically stable and reduces optimizer-state memory by $\sim 8\times$ compared to $\text{fp32}$ AdamW. PowerStep thus provides a simple, memory-efficient alternative for large-scale training.