PowerStep: Memory-Efficient Adaptive Optimization via $\ell_p$-Norm Steepest Descent

Adaptive optimizers, most notably Adam, have become the default standard for training large-scale neural networks such as Transformers. These methods maintain running estimates of gradient first and second moments, incurring substantial memory overhead. We introduce PowerStep, a memory-efficient optimizer that achieves coordinate-wise adaptivity without storing second-moment statistics. Motivated by steepest descent under an $\ell_p$-norm geometry, we show that applying a nonlinear transform directly to a momentum buffer yields coordinate-wise adaptivity. We prove that PowerStep converges at the optimal $O(1/\sqrt{T})$ rate for non-convex stochastic optimization. Extensive experiments on Transformer models ranging from 124M to 235B parameters demonstrate that PowerStep matches Adam's convergence speed while halving optimizer memory. Furthermore, when combined with aggressive \texttt{int8} quantization, PowerStep remains numerically stable and reduces optimizer memory by $\sim\!8\times$ compared to full-precision Adam. PowerStep thus provides a principled, scalable and resource-efficient alternative for large-scale training. Code is available at https://github.com/yaolubrain/PowerStep.