On the $O(\frac{\sqrt{d}}{K^{1/4}})$ Convergence Rate of AdamW Measured by $\ell_1$ Norm

As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood. This paper establishes the convergence rate $\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_1\right]\leq O(\frac{\sqrt{d}C}{K^{1/4}})$ for AdamW measured by $\ell_1$ norm, where $K$ represents the iteration number, $d$ denotes the model dimension, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $||\nabla f(x)||_2\ll ||\nabla f(x)||_1\leq \sqrt{d}||\nabla f(x)||_2$ for any high-dimensional vector $x$ and $E\left[||\nabla f(x)||_1\right]\geq\sqrt{\frac{2d}{\pi}}E\left[||\nabla f(x)||_2\right]$ when each element of $\nabla f(x)$ is generated from Gaussian distribution $\mathcal N(0,1)$. Empirically, our experimental results on real-world deep learning tasks reveal $||\nabla f(x)||_1=\varTheta(\sqrt{d})||\nabla f(x)||_2$. Both support that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[||\nabla f(x^k)||_2\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case. We also extend our result to NAdamW, an AdamW variant that employs a double-momentum mechanism, and demonstrate that it maintains the same convergence rate.