  Adam has been widely adopted for training deep neural networks due to less
hyperparameter tuning and remarkable performance. To improve generalization,
Adam is typically used in tandem with a squared $\ell_2$ regularizer (referred
to as Adam-$\ell_2$). However, even better performance can be obtained with
AdamW, which decouples the gradient of the regularizer from the update rule of
Adam-$\ell_2$. Yet, we are still lacking a complete explanation of the
advantages of AdamW. In this paper, we tackle this question from both an
optimization and an empirical point of view. First, we show how to re-interpret
AdamW as an approximation of a proximal gradient method, which takes advantage
of the closed-form proximal mapping of the regularizer instead of only
utilizing its gradient information as in Adam-$\ell_2$. Next, we consider the
property of "scale-freeness" enjoyed by AdamW and by its proximal counterpart:
their updates are invariant to component-wise rescaling of the gradients. We
provide empirical evidence across a wide range of deep learning experiments
showing a correlation between the problems in which AdamW exhibits an advantage
over Adam-$\ell_2$ and the degree to which we expect the gradients of the
network to exhibit multiple scales, thus motivating the hypothesis that the
advantage of AdamW could be due to the scale-free updates.
