  Adam with decoupled weight decay, also known as AdamW, is widely acclaimed
for its superior performance in language modeling tasks, surpassing Adam with
$\ell_2$ regularization in terms of generalization and optimization. However,
this advantage is not theoretically well-understood. One challenge here is that
though intuitively Adam with $\ell_2$ regularization optimizes the $\ell_2$
regularized loss, it is not clear if AdamW optimizes a specific objective. In
this work, we make progress toward understanding the benefit of AdamW by
showing that it implicitly performs constrained optimization. More concretely,
we show in the full-batch setting, if AdamW converges with any non-increasing
learning rate schedule whose partial sum diverges, it must converge to a KKT
point of the original loss under the constraint that the $\ell_\infty$ norm of
the parameter is bounded by the inverse of the weight decay factor. This result
is built on the observation that Adam can be viewed as a smoothed version of
SignGD, which is the normalized steepest descent with respect to $\ell_\infty$
norm, and a surprising connection between normalized steepest descent with
weight decay and Frank-Wolfe.
