Towards Understanding Convergence and Generalization of AdamW
Pan Zhou, Xingyu Xie, Zhouchen Lin, Shuicheng Yan
Abstract
AdamW modifies Adam by adding a decoupled weight decay to decay network weights per training iteration. For adaptive algorithms, this decoupled weight decay does not affect specific optimization steps, and differs from the widely used <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -regularizer which changes optimization steps via changing the first- and second-order gradient moments. Despite its great practical success, for AdamW, its convergence behavior and generalization improvement over Adam and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -regularized Adam ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -Adam) remain absent yet. To solve this issue, we prove the convergence of AdamW and justify its generalization advantages over Adam and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -Adam. Specifically, AdamW provably converges but minimizes a dynamically regularized loss that combines vanilla loss and a dynamical regularization induced by decoupled weight decay, thus yielding different behaviors with Adam and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -Adam. Moreover, on both general nonconvex problems and PŁ-conditioned problems, we establish stochastic gradient complexity of AdamW to find a stationary point. Such complexity is also applicable to Adam and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -Adam, and improves their previously known complexity, especially for over-parametrized networks. Besides, we prove that AdamW enjoys smaller generalization errors than Adam and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -Adam from the Bayesian posterior aspect. This result, for the first time, explicitly reveals the benefits of decoupled weight decay in AdamW. Experimental results validate our theory.