ADOPT: Modified Adam Can Converge with the Optimal Rate with Any Hyperparameters

Adaptive gradient methods based on exponential moving averages, such as Adam and RMSprop, are widely used for deep learning. However, it is known that they do not converge unless choosing hyperparameters in a problem-dependent manner. There have been many attempts to fix their convergence (e.g., AMSGrad), but they require an impractical assumption that the gradient noise is uniformly bounded. In this paper, we propose a new adaptive gradient method named ADOPT, which achieves the optimal convergence rate of $\mathcal{O} ( 1 / \sqrt{T} )$ with any hyperparameter choice without the bounded noise assumption. ADOPT addresses the non-convergence issue of Adam by removing the current gradient from the second moment estimate and changing the order of the momentum calculation and the scaling operation by the second moment estimate. We also conduct intensive numerical experiments, and verify that our ADOPT achieves competitive or even better results compared to Adam and its variants across a wide range of tasks, including image classification, generative modeling, natural language processing, and deep reinforcement learning.