Adaptive first-order optimizers are fundamental tools in deep learning, although they may suffer from poor generalization due to the nonuniform gradient scaling. In this work, we propose AdamL, a novel variant of the Adam optimizer, that takes into account the loss function information to attain better generalization results. We provide sufficient conditions that together with the Polyak-Lojasiewicz inequality, ensure the linear convergence of AdamL. As a byproduct of our analysis, we prove similar convergence properties for the EAdam, and AdaBelief optimizers. Experimental results on benchmark functions show that AdamL typically achieves either the fastest convergence or the lowest objective function values when compared to Adam, EAdam, and AdaBelief. These superior performances are confirmed when considering deep learning tasks such as training convolutional neural networks, training generative adversarial networks using vanilla convolutional neural networks, and long short-term memory networks. Finally, in the case of vanilla convolutional neural networks, AdamL stands out from the other Adam's variants and does not require the manual adjustment of the learning rate during the later stage of the training.

Many adaptive optimization methods have been proposed and used in deep learning, in which Adam is regarded as the default algorithm and widely used in many deep learning frameworks. Recently, many variants of Adam, such as Adabound, RAdam and Adabelief, have been proposed and show better performance than Adam. However, these variants mainly focus on changing the stepsize by making differences on the gradient or the square of it. Motivated by the fact that suitable damping is important for the success of powerful second-order optimizers, we discuss the impact of the constant $\epsilon$ for Adam in this paper. Surprisingly, we can obtain better performance than Adam simply changing the position of $\epsilon$. Based on this finding, we propose a new variant of Adam called EAdam, which doesn't need extra hyper-parameters or computational costs. We also discuss the relationships and differences between our method and Adam. Finally, we conduct extensive experiments on various popular tasks and models. Experimental results show that our method can bring significant improvement compared with Adam. Our code is available at https://github.com/yuanwei2019/EAdam-optimizer.