We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. 
 We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) 
decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter).
 We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-of-the-art results on CIFAR-10 and ImageNet32x32. 
 Our source code will become available after the review process.
Fixing weight decay regularization in adaptive gradient methods such as Adam Proposes idea to decouple the weight decay from the number of steps taken by the optimization process. The paper presents an alternative way to implement weight decay in Adam with empirical results shown Investigates weight decay issues lied in the SGD variants and proposes the decoupling method between weight decay and the gradient-based update.