Weight decay is a simple yet powerful regularization technique that has been very widely used in training of deep neural networks (DNNs). While weight decay has attracted much attention, previous studies fail to discover some overlooked pitfalls on large gradient norms resulted by weight decay. In this paper, we discover that, weight decay can unfortunately lead to large gradient norms at the final phase (or the terminated solution) of training, which often indicates bad convergence and poor generalization. To mitigate the gradient-norm-centered pitfalls, we present the first practical scheduler for weight decay, called the Scheduled Weight Decay (SWD) method that can dynamically adjust the weight decay strength according to the gradient norm and significantly penalize large gradient norms during training. Our experiments also support that SWD indeed mitigates large gradient norms and often significantly outperforms the conventional constant weight decay strategy for Adaptive Moment Estimation (Adam).