Weight decay is one of the standard tricks in the neural network toolbox, but the reasons for its regularization effect are poorly understood, and recent results have cast doubt on the traditional interpretation in terms of $L_2$ regularization.
 Literal weight decay has been shown to outperform $L_2$ regularization for optimizers for which they differ. 
 We empirically investigate weight decay for three optimization algorithms (SGD, Adam, and K-FAC) and a variety of network architectures. We identify three distinct mechanisms by which weight decay exerts a regularization effect, depending on the particular optimization algorithm and architecture: (1) increasing the effective learning rate, (2) approximately regularizing the input-output Jacobian norm, and (3) reducing the effective damping coefficient for second-order optimization. 
 Our results provide insight into how to improve the regularization of neural networks.
We investigate weight decay regularization for different optimizers and identify three distinct mechanisms by which weight decay improves generalization. Discusses the effect of weight decay on the training of deep network models with and without batch normalization and when using first/second order optimization methods and hypothesizes that a larger learning rate has a regularization effect.