Before revision:  Moreover, we reveal the relation between the flatness of the loss function with respect to input data and the smoothness of the adversarial loss with respect to parameters. Our analysis implies that if we flatten the loss function with respect to input data, the smoothness of adversarial loss tends to decrease .
After revision:  Moreover, we reveal the relation between the flatness of the loss function with respect to input data , the Lipschitz constant of the gradient of adversarial loss tends to increase. To address the non-smoothness, we show that EntropySGD smoothens the non-smooth loss and improves the performance of adversarial training .