Title: Is PGD-Adversarial Training Necessary? Alternative Training via a Soft-Quantization Network with Noisy-Natural Samples Only. Abstract: Recent work on adversarial attack and defense suggests that projected gradient descent (PGD) is a universal $l_\infty$ first-order attack, and PGD adversarial training can significantly improve network robustness against a wide range of first-order $l_\infty$-bounded attacks, represented as the state-of-the-art defense method. However, an obvious weakness of PGD adversarial training is its highly-computational cost in generating adversarial samples, making it computationally infeasible for large and high-resolution real datasets such as the ImageNet dataset. In addition, recent work also has suggested a simple ``close-form'' solution to a robust model on MNIST. Therefore, a natural question raised is that is PGD adversarial training really necessary for robust defense? In this paper, surprisingly, we give a negative answer by proposing a training paradigm that is comparable to PGD adversarial training on several standard datasets, while only using noisy-natural samples. Specifically, we reformulate the min-max objective in PGD adversarial training by a minimization problem to minimize the original network loss plus $l_1$ norms of its gradients evaluated on the inputs (including adversarial samples). The original loss can be solved by natural training; for the $l_1$-norm loss, we propose a computationally-feasible solution by embedding a differentiable soft-quantization layer after the input layer of a network. We show formally that the soft-quantization layer trained with noisy-natural samples is an alternative approach to minimizing the $l_1$-gradient norms as in PGD adversarial training. Extensive empirical evaluations on three standard datasets including MNIST, CIFAR-10 and ImageNet show that our proposed models are comparable to PGD-adversarially-trained models under PGD and BPDA attacks using both cross-entropy and $CW_\infty$ losses. Remarkably, our method achieves a 24X speed-up on MNIST while maintaining a comparable defensive ability, and for the first time fine-tunes a robust Imagenet model within only two days. Code for the experiments will be released on Github.