Title: Theoretical and Empirical Study of Adversarial Examples. Abstract: Many techniques are developed to defend against adversarial examples at scale. So far, the most successful defenses generate adversarial examples during each training step and add them to the training data. Yet, this brings significant computational overhead.  In this paper, we investigate defenses against adversarial attacks. First, we propose feature smoothing, a simple data augmentation method with little computational overhead. Essentially, feature smoothing trains a neural network on virtual training data as an interpolation of features from a pair of samples, with the new label remaining the same as the dominant data point.  The intuition behind feature smoothing is to generate virtual data points as close as adversarial examples, and to avoid the computational burden of generating data during training. Our experiments on MNIST and CIFAR10 datasets explore different combinations of known regularization and data augmentation methods and show that feature smoothing with logit squeezing performs best for both adversarial and clean accuracy. Second, we propose an unified framework to understand the connections and differences among different efficient methods by analyzing the biases and variances of decision boundary. We show that under some symmetrical assumptions, label smoothing, logit squeezing, weight decay, mix up and feature smoothing all produce an unbiased estimation of the decision boundary with smaller estimated variance. All of those methods except weight decay are also stable when the assumptions no longer hold.