  Developing techniques for adversarial attack and defense is an important
research field for establishing reliable machine learning and its applications.
Many existing methods employ Gaussian random variables for exploring the data
space to find the most adversarial (for attacking) or least adversarial (for
defense) point. However, the Gaussian distribution is not necessarily the
optimal choice when the exploration is required to follow the complicated
structure that most real-world data distributions exhibit. In this paper, we
investigate how statistics of random variables affect such random walk
exploration. Specifically, we generalize the Boundary Attack, a
state-of-the-art black-box decision based attacking strategy, and propose the
L\'evy-Attack, where the random walk is driven by symmetric $\alpha$-stable
random variables. Our experiments on MNIST and CIFAR10 datasets show that the
L\'evy-Attack explores the image data space more efficiently, and significantly
improves the performance. Our results also give an insight into the recently
found fact in the whitebox attacking scenario that the choice of the norm for
measuring the amplitude of the adversarial patterns is essential.
