  This paper investigates a family of methods for defending against adversarial
attacks that owe part of their success to creating a noisy, discontinuous, or
otherwise rugged loss landscape that adversaries find difficult to navigate. A
common, but not universal, way to achieve this effect is via the use of
stochastic neural networks. We show that this is a form of gradient
obfuscation, and propose a general extension to gradient-based adversaries
based on the Weierstrass transform, which smooths the surface of the loss
function and provides more reliable gradient estimates. We further show that
the same principle can strengthen gradient-free adversaries. We demonstrate the
efficacy of our loss-smoothing method against both stochastic and
non-stochastic adversarial defences that exhibit robustness due to this type of
obfuscation. Furthermore, we provide analysis of how it interacts with
Expectation over Transformation; a popular gradient-sampling method currently
used to attack stochastic defences.
