  Methods to certify the robustness of neural networks in the presence of input
uncertainty are vital in safety-critical settings. Most certification methods
in the literature are designed for adversarial or worst-case inputs, but
researchers have recently shown a need for methods that consider random input
noise. In this paper, we examine the setting where inputs are subject to random
noise coming from an arbitrary probability distribution. We propose a
robustness certification method that lower-bounds the probability that network
outputs are safe. This bound is cast as a chance-constrained optimization
problem, which is then reformulated using input-output samples to make the
optimization constraints tractable. We develop sufficient conditions for the
resulting optimization to be convex, as well as on the number of samples needed
to make the robustness bound hold with overwhelming probability. We show for a
special case that the proposed optimization reduces to an intuitive closed-form
solution. Case studies on synthetic, MNIST, and CIFAR-10 networks
experimentally demonstrate that this method is able to certify robustness
against various input noise regimes over larger uncertainty regions than prior
state-of-the-art techniques.
