Adversarial Robustness Certification for Bayesian Neural Networks
Matthew Wicker Andrea Patane Luca Laurenti Marta Kwiatkowska
We study the problem of certifying the robustness of Bayesian neural networks
(BNNs) to adversarial input perturbations. Given a compact set of input points
$T \subseteq \mathbb{R}^m$ and a set of output points $S \subseteq
\mathbb{R}^n$, we define two notions of robustness for BNNs in an adversarial
setting: probabilistic robustness and decision robustness. Probabilistic
robustness is the probability that for all points in $T$ the output of a BNN
sampled from the posterior is in $S$. On the other hand, decision robustness
considers the optimal decision of a BNN and checks if for all points in $T$ the
optimal decision of the BNN for a given loss function lies within the output
set $S$. Although exact computation of these robustness properties is
challenging due to the probabilistic and non-convex nature of BNNs, we present
a unified computational framework for efficiently and formally bounding them.
Our approach is based on weight interval sampling, integration, and bound
propagation techniques, and can be applied to BNNs with a large number of
parameters, and independently of the (approximate) inference method employed to
train the BNN. We evaluate the effectiveness of our methods on various
regression and classification tasks, including an industrial regression
benchmark, MNIST, traffic sign recognition, and airborne collision avoidance,
and demonstrate that our approach enables certification of robustness and
uncertainty of BNN predictions.