  In tasks like node classification, image segmentation, and named-entity
recognition we have a classifier that simultaneously outputs multiple
predictions (a vector of labels) based on a single input, i.e. a single graph,
image, or document respectively. Existing adversarial robustness certificates
consider each prediction independently and are thus overly pessimistic for such
tasks. They implicitly assume that an adversary can use different perturbed
inputs to attack different predictions, ignoring the fact that we have a single
shared input. We propose the first collective robustness certificate which
computes the number of predictions that are simultaneously guaranteed to remain
stable under perturbation, i.e. cannot be attacked. We focus on Graph Neural
Networks and leverage their locality property - perturbations only affect the
predictions in a close neighborhood - to fuse multiple single-node certificates
into a drastically stronger collective certificate. For example, on the
Citeseer dataset our collective certificate for node classification increases
the average number of certifiable feature perturbations from $7$ to $351$.
