  The robustness of a neural network to adversarial examples can be provably
certified by solving a convex relaxation. If the relaxation is loose, however,
then the resulting certificate can be too conservative to be practically
useful. Recently, a less conservative robustness certificate was proposed,
based on a semidefinite programming (SDP) relaxation of the ReLU activation
function. In this paper, we describe a geometric technique that determines
whether this SDP certificate is exact, meaning whether it provides both a
lower-bound on the size of the smallest adversarial perturbation, as well as a
globally optimal perturbation that attains the lower-bound. Concretely, we
show, for a least-squares restriction of the usual adversarial attack problem,
that the SDP relaxation amounts to the nonconvex projection of a point onto a
hyperbola. The resulting SDP certificate is exact if and only if the projection
of the point lies on the major axis of the hyperbola. Using this geometric
technique, we prove that the certificate is exact over a single hidden layer
under mild assumptions, and explain why it is usually conservative for several
hidden layers. We experimentally confirm our theoretical insights using a
general-purpose interior-point method and a custom rank-2 Burer-Monteiro
algorithm.
