  Interval analysis (or interval bound propagation, IBP) is a popular technique
for verifying and training provably robust deep neural networks, a fundamental
challenge in the area of reliable machine learning. However, despite
substantial efforts, progress on addressing this key challenge has stagnated,
calling into question whether interval arithmetic is a viable path forward.
  In this paper we present two fundamental results on the limitations of
interval arithmetic for analyzing neural networks. Our main impossibility
theorem states that for any neural network classifying just three points, there
is a valid specification over these points that interval analysis can not
prove. Further, in the restricted case of one-hidden-layer neural networks we
show a stronger impossibility result: given any radius $\alpha < 1$, there is a
set of $O(\alpha^{-1})$ points with robust radius $\alpha$, separated by
distance $2$, that no one-hidden-layer network can be proven to classify
robustly via interval analysis.
