Provable Certificates for Adversarial Examples: Fitting a Ball in the Union of Polytopes

We propose a novel method for computing exact pointwise robustness of deep neural networks for all convex ℓ_p norms. Our algorithm, GeoCert, finds the largest ℓ_p ball centered at an input point x_0, within which the output class of a given neural network with ReLU nonlinearities remains unchanged. We relate the problem of computing pointwise robustness of these networks to that of computing the maximum norm ball with a fixed center that can be contained in a non-convex polytope. This is a challenging problem in general, however we show that there exists an efficient algorithm to compute this for polyhedral complices. Further we show that piecewise linear neural networks partition the input space into a polyhedral complex. Our algorithm has the ability to almost immediately output a nontrivial lower bound to the pointwise robustness which is iteratively improved until it ultimately becomes tight. We empirically show that our approach generates distance lower bounds that are tighter compared to prior work, under moderate time constraints.