  Binarized Neural Networks (BNNs) have recently attracted significant interest
due to their computational efficiency. Concurrently, it has been shown that
neural networks may be overly sensitive to "attacks" - tiny adversarial changes
in the input - which may be detrimental to their use in safety-critical
domains. Designing attack algorithms that effectively fool trained models is a
key step towards learning robust neural networks. The discrete,
non-differentiable nature of BNNs, which distinguishes them from their
full-precision counterparts, poses a challenge to gradient-based attacks. In
this work, we study the problem of attacking a BNN through the lens of
combinatorial and integer optimization. We propose a Mixed Integer Linear
Programming (MILP) formulation of the problem. While exact and flexible, the
MILP quickly becomes intractable as the network and perturbation space grow. To
address this issue, we propose IProp, a decomposition-based algorithm that
solves a sequence of much smaller MILP problems. Experimentally, we evaluate
both proposed methods against the standard gradient-based attack (FGSM) on
MNIST and Fashion-MNIST, and show that IProp performs favorably compared to
FGSM, while scaling beyond the limits of the MILP.
