Universal Approximation with Certified Networks
Maximilian Baader, Matthew Mirman, Martin Vechev · Repository for Publications and Research Data (ETH Zurich) · 2020
Training neural networks to be certifiably robust is critical to ensure their safety against adversarial attacks. However, while promising, state-of-the-art results with certified training are far from satisfactory. Currently, it is very difficult to train a neural network that is both accurate and certifiably robust. In this work we take a step towards addressing this challenge. We prove that for every continuous function $f$, there exists a network $n$ such that: (i) $n$ approximates $f$ arbitrarily close, and (ii) simple interval bound propagation of a region $B$ through $n$ yields a result that is arbitrarily close to the optimal output of $f$ on $B$. Our result can be seen as a Universal Approximation Theorem for interval-certified ReLU networks. To the best of our knowledge, this is the first work to prove the existence of accurate, interval-certified networks.