Abstract
We consider the problem of verifying linear properties of neural networks. Despite their success in many classification and prediction tasks, neural networks may return unexpected results for certain inputs. This is highly problematic with respect to the application of neural networks for safety-critical tasks, e.g. in autonomous driving. We provide an overview of algorithmic approaches that aim to provide formal guarantees on the behaviour of neural networks. Moreover, we present new theoretical results with respect to the approximation of ReLU neural networks. On the other hand, we implement a solver for verification of ReLU neural networks which combines mixed integer programming with specialized branching and approximation techniques. To evaluate its performance, we conduct an extensive computational study. For that we use test instances based on the ACAS Xu system and the MNIST handwritten digit data set. The results indicate that our approach is very competitive with others, i.e. it outperforms the solvers of Bunel et al. (in: Bengio, Wallach, Larochelle, Grauman, Cesa-Bianchi, Garnett (eds) Advances in neural information processing systems (NIPS 2018), 2018) and Reluplex (Katz et al. in: Computer aided verification—29th international conference, CAV 2017, Heidelberg, Germany, July 24–28, 2017, Proceedings, 2017). In comparison to the solvers ReluVal (Wang et al. in: 27th USENIX security symposium (USENIX Security 18), USENIX Association, Baltimore, 2018a) and Neurify (Wang et al. in: 32nd Conference on neural information processing systems (NIPS), Montreal, 2018b), the number of necessary branchings is much smaller. Our solver is publicly available and able to solve the verification problem for instances which do not have independent bounds for each input neuron.
On a Hybridization of Deep Learning and Rough Set Based Granular Computing