Robustness Verifcation in Neural Networks

Neural networks are widely used in all kinds of data processing, especially on seemingly unfeasible tasks such as image [15] and language recognition [10], as well as applications in medicine [16], and prediction of stock markets [6], just to mention a few. Khan et al. [14] provide a survey of such applications, a mathematically oriented textbook concerning structural issues related to Deep Neural Networks is provided by [3]. Neural networks are nowadays also made use of in safety-critical systems like autonomous driving [8] or power grid management. In such a setting, when security issues become important, aspects of certification come into play [7, 11, 17]. If we for example want provable guarantees for certain scenarios to be unreachable, we first need to formulate them as constraints and precisely state for which property of a network we want verification. In the present paper we are interested in studying certain verification problems for NNs in form of particular robustness and minimization problems such as: How will a network react to a small perturbation of the input [9]? And how likely is a network to change the classification of an input that is altered a little? These probabilities are crucial when for example a self-driving car is supposed to recognize a speed limit, and they have already been tackled in practical settings by simulations and heuristic algorithms.