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The unprecedented success of deep learning (DL) makes it unchallenged when itcomes to classification problems. However, it is well established that thecurrent DL methodology produces universally unstable neural networks (NNs). Theinstability problem has caused an enormous research effort -- with a vastliterature on so-called adversarial attacks -- yet there has been no solutionto the problem. Our paper addresses why there has been no solution to theproblem, as we prove the following mathematical paradox: any training procedurebased on training neural networks for classification problems with a fixedarchitecture will yield neural networks that are either inaccurate or unstable(if accurate) -- despite the provable existence of both accurate and stableneural networks for the same classification problems. The key is that thestable and accurate neural networks must have variable dimensions depending onthe input, in particular, variable dimensions is a necessary condition forstability.  Our result points towards the paradox that accurate and stable neuralnetworks exist, however, modern algorithms do not compute them. This yields thequestion: if the existence of neural networks with desirable properties can beproven, can one also find algorithms that compute them? There are cases inmathematics where provable existence implies computability, but will this bethe case for neural networks? The contrary is true, as we demonstrate howneural networks can provably exist as approximate minimisers to standardoptimisation problems with standard cost functions, however, no randomisedalgorithm can compute them with probability better than 1/2.

The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks