Bounding the Complexity of Formally Verifying Neural Networks: A Geometric Approach

In this paper, we consider the computational complexity of formally verifying the behavior of Rectified Linear Unit (ReLU) Neural Networks (NNs), where verification entails determining whether the NN satisfies convex polytopic specifications. Specifically, we show that for two different NN architectures -- shallow NNs and Two-Level Lattice (TLL) NNs -- the verification problem with (convex) polytopic constraints is polynomial in the number of neurons in the NN to be verified, when all other aspects of the verification problem held fixed. We achieve these complexity results by exhibiting explicit (but similar) verification algorithms for each type of architecture. Both algorithms efficiently translate the NN parameters into a partitioning of the NN's input space by means of hyperplanes; this has the effect of partitioning the original verification problem into polynomially many sub-verification problems derived from the geometry of the neurons. We show that these sub-problems may be chosen so that the NN is purely affine within each, and hence each sub-problem is solvable in polynomial time by means of a Linear Program (LP). Thus, a polynomial-time algorithm for the original verification problem can be obtained using known algorithms for enumerating the regions in a hyperplane arrangement. Finally, we adapt our proposed algorithms to the verification of dynamical systems, specifically when these NN architectures are used as state-feedback controllers for LTI systems. We further evaluate the viability of this approach numerically.