Abstract: Given a neural network, training data, and a threshold, finding weights for the neural network such that the total error is below the threshold is known to be NP-hard. We determine the algorithmic complexity of this fundamental problem precisely, by showing that it is $\exists\mathbb R$-complete. This means that the problem is equivalent, up to polynomial time reductions, to deciding whether a system of polynomial equations and inequalities with integer coefficients and real unknowns has a solution. If, as widely expected, $\exists\mathbb R$ is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP.Neural networks are usually trained using some variation of backpropagation. The result of this paper gives an explanation why techniques commonly used to solve big instances of NP-complete problems (such as SAT solvers, IP solvers, local search, dynamic programming, etc.) seem to be of no use to this task.