Given a neural network, training data, and a threshold, it was known that it is NP-hard to find weights for the neural network such that the total error is below the threshold. We determine the algorithmic complexity of this fundamental problem precisely, by showing that it is ER-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, ER is strictly larger than NP, our work implies that the problem of training neural networks is not even in NP.

中文翻译：

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训练神经网络是ER完成的

给定一个神经网络，训练数据和一个阈值，已知很难找到神经网络的权重以使总误差低于阈值是NP。通过证明它是ER完全的，我们可以精确地确定此基本问题的算法复杂性。这意味着直到确定多项式时间减少为止，问题等同于确定具有整数系数和实数未知数的多项式方程和不等式系统是否具有解决方案。如果人们普遍认为ER严格大于NP，则我们的工作意味着训练神经网络的问题甚至不在NP中。