The stunning empirical successes of neural networks currently lack rigorous theoretical explanation. What form would such an explanation take, in the face of existing complexity-theoretic lower bounds? A first step might be to show that data generated by neural networks with a single hidden layer, smooth activation functions and benign input distributions can be learned efficiently. We demonstrate here a comprehensive lower bound ruling out this possibility: for a wide class of activation functions (including all currently used), and inputs drawn from any logconcave distribution, there is a family of one-hidden-layer functions whose output is a sum gate, that are hard to learn in a precise sense: any statistical query algorithm (which includes all known variants of stochastic gradient descent with any loss function) needs an exponential number of queries even using tolerance inversely proportional to the input dimensionality. Moreover, this hard family of functions is realizable with a small (sublinear in dimension) number of activation units in the single hidden layer. The lower bound is also robust to small perturbations of the true weights. Systematic experiments illustrate a phase transition in the training error as predicted by the analysis.

本文研究神经网络的理论解释，针对单个隐藏层、平滑激活函数和良好输入分布条件下生成的数据可否进行有效学习，证明了对于广泛的激活函数和任何对数凹分布的输入，存在一类单隐藏层函数，其输出为和门，难以以任何精度有效地学习，这一下界对权重的微小扰动具有鲁棒性，且通过实验验证了训练误差的相变现象。

神经网络学习复杂性