Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. Our bounds also shed light on the advantage of using ResNet over the fully connected feedforward architecture; our bound requires the number of neurons per layer scaling exponentially with depth for feedforward networks whereas for ResNet the bound only requires the number of neurons per layer scaling polynomially with depth. We further extend our analysis to deep residual convolutional neural networks and obtain a similar convergence result.

通过分析神经网络架构的格拉姆矩阵的结构，证明了梯度下降法在针对深度超参数神经网络ResNet的多项式时间内实现零训练损失，并且进一步将该分析扩展到了深度残差卷积神经网络并获得了类似的收敛结果。

梯度下降找到深度神经网络的全局最小值