Towards an Understanding of Residual Networks Using Neural Tangent Hierarchy (NTH) (CSIAM Transactions on Applied Mathematics, 2022)
Gradient descent yields zero training loss in polynomial time for deep neural networks despite non-convex nature of the objective function. The behavior ofnetwork in the infinite width limit trained by gradient descent can be described by theNeural Tangent Kernel (NTK) introduced in [25]. In this paper, we study dynamics ofthe NTK for finite width Deep Residual Network (ResNet) using the neural tangenthierarchy (NTH) proposed in [24]. For a ResNet with smooth and Lipschitz activationfunction, we reduce the requirement on the layer width $m$ with respect to the numberof training samples $n$ from quartic to cubic. Our analysis suggests strongly that theparticular skip-connection structure of ResNet is the main reason for its triumph overfully-connected network.