The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in the paper "Neural tangent kernel: Convergence and generalization in neural networks" by A. Jacot, F. Gabriel, and C. Hongle, where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in some other recent papers. In the overparametrization regime, a fully-trained deep neural network is indeed equivalent to the kernel regression predictor using the limiting NTK. And the gradient descent achieves zero training loss for a deep overparameterized neural network. However, it was observed in the paper " Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks" that there is a performance gap between the kernel regression using the limiting NTK and the deep neural networks. This performance gap is likely to originate from the change of the NTK along training due to the finite width effect. The change of the NTK along the training is central to describe the generalization features of deep neural networks.

In the current paper, we study the dynamic of the NTK for finite width deep fully-connected neural networks. We derive an infinite hierarchy of ordinary differential equations, the neural tangent hierarchy (NTH) which captures the gradient descent dynamic of the deep neural network. Moreover, under certain conditions on the neural network width and the data set dimension, we prove that the truncated hierarchy of NTH approximates the dynamic of the NTK up to arbitrary precision. This description makes it possible to directly study the change of the NTK for deep neural networks, and sheds light on the observation that deep neural networks outperform kernel regressions using the corresponding limiting NTK.