Neural Tangent Kernel: Convergence and Generalization in Neural Networks

TL;DR

This paper introduces the Neural Tangent Kernel (NTK), a theoretical framework that describes the training dynamics and generalization of infinitely wide neural networks, showing they follow kernel gradient descent and converge under certain conditions.

Contribution

It proves the convergence and constancy of NTK during training in the infinite-width limit, linking neural network training to kernel methods and providing insights into generalization and early stopping.

Findings

NTK converges to a fixed kernel in the infinite-width limit.

Training dynamics follow a linear differential equation.

NTK's positive-definiteness is proven for data on the sphere with non-polynomial non-linearity.

Abstract

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function $f_\theta$ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function $f_\theta$ follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.