Understanding the role of depth in the neural tangent kernel for overparameterized neural networks
This work provides theoretical insights into the role of depth in neural network behavior for researchers in machine learning theory, though it is incremental as it builds on existing kernel-based analyses.
The paper analyzes how increasing depth affects the neural tangent kernel in overparameterized ReLU networks, showing that the normalized limiting kernel approaches a matrix of ones while the closed-form solution converges to a fixed limit on the sphere.
Overparameterized fully-connected neural networks have been shown to behave like kernel models when trained with gradient descent, under mild conditions on the width, the learning rate, and the parameter initialization. In the limit of infinitely large widths and small learning rate, the kernel that is obtained allows to represent the output of the learned model with a closed-form solution. This closed-form solution hinges on the invertibility of the limiting kernel, a property that often holds on real-world datasets. In this work, we analyze the sensitivity of large ReLU networks to increasing depths by characterizing the corresponding limiting kernel. Our theoretical results demonstrate that the normalized limiting kernel approaches the matrix of ones. In contrast, they show the corresponding closed-form solution approaches a fixed limit on the sphere. We empirically evaluate the order of magnitude in network depth required to observe this convergent behavior, and we describe the essential properties that enable the generalization of our results to other kernels.