Neural Kernels Without Tangents

TL;DR

This paper explores the relationship between neural networks and kernel methods, introducing an algebra for compositional kernels that correlates with neural tangent kernels and demonstrates competitive performance on CIFAR10.

Contribution

It develops an algebraic framework for creating compositional kernels from features, linking them to neural tangent kernels and neural networks, and compares their performance.

Findings

Compositional kernels achieve 90% accuracy on CIFAR10.

Neural networks outperform both NTKs and compositional kernels on small datasets.

A simple neural network architecture reaches 96% accuracy on CIFAR10.

Abstract

We investigate the connections between neural networks and simple building blocks in kernel space. In particular, using well established feature space tools such as direct sum, averaging, and moment lifting, we present an algebra for creating "compositional" kernels from bags of features. We show that these operations correspond to many of the building blocks of "neural tangent kernels (NTK)". Experimentally, we show that there is a correlation in test error between neural network architectures and the associated kernels. We construct a simple neural network architecture using only 3x3 convolutions, 2x2 average pooling, ReLU, and optimized with SGD and MSE loss that achieves 96% accuracy on CIFAR10, and whose corresponding compositional kernel achieves 90% accuracy. We also use our constructions to investigate the relative performance of neural networks, NTKs, and compositional kernels in the small dataset regime. In particular, we find that compositional kernels outperform NTKs and neural networks outperform both kernel methods.