[Paper Review] Neural Kernels Without Tangents

Vaishaal Shankar, Alex Chengyu Fang+5 more|arXiv (Cornell University)|Mar 4, 2020

Advanced Neural Network Applications27 references35 citations

TL;DR

The paper builds compositional kernels from bags of features that emulate neural network operations (convolution, pooling, nonlinearity) and compares kernel performance to neural nets and NTKs on vision and tabular data, showing strong correlations and practical kernel design insights.

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.

Motivation & Objective

• Investigate empirical links between neural network architectures and kernel spaces through compositional kernels.
• Develop a set of feature-space operations (concatenation, downsampling, embedding) to build expressive kernels from bags of features.
• Compute these compositional kernels directly from data without random feature approximations.
• Compare performance of neural networks, NTKs, and compositional kernels across CIFAR-10, CIFAR-100, MNIST, CIFAR-10.1, and 90 UCI datasets.
• Assess how preprocessing and data augmentation affect kernel vs. neural network performance.

Proposed method

• Define a bag-of-features framework with index set B and feature space H (e.g., an image as a bag of 3D color vectors).
• Introduce three kernel-preserving operations on bags: concatenation, downsampling (pooling), embedding.
• Derive kernel computations for each operation, enabling exact kernel construction from image data without explicit feature maps.
• Specialize the operations to images via input kernels, convolutions, average pooling, and ReLU or Gaussian embeddings.
• Relate compositional kernels to neural network architectures through depth-wise cascades and expected value calculations using random features, following Daniely et al. and Jacot et al.

Experimental results

Research questions

• RQ1 Do compositional kernels built from simple feature-space operations approximate the expressive power of neural networks and NTKs?
• RQ2 How do convolution, pooling, and nonlinearity in kernel space correspond to their neural network counterparts?
• RQ3 What is the empirical performance of Myrtle-family convolutional kernels versus NTKs and CNNs on CIFAR-10, CIFAR-100, and MNIST?
• RQ4 How does preprocessing (e.g., ZCA) and data augmentation influence kernel vs. neural network performance in small-data and standard regimes?
• RQ5 Can these kernel constructions extend to large-scale datasets (and what are the computational bottlenecks)?

Key findings

• A simple network with 3x3 convolutions, 2x2 average pooling, and ReLU reaches 96% accuracy on CIFAR-10 with SGD and MSE loss, while the corresponding compositional kernel reaches 90%.
• On CIFAR-10, compositional kernels outperform NTKs by a large margin under the same architecture family.
• In the small data regime, compositional kernels outperform NTKs, and neural networks outperform both kernel methods when properly tuned.
• On MNIST, convolutional kernels and networks achieve similar high accuracy, all outperforming non-convolution baselines (NTK, arccosine, Gaussian).
• On CIFAR-100, the best compositional kernel matches a CNN without augmentation, with improvements when augmentation or BatchNorm are applied; cross-entropy loss was easier to optimize than MSE for CNNs here.
• Across 90 UCI datasets, Gaussian kernels perform comparably or better than NTKs under a revised cross-validation evaluation protocol.