Learning with Neural Tangent Kernels in Near Input Sparsity Time

TL;DR

This paper introduces a fast algorithm for approximating Neural Tangent Kernels using randomized feature maps, enabling scalable training of kernel methods that outperform neural nets and existing approximations in large-scale tasks.

Contribution

The paper presents a near input sparsity time algorithm for approximating NTK with randomized features, including a convolutional extension, improving scalability and performance.

Findings

Outperforms neural nets and Nystrom methods on large-scale tasks

Provides a linear-time feature map for convolutional NTK

Enables scalable kernel approximation with high accuracy

Abstract

The Neural Tangent Kernel (NTK) characterizes the behavior of infinitely wide neural nets trained under least squares loss by gradient descent. However, despite its importance, the super-quadratic runtime of kernel methods limits the use of NTK in large-scale learning tasks. To accelerate kernel machines with NTK, we propose a near input sparsity time algorithm that maps the input data to a randomized low-dimensional feature space so that the inner product of the transformed data approximates their NTK evaluation. Our transformation works by sketching the polynomial expansions of arc-cosine kernels. Furthermore, we propose a feature map for approximating the convolutional counterpart of the NTK, which can transform any image using a runtime that is only linear in the number of pixels. We show that in standard large-scale regression and classification tasks a linear regressor trained on our features outperforms trained Neural Nets and Nystrom approximation of NTK kernel.