Scaling Neural Tangent Kernels via Sketching and Random Features

The Neural Tangent Kernel (NTK) characterizes the behavior of infinitely-wide neural networks trained under least squares loss by gradient descent. Recent works also report that NTK regression can outperform finitely-wide neural networks trained on small-scale datasets. However, the computational complexity of kernel methods has limited its use in large-scale learning tasks. To accelerate learning with NTK, we design a near input-sparsity time approximation algorithm for NTK, by sketching the polynomial expansions of arc-cosine kernels: our sketch for the convolutional counterpart of NTK (CNTK) can transform any image using a linear runtime in the number of pixels. Furthermore, we prove a spectral approximation guarantee for the NTK matrix, by combining random features (based on leverage score sampling) of the arc-cosine kernels with a sketching algorithm. We benchmark our methods on various large-scale regression and classification tasks and show that a linear regressor trained on our CNTK features matches the accuracy of exact CNTK on CIFAR-10 dataset while achieving 150× speedup. However, the NTK-based approaches encounter the computational bottlenecks of kernel learning. In particular, for a dataset of n images x 1 , x 2 , . . . x n ∈ R d×d , only writing down the CNTK kernel matrix requires Ω d 4 • n 2 operations [5] . Running regression or PCA on the resulting kernel matrix takes additional cubic time in n, which is infeasible in large-scale setups.