On Exact Computation with an Infinitely Wide Neural Net

TL;DR

This paper introduces an efficient exact algorithm and GPU implementation for the Convolutional Neural Tangent Kernel (CNTK), enabling kernel-based classification of infinite-width CNNs with performance close to finite deep nets on CIFAR-10.

Contribution

It provides the first efficient exact computation method for CNTK and demonstrates its high performance as a kernel method, bridging deep learning and kernel theory.

Findings

CNTK achieves 10% higher accuracy than previous kernel methods on CIFAR-10.

Performance of CNTK is within 6% of finite deep CNNs without batch normalization.

The paper offers the first non-asymptotic proof linking wide neural nets to NTK-based kernel regression.

Abstract

How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its width --- namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers --- is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. An attraction of such ideas is that a pure kernel-based method is used to capture the power of a fully-trained deep net of infinite width. The current paper gives the first efficient exact algorithm for computing the extension of NTK to convolutional neural nets, which we call Convolutional NTK (CNTK), as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for the performance of a pure kernel-based method on CIFAR-10, being $10\%$ higher than the methods reported in [Novak et al., 2019], and only $6\%$ lower than the performance of the corresponding finite deep net architecture (once batch normalization, etc. are turned off). Theoretically, we also give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK.