  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.
