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.

This paper has two main contributions. First, the convolutional extension of the neural tangent kernel (CNTK) was proposed and then an algorithm using CNTKs for "exact computation with an infinitely wide neural net" was designed. The algorithm allows squared-loss kernel regression with CNTKs corresponding to infinitely wide vanilla CNNs with ReLU activation as well as those also with global average pooling. Its time complexity is linear in the depth and quadratic in the amount of data and the height and width of the images. This time complexity, in previous papers, was believed to be impossible.

The claim of this paper is clear and easy to read. Although the composition of CNTK is straightforward and its exact computation in ReLU networks seems to be a bit trivial, this is a natural direction to be explored and some researchers will be interested in experimental results reported in this paper. The non-asymptotic proof is also a novel contribution and will be helpful to further develop the theory of NTK. Basically, I believe that this paper is worth to be published. I think that clarifying the following points will further increase the significance of this paper.

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.

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.