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How well does a classic deep net architecture like AlexNet or VGG19 classifyon a standard dataset such as CIFAR-10 when its width --- namely, number ofchannels in convolutional layers, and number of nodes in fully-connectedinternal layers --- is allowed to increase to infinity? Such questions havecome to the forefront in the quest to theoretically understand deep learningand its mysteries about optimization and generalization. They also connect deeplearning to notions such as Gaussian processes and kernels. A recent paper[Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which capturesthe behavior of fully-connected deep nets in the infinite width limit trainedby gradient descent; this object was implicit in some other recent papers. Anattraction of such ideas is that a pure kernel-based method is used to capturethe power of a fully-trained deep net of infinite width.  The current paper gives the first efficient exact algorithm for computing theextension of NTK to convolutional neural nets, which we call Convolutional NTK(CNTK), as well as an efficient GPU implementation of this algorithm. Thisresults in a significant new benchmark for the performance of a purekernel-based method on CIFAR-10, being $10\%$ higher than the methods reportedin [Novak et al., 2019], and only $6\%$ lower than the performance of thecorresponding finite deep net architecture (once batch normalization, etc. areturned off). Theoretically, we also give the first non-asymptotic proof showingthat a fully-trained sufficiently wide net is indeed equivalent to the kernelregression predictor using NTK.

On Exact Computation with an Infinitely Wide Neural Net