What Can ResNet Learn Efficiently, Going Beyond Kernels?

How can neural networks such as ResNet \emph{efficiently} learn CIFAR-10 with test accuracy more than $96 \%$, while other methods, especially kernel methods, fall relatively behind? Can we more provide theoretical justifications for this gap? Recently, there is an influential line of work relating neural networks to kernels in the over-parameterized regime, proving they can learn certain concept class that is also learnable by kernels with similar test error. Yet, can neural networks provably learn some concept class \emph{better} than kernels? We answer this positively in the distribution-free setting. We prove neural networks can efficiently learn a notable class of functions, including those defined by three-layer residual networks with smooth activations, without any distributional assumption. At the same time, we prove there are simple functions in this class such that with the same number of training examples, the test error obtained by neural networks can be \emph{much smaller} than \emph{any} kernel method, including neural tangent kernels (NTK). The main intuition is that \emph{multi-layer} neural networks can implicitly perform hierarchal learning using different layers, which reduces the sample complexity comparing to ``one-shot'' learning algorithms such as kernel methods. In the end, we also prove a computation complexity advantage of ResNet with respect to other learning methods including linear regression over arbitrary feature mappings.