Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK

We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $xinmathbb{R}^d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $f^{star}(x) = a^{top}|W^{star}x|$, where $ainmathbb{R}^d$ is a nonnegative vector and $W^{star} inmathbb{R}^{dtimes d}$ is an orthonormal matrix. We show that an over-parametrized two-layer neural network with ReLU activation, trained by gradient descent from random initialization, can provably learn the ground truth network with population loss at most $o(1/d)$ in polynomial time with polynomial samples. On the other hand, we prove that any kernel method, including Neural Tangent Kernel, with a polynomial number of samples in $d$, has population loss at least $Omega(1 / d)$.