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

Abstract: We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $x\in\mathbb{R}<sup>d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $$f^{\star}(x) = a<sup>{\top}|W<sup>{\star}x|$$, where $a\in\mathbb{R}<sup>d$ is a nonnegative vector and $$W^{\star} \in\mathbb{R}^{d\times d}$$ 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 $\Omega(1 / d)$.