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

; Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:2613-2682, 2020.

#### Abstract

We consider the dynamic of gradient descent for learning a two-layer neural network. We assume the input $x\in\mathbb{R}^d$ is drawn from a Gaussian distribution and the label of $x$ satisfies $f^{\star}(x) = a^{\top}|W^{\star}x|$, where $a\in\mathbb{R}^d$ is a nonnegative vector and $W^{\star} \in\mathbb{R}^{d\times d}$ is an orthonormal matrix. We show that an \emph{over-parameterized} two layer neural network with ReLU activation, trained by gradient descent from \emph{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 $\Omega(1 / d)$.