  While deep learning has outperformed other methods for various tasks,
theoretical frameworks that explain its reason have not been fully established.
To address this issue, we investigate the excess risk of two-layer ReLU neural
networks in a teacher-student regression model, in which a student network
learns an unknown teacher network through its outputs. Especially, we consider
the student network that has the same width as the teacher network and is
trained in two phases: first by noisy gradient descent and then by the vanilla
gradient descent. Our result shows that the student network provably reaches a
near-global optimal solution and outperforms any kernel methods estimator (more
generally, linear estimators), including neural tangent kernel approach, random
feature model, and other kernel methods, in a sense of the minimax optimal
rate. The key concept inducing this superiority is the non-convexity of the
neural network models. Even though the loss landscape is highly non-convex, the
student network adaptively learns the teacher neurons.
