Excess Risk of Two-Layer ReLU Neural Networks in Teacher-Student Settings and its Superiority to Kernel Methods

TL;DR

This paper demonstrates that two-layer ReLU neural networks trained in a teacher-student setting can achieve near-optimal performance and outperform kernel methods due to their non-convex learning dynamics, despite complex loss landscapes.

Contribution

It provides a theoretical analysis showing the superiority of neural networks over kernel methods in a teacher-student regression model, highlighting the role of non-convexity.

Findings

Student networks reach near-global optimal solutions.

Neural networks outperform kernel methods in minimax rate.

Non-convexity enables adaptive learning of teacher neurons.

Abstract

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.