Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers

TL;DR

This paper proves that overparameterized neural networks, including two and three-layer models with smooth activations, can learn complex functions efficiently with polynomial time and sample complexity, surpassing NTK limitations.

Contribution

It introduces a new quadratic approximation framework for neural networks, enabling analysis beyond NTK and demonstrating learnability of certain classes with fewer parameters.

Findings

Overparameterized networks can learn notable concept classes.

SGD can train these networks efficiently in polynomial time.

Sample complexity is nearly independent of network size.

Abstract

The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesn't the trained network overfit when it is overparameterized? In this work, we prove that overparameterized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network. On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic approximation of the neural network (that can be viewed as a second-order variant of NTK), and connect it to the SGD theory of escaping saddle points.