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

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