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

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