Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers
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Neural networks have great success in many machine learning applications, but the fundamental learning theory behind them remains largely unsolved. Learning neural networks is NP-hard, but in practice, simple algorithms like stochastic gradient descent (SGD) often produce good solutions. Moreover, it is observed that overparameterization --- designing networks whose number of parameters is larger than statistically needed to perfectly fit the data --- improves both optimization and generalization, appearing to contradict traditional learning theory. In this work, we extend the theoretical understanding of two and three-layer neural networks in the overparameterized regime. We prove that, using overparameterized neural networks, one can (improperly) learn some notable hypothesis classes, including two and three-layer neural networks with fewer parameters. Moreover, the learning process can be simply done by SGD or its variants in polynomial time using polynomially many samples. We also show that for a fixed sample size, the generalization error of the solution found by some SGD variant can be made almost independent of the number of parameters in the overparameterized network.
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