Approximation and nonparametric estimation of ResNettype convolutional neural networks
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Proceedings of the 36th International Conference on Machine Learning, PMLR 97:49224931, 2019.
Abstract
Convolutional neural networks (CNNs) have been shown to achieve optimal approximation and estimation error rates (in minimax sense) in several function classes. However, previous analyzed optimal CNNs are unrealistically wide and difficult to obtain via optimization due to sparse constraints in important function classes, including the Hölder class. We show a ResNettype CNN can attain the minimax optimal error rates in these classes in more plausible situations – it can be dense, and its width, channel size, and filter size are constant with respect to sample size. The key idea is that we can replicate the learning ability of Fullyconnected neural networks (FNNs) by tailored CNNs, as long as the FNNs have blocksparse structures. Our theory is general in a sense that we can automatically translate any approximation rate achieved by blocksparse FNNs into that by CNNs. As an application, we derive approximation and estimation error rates of the aformentioned type of CNNs for the Barron and Hölder classes with the same strategy.
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