Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks
This addresses the challenge of designing practical CNNs with theoretical guarantees for machine learning practitioners, though it is incremental as it builds on existing block-sparse FNN theory.
The paper tackles the problem of unrealistic width and sparse constraints in CNNs for achieving optimal error rates in function classes like Hölder, showing that a ResNet-type CNN can attain minimax optimal rates with constant width, channel size, and filter size, making it more plausible for optimization.
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 ResNet-type 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 Fully-connected neural networks (FNNs) by tailored CNNs, as long as the FNNs have \textit{block-sparse} structures. Our theory is general in a sense that we can automatically translate any approximation rate achieved by block-sparse 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.