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We develop a minimax rate analysis to describe the reason that deep neuralnetworks (DNNs) perform better than other standard methods. For nonparametricregression problems, it is well known that many standard methods attain theminimax optimal rate of estimation errors for smooth functions, and thus, it isnot straightforward to identify the theoretical advantages of DNNs. This studytries to fill this gap by considering the estimation for a class of non-smoothfunctions that have singularities on hypersurfaces. Our findings are asfollows: (i) We derive the generalization error of a DNN estimator and provethat its convergence rate is almost optimal. (ii) We elucidate a phase diagramof estimation problems, which describes the situations where the DNNsoutperform a general class of estimators, including kernel methods, Gaussianprocess methods, and others. We additionally show that DNNs outperform harmonicanalysis based estimators. This advantage of DNNs comes from the fact that ashape of singularity can be successfully handled by their multi-layeredstructure.

Advantage of Deep Neural Networks for Estimating Functions with Singularity on Hypersurfaces