  We develop a minimax rate analysis to describe the reason that deep neural
networks (DNNs) perform better than other standard methods. For nonparametric
regression problems, it is well known that many standard methods attain the
minimax optimal rate of estimation errors for smooth functions, and thus, it is
not straightforward to identify the theoretical advantages of DNNs. This study
tries to fill this gap by considering the estimation for a class of non-smooth
functions that have singularities on hypersurfaces. Our findings are as
follows: (i) We derive the generalization error of a DNN estimator and prove
that its convergence rate is almost optimal. (ii) We elucidate a phase diagram
of estimation problems, which describes the situations where the DNNs
outperform a general class of estimators, including kernel methods, Gaussian
process methods, and others. We additionally show that DNNs outperform harmonic
analysis based estimators. This advantage of DNNs comes from the fact that a
shape of singularity can be successfully handled by their multi-layered
structure.
