  It is shown that over-parameterized neural networks can achieve minimax
optimal rates of convergence (up to logarithmic factors) for learning functions
from certain smooth function classes, if the weights are suitably constrained
or regularized. Specifically, we consider the nonparametric regression of
estimating an unknown $d$-variate function by using shallow ReLU neural
networks. It is assumed that the regression function is from the H\"older space
with smoothness $\alpha<(d+3)/2$ or a variation space corresponding to shallow
neural networks, which can be viewed as an infinitely wide neural network. In
this setting, we prove that least squares estimators based on shallow neural
networks with certain norm constraints on the weights are minimax optimal, if
the network width is sufficiently large. As a byproduct, we derive a new
size-independent bound for the local Rademacher complexity of shallow ReLU
neural networks, which may be of independent interest.
