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We study the approximation capacity of some variation spaces corresponding toshallow ReLU$^k$ neural networks. It is shown that sufficiently smoothfunctions are contained in these spaces with finite variation norms. Forfunctions with less smoothness, the approximation rates in terms of thevariation norm are established. Using these results, we are able to prove theoptimal approximation rates in terms of the number of neurons for shallowReLU$^k$ neural networks. It is also shown how these results can be used toderive approximation bounds for deep neural networks and convolutional neuralnetworks (CNNs). As applications, we study convergence rates for nonparametricregression using three ReLU neural network models: shallow neural network,over-parameterized neural network, and CNN. In particular, we show that shallowneural networks can achieve the minimax optimal rates for learning H\"olderfunctions, which complements recent results for deep neural networks. It isalso proven that over-parameterized (deep or shallow) neural networks canachieve nearly optimal rates for nonparametric regression.

Optimal rates of approximation by shallow ReLU$^k$ neural networks and applications to nonparametric regression