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We study the properties of nonparametric least squares regression using deepneural networks. We derive non-asymptotic upper bounds for the prediction errorof the empirical risk minimizer of feedforward deep neural regression. Ourerror bounds achieve minimax optimal rate and significantly improve over theexisting ones in the sense that they depend polynomially on the dimension ofthe predictor, instead of exponentially on dimension. We show that the neuralregression estimator can circumvent the curse of dimensionality under theassumption that the predictor is supported on an approximate low-dimensionalmanifold or a set with low Minkowski dimension. We also establish the optimalconvergence rate under the exact manifold support assumption. We investigatehow the prediction error of the neural regression estimator depends on thestructure of neural networks and propose a notion of network relativeefficiency between two types of neural networks, which provides a quantitativemeasure for evaluating the relative merits of different network structures. Toestablish these results, we derive a novel approximation error bound for theH\"older smooth functions with a positive smoothness index using ReLU activatedneural networks, which may be of independent interest. Our results are derivedunder weaker assumptions on the data distribution and the neural networkstructure than those in the existing literature.

Deep Nonparametric Regression on Approximate Manifolds: Non-Asymptotic Error Bounds with Polynomial Prefactors