Deep Nonparametric Regression on Approximately Low-dimensional Manifolds

In this paper, we study the properties of nonparametric least squares regression using deep neural networks. We derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer for feedforward deep neural regression. Our error bounds achieve the minimax optimal rate and significantly improve over the existing ones in the sense that they depend linearly or quadratically on the dimension d of the predictor, instead of exponentially on d. We show that the neural regression estimator can circumvent the curse of dimensionality under the assumption that the predictor is supported on an approximate low-dimensional manifold. This assumption differs from the structural condition imposed on the target regression function and is weaker and more realistic than the exact low-dimensional manifold support assumption in the existing literature. We investigate how the prediction error of the neural regression estimator depends on the structure of neural networks and propose a notion of network relative efficiency between two types of neural networks, which provides a quantitative measure for evaluating the relative merits of different network structures. Our results are derived under weaker assumptions on the data distribution, the target regression function and the neural network structure than those in the existing literature.