  This paper considers the problem of nonparametric quantile regression under
the assumption that the target conditional quantile function is a composition
of a sequence of low-dimensional functions. We study the nonparametric quantile
regression estimator using deep neural networks to approximate the target
conditional quantile function. For convenience, we shall refer to such an
estimator as a deep quantile regression (DQR) estimator. We show that the DQR
estimator achieves the nonparametric optimal convergence rate up to a
logarithmic factor determined by the intrinsic dimension of the underlying
compositional structure of the conditional quantile function, not the ambient
dimension of the predictor. Therefore, DQR is able to mitigate the curse of
dimensionality under the assumption that the conditional quantile function has
a compositional structure. To establish these results, we analyze the
approximation error of a composite function by neural networks and show that
the error rate only depends on the dimensions of the component functions. We
apply our general results to several important statistical models often used in
mitigating the curse of dimensionality, including the single index, the
additive, the projection pursuit, the univariate composite, and the generalized
hierarchical interaction models. We explicitly describe the prefactors in the
error bounds in terms of the dimensionality of the data and show that the
prefactors depends on the dimensionality linearly or quadratically in these
models. We also conduct extensive numerical experiments to evaluate the
effectiveness of DQR and demonstrate that it outperforms a kernel-based method
for nonparametric quantile regression.
