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This paper considers the problem of nonparametric quantile regression underthe assumption that the target conditional quantile function is a compositionof a sequence of low-dimensional functions. We study the nonparametric quantileregression estimator using deep neural networks to approximate the targetconditional quantile function. For convenience, we shall refer to such anestimator as a deep quantile regression (DQR) estimator. We show that the DQRestimator achieves the nonparametric optimal convergence rate up to alogarithmic factor determined by the intrinsic dimension of the underlyingcompositional structure of the conditional quantile function, not the ambientdimension of the predictor. Therefore, DQR is able to mitigate the curse ofdimensionality under the assumption that the conditional quantile function hasa compositional structure. To establish these results, we analyze theapproximation error of a composite function by neural networks and show thatthe error rate only depends on the dimensions of the component functions. Weapply our general results to several important statistical models often used inmitigating the curse of dimensionality, including the single index, theadditive, the projection pursuit, the univariate composite, and the generalizedhierarchical interaction models. We explicitly describe the prefactors in theerror bounds in terms of the dimensionality of the data and show that theprefactors depends on the dimensionality linearly or quadratically in thesemodels. We also conduct extensive numerical experiments to evaluate theeffectiveness of DQR and demonstrate that it outperforms a kernel-based methodfor nonparametric quantile regression.

Deep Quantile Regression: Mitigating the Curse of Dimensionality Through Composition