Jian Huang

Guohao Shen, Yuling Jiao, Yuanyuan Lin et al.

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >0$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.

Wenlu Tang, Guohao Shen, Yuanyuan Lin et al.

We propose a nonparametric quantile regression method using deep neural networks with a rectified linear unit penalty function to avoid quantile crossing. This penalty function is computationally feasible for enforcing non-crossing constraints in multi-dimensional nonparametric quantile regression. We establish non-asymptotic upper bounds for the excess risk of the proposed nonparametric quantile regression function estimators. Our error bounds achieve optimal minimax rate of convergence for the Holder class, and the prefactors of the error bounds depend polynomially on the dimension of the predictor, instead of exponentially. Based on the proposed non-crossing penalized deep quantile regression, we construct conformal prediction intervals that are fully adaptive to heterogeneity. The proposed prediction interval is shown to have good properties in terms of validity and accuracy under reasonable conditions. We also derive non-asymptotic upper bounds for the difference of the lengths between the proposed non-crossing conformal prediction interval and the theoretically oracle prediction interval. Numerical experiments including simulation studies and a real data example are conducted to demonstrate the effectiveness of the proposed method.

Guohao Shen, Yuling Jiao, Yuanyuan Lin et al.

We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improved approximation error bounds when data has an approximate low-dimensional support, demonstrating the ability of RePU networks to mitigate the curse of dimensionality. To illustrate the usefulness of our results, we consider a deep score matching estimator (DSME) and propose a penalized deep isotonic regression (PDIR) using RePU networks. We establish non-asymptotic excess risk bounds for DSME and PDIR under the assumption that the target functions belong to a class of $C^s$ smooth functions. We also show that PDIR achieves the minimax optimal convergence rate and has a robustness property in the sense it is consistent with vanishing penalty parameters even when the monotonicity assumption is not satisfied. Furthermore, if the data distribution is supported on an approximate low-dimensional manifold, we show that DSME and PDIR can mitigate the curse of dimensionality.