Quantile regression and conditional density estimation can reveal structure that is missed by mean regression, such as multimodality and skewness. In this paper, we introduce a deep learning generative model for joint quantile estimation called Penalized Generative Quantile Regression (PGQR). Our approach simultaneously generates samples from many random quantile levels, allowing us to infer the conditional distribution of a response variable given a set of covariates. Our method employs a novel variability penalty to avoid the problem of vanishing variability, or memorization, in deep generative models. Further, we introduce a new family of partial monotonic neural networks (PMNN) to circumvent the problem of crossing quantile curves. A major benefit of PGQR is that it can be fit using a single optimization, thus bypassing the need to repeatedly train the model at multiple quantile levels or use computationally expensive cross-validation to tune the penalty parameter. We illustrate the efficacy of PGQR through extensive simulation studies and analysis of real datasets. Code to implement our method is available at https://github.com/shijiew97/PGQR.

翻译：分位数回归和条件密度估计能够揭示均值回归难以捕捉的结构，例如多模态性与偏态性。本文提出一种用于联合分位数估计的深度学习生成模型——带惩罚的生成式分位数回归（Penalized Generative Quantile Regression, PGQR）。该方法可同时从多个随机分位水平生成样本，从而推断给定协变量条件下响应变量的条件分布。我们采用新型变异性惩罚机制，以规避深度生成模型中的变异性消失（即记忆化现象）问题。此外，我们提出部分单调神经网络（PMNN）家族来避免分位数曲线交叉问题。PGQR的主要优势在于可通过单次优化完成训练，无需针对多个分位水平重复训练模型，也无需使用计算成本高昂的交叉验证来调优惩罚参数。通过大规模仿真实验和真实数据集分析，我们验证了PGQR的有效性。方法实现代码见 https://github.com/shijiew97/PGQR。