发表机构
Université Jean Monnet, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Universite Claude Bernard Lyon 1（里昂 Jean Monnet 大学，ICJ UMR5208，国家科学研究中心，里昂中央理工大学，里昂国家理工学校，里昂大学 Claude Bernard 1）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究协变量转移下基于相关数据的深度非参数回归，提出稀疏惩罚深度神经网络（SPDNN）估计器，当密度比未知时经两步预训练，在Hölder光滑函数类中建立其非渐近误差界，能自适应达到最优收敛速率。
AI 中文摘要
协变量转移在许多实际应用中经常出现，因为源观测和目标观测可能来自不同分布，此时源分布下的标准度量不合适。本文考虑了协变量转移下基于相关观测的深度神经网络非参数分位数和Huber回归估计器。处理了许多经典模型满足的广义伯恩斯坦型不等式。提出了一种稀疏惩罚深度神经网络（SPDNN）估计器来处理协变量转移现象。当协变量的源分布和目标分布之间的密度比未知时，进行两步预训练过程：第一步构建密度比的最小二乘SPDNN估计器，第二步用于对回归函数进行预训练重加权SPDNN估计器。对于分位数和Huber回归，在Hölder光滑函数类中建立了所提出的SPDNN估计器的非渐近误差界。这些估计器可以自适应地（至多一个对数因子）达到来自独立同分布数据以及几个经典时间序列模型的极小极大最优收敛速率。
英文摘要
Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $ϕ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.
