论文标题
带有深度relu网络的分位数回归：估计速率和最小值速率
Quantile regression with deep ReLU Networks: Estimators and minimax rates
论文作者
论文摘要
分位数回归是从已知协变量集合中估算指定百分位反应（例如中位数）的任务。我们以整流的线性单元（Relu）神经网络为选定的模型类研究分数回归。我们在用于估算一组协变量上的任何分位数的RELU网络的预期平方误差上得出了上限。该上限仅取决于最佳可能的近似误差，网络中的层数以及每层节点的数量。我们进一步显示了两个大型功能的上限：Hölder函数的组成和BESOV空间的成员。这些紧密的界限意味着具有分位数回归的Relu网络可实现最小功能类型收集的最小值。与现有的工作不同，理论结果在最少的假设下成立，并适用于一般错误分布，包括重型分布。一系列合成响应函数的经验模拟证明了理论结果转化为Relu网络的实际实现。总体而言，理论和经验结果提供了洞察Relu神经网络在广泛的函数类别和误差分布之间进行分位回归的强劲性能。本文的所有代码均在https://github.com/tansey/quantile-regression上公开获取。
Quantile regression is the task of estimating a specified percentile response, such as the median, from a collection of known covariates. We study quantile regression with rectified linear unit (ReLU) neural networks as the chosen model class. We derive an upper bound on the expected mean squared error of a ReLU network used to estimate any quantile conditional on a set of covariates. This upper bound only depends on the best possible approximation error, the number of layers in the network, and the number of nodes per layer. We further show upper bounds that are tight for two large classes of functions: compositions of Hölder functions and members of a Besov space. These tight bounds imply ReLU networks with quantile regression achieve minimax rates for broad collections of function types. Unlike existing work, the theoretical results hold under minimal assumptions and apply to general error distributions, including heavy-tailed distributions. Empirical simulations on a suite of synthetic response functions demonstrate the theoretical results translate to practical implementations of ReLU networks. Overall, the theoretical and empirical results provide insight into the strong performance of ReLU neural networks for quantile regression across a broad range of function classes and error distributions. All code for this paper is publicly available at https://github.com/tansey/quantile-regression.
