Optimal Nonparametric Inference via Deep Neural Network
Optimal Nonparametric Inference via Deep Neural Network
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DOI：
10.1016/j.jmaa.2021.125561
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发表时间：
2019-02
期刊：
影响因子：
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通讯作者：
Ruiqi Liu;B. Boukai;Zuofeng Shang
中科院分区：
文献类型：
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作者：
Ruiqi Liu;B. Boukai;Zuofeng Shang
深度神经网络是现代科学技术中最先进的方法。许多统计文献都致力于理解其在非参数估计中的性能，然而由于冗余的对数牺牲，结果不是最优的。在本文中，我们证明了这些对数因子是不必要的。我们导出了非参数估计中l2极大极小风险的上界。在网络架构上提供了足够的条件，使上界成为最优的（没有日志牺牲）。我们的证明依赖于一个基于张量积b样条的显式构造的网络估计量。我们还推导了所构建网络的渐近分布和相关的假设检验过程。在适当的网络结构下，进一步证明了测试过程是最小最大最优的。
Deep neural network is a state-of-art method in modern science and technology. Much statistical literature have been devoted to understanding its performance in nonparametric estimation, whereas the results are suboptimal due to a redundant logarithmic sacrifice. In this paper, we show that such log-factors are not necessary. We derive upper bounds for the L 2 minimax risk in nonparametric estimation. Sufficient conditions on network architectures are provided such that the upper bounds become optimal (without log-sacrifice). Our proof relies on an explicitly constructed network estimator based on tensor product B-splines. We also derive asymptotic distributions for the constructed network and a relating hypothesis testing procedure. The testing procedure is further proved as minimax optimal under suitable network architectures.