为依赖数据下的深度神经网络提供了收敛性理论保障。
Statistical Properties of Deep Neural Networks with Dependent Data
- 在非平稳数据下给出概率收敛速率，适用于常见DNN结构。
- 在平稳β-混合数据中，给出L²误差的非渐近概率界。
- 适合研究时序建模与新架构的理论分析者参考。
本文建立了深度神经网络（DNN）估计器在依赖数据下的统计性质。给出了两个可直接应用于DNN估计器的非参数筛法估计器的一般结果：第一个在非平稳数据下建立了概率收敛速率；第二个在平稳β-混合数据下提供了L²误差的非渐近概率界。将这些结果应用于回归和分类场景中的DNN估计器，仅需标准Hölder光滑性假设。所考虑的DNN架构为实际应用中常见的全连接前馈网络，采用任意连续分段线性激活函数，权重无界，宽度与深度随样本量增长。该框架也为其他DNN架构及时间序列应用的研究提供可能。
原文摘要 · Abstract (English)
This paper establishes statistical properties of deep neural network (DNN) estimators under dependent data. Two general results for nonparametric sieve estimators directly applicable to DNN estimators are given. The first establishes rates for convergence in probability under nonstationary data. The second provides non-asymptotic probability bounds on $\mathcal{L}^{2}$-errors under stationary $β$-mixing data. I apply these results to DNN estimators in both regression and classification contexts imposing only a standard Hölder smoothness assumption. The DNN architectures considered are common in applications, featuring fully connected feedforward networks with any continuous piecewise linear activation function, unbounded weights, and a width and depth that grows with sample size. The framework provided also offers potential for research into other DNN architectures and time-series applications.
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