ON THE RATE OF CONVERGENCE OF FULLY CONNECTED DEEP NEURAL NETWORK REGRESSION ESTIMATES

• 成果类型: Article
• 署名作者: Kohler, Michael; Langer, Sophie
• 署名单位: Technical University of Darmstadt
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS2034
• 发表日期: 2021
• 页码: 2231-2249
• 关键词: Approximation; bounds

摘要

Recent results in nonparametric regression show that deep learning, that is, neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this paper, we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Here, either the number of neurons per hidden layer is fixed and the number of hidden layers tends to infinity suitably fast for sample size tending to infinity, or the number of hidden layers is bounded by some logarithmic factor in the sample size and the number of neurons per hidden layer tends to infinity suitably fast for sample size tending to infinity. The proof is based on new approximation results concerning deep neural networks.