Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the distribution of the predictor variable is used as the error criterion. The rate of convergence of least squares estimates based on fully connected spaces of deep neural networks with ReLU activation function is analyzed for smooth regression functions. It is shown that in case that the distribution of the predictor variable is concentrated on a manifold, these estimates achieve a rate of convergence which depends on the dimension of the manifold and not on the number of components of the predictor variable.
翻译:考虑从独立和相同分布的数据中估算回归函数。在预测变量分布方面,以$L_2美元整合差作为错误标准。根据与RELU激活功能完全相连的深神经网络空间,对最小平方估计数的趋同率进行了分析,以利平稳回归功能。显示如果预测变量的分布集中在多个方面,这些估计达到的趋同率取决于元的尺寸,而不是取决于预测变量的组件数量。