论文标题
正规神经网络的统计保证
Statistical Guarantees for Regularized Neural Networks
论文作者
论文摘要
神经网络已成为数据分析的标准工具，但它们缺乏全面的数学理论。例如，从数据中学习神经网络的统计保证很少，尤其是用于实践中或至少与此类似的估计器类别。在本文中，我们针对由最小二乘项和正规器组成的估计器开发了一般统计保证。然后，我们用$ \ ell_1 $ - 重新定性来体现此保证，表明相应的预测误差最多在层数中最多增加，最多在参数总数中最多增加。我们的结果为神经网络的正则估计建立了数学基础，并更加一般地对神经网络和深度学习加深了我们的数学理解。
Neural networks have become standard tools in the analysis of data, but they lack comprehensive mathematical theories. For example, there are very few statistical guarantees for learning neural networks from data, especially for classes of estimators that are used in practice or at least similar to such. In this paper, we develop a general statistical guarantee for estimators that consist of a least-squares term and a regularizer. We then exemplify this guarantee with $\ell_1$-regularization, showing that the corresponding prediction error increases at most sub-linearly in the number of layers and at most logarithmically in the total number of parameters. Our results establish a mathematical basis for regularized estimation of neural networks, and they deepen our mathematical understanding of neural networks and deep learning more generally.
