Statistical Guarantees for Regularized Neural Networks
This provides a mathematical foundation for regularized neural network estimation, addressing a core theoretical gap in deep learning for researchers and practitioners.
The paper tackles the lack of statistical guarantees for neural networks by developing a general guarantee for estimators with least-squares and regularization, exemplified with ℓ₁-regularization showing prediction error increases sub-linearly in layers and logarithmically in parameters.
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.