Towards a Mathematical Understanding of Neural Network-Based Machine Learning: What We Know and What We Don’t

Abstract

The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied math-ematics, we will not only give attention to rigorous mathematical results, but also the insight we have gained from careful numerical experiments as well as the analysis of simplified models. Along the way, we also list the open problems which we believe to be the most important topics for further study. This is not a complete overview over this quickly moving field, but we hope to provide a perspective which may be helpful especially to new researchers in the area.

Keywords

Neural networks / machine learning / supervised learning / regression problems / ap-proximation / optimization / estimation / a priori estimates / Barron space / multi-layer space / flow-induced function space