Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
It provides a perspective to help new researchers grasp foundational issues in neural network theory, but it is incremental as it synthesizes existing knowledge rather than presenting new findings.
This paper reviews recent progress in mathematically understanding the success and subtleties of neural network-based machine learning, highlighting both rigorous results and insights from experiments, while also identifying key open problems for future research.
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 mathematics, 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.