Computational Nonsmooth Analysis: Modern Applications and Recent Developments [special Issue on the Mathematics of Deep Learning]

Neural networks (NNs) have transformed the field of artificial intelligence, showcasing remarkable abilities in learning complex representations, scaling to large datasets, and driving advancements across a wide range of applications. The training of these networks often involves optimizing functions that are not only nonconvex but possibly also nonsmooth, which introduces significant challenges in algorithm design and analysis. In this article, we provide an overview of recent advancements in the theoretical and computational aspects of nonsmooth optimization, thereby highlighting an area of growing importance that we refer to as “computational nonsmooth analysis.” We also demonstrate the relevance of these advancements in some representative applications. Lastly, we discuss some fundamental open questions in this area, the answers to which will impact the use and understanding of nonsmooth models in modern deep learning and signal processing.