  In the past decade the mathematical theory of machine learning has lagged far
behind the triumphs of deep neural networks on practical challenges. However,
the gap between theory and practice is gradually starting to close. In this
paper I will attempt to assemble some pieces of the remarkable and still
incomplete mathematical mosaic emerging from the efforts to understand the
foundations of deep learning. The two key themes will be interpolation, and its
sibling, over-parameterization. Interpolation corresponds to fitting data, even
noisy data, exactly. Over-parameterization enables interpolation and provides
flexibility to select a right interpolating model.
  As we will see, just as a physical prism separates colors mixed within a ray
of light, the figurative prism of interpolation helps to disentangle
generalization and optimization properties within the complex picture of modern
Machine Learning. This article is written with belief and hope that clearer
understanding of these issues brings us a step closer toward a general theory
of deep learning and machine learning.
