  Deep learning networks have been trained to recognize speech, caption
photographs and translate text between languages at high levels of performance.
Although applications of deep learning networks to real world problems have
become ubiquitous, our understanding of why they are so effective is lacking.
These empirical results should not be possible according to sample complexity
in statistics and non-convex optimization theory. However, paradoxes in the
training and effectiveness of deep learning networks are being investigated and
insights are being found in the geometry of high-dimensional spaces. A
mathematical theory of deep learning would illuminate how they function, allow
us to assess the strengths and weaknesses of different network architectures
and lead to major improvements. Deep learning has provided natural ways for
humans to communicate with digital devices and is foundational for building
artificial general intelligence. Deep learning was inspired by the architecture
of the cerebral cortex and insights into autonomy and general intelligence may
be found in other brain regions that are essential for planning and survival,
but major breakthroughs will be needed to achieve these goals.
