  Deep learning algorithms are responsible for a technological revolution in a
variety of tasks including image recognition or Go playing. Yet, why they work
is not understood. Ultimately, they manage to classify data lying in high
dimension -- a feat generically impossible due to the geometry of high
dimensional space and the associated curse of dimensionality. Understanding
what kind of structure, symmetry or invariance makes data such as images
learnable is a fundamental challenge. Other puzzles include that (i) learning
corresponds to minimizing a loss in high dimension, which is in general not
convex and could well get stuck bad minima. (ii) Deep learning predicting power
increases with the number of fitting parameters, even in a regime where data
are perfectly fitted. In this manuscript, we review recent results elucidating
(i,ii) and the perspective they offer on the (still unexplained) curse of
dimensionality paradox. We base our theoretical discussion on the $(h,\alpha)$
plane where $h$ is the network width and $\alpha$ the scale of the output of
the network at initialization, and provide new systematic measures of
performance in that plane for MNIST and CIFAR 10. We argue that different
learning regimes can be organized into a phase diagram. A line of critical
points sharply delimits an under-parametrised phase from an over-parametrized
one. In over-parametrized nets, learning can operate in two regimes separated
by a smooth cross-over. At large initialization, it corresponds to a kernel
method, whereas for small initializations features can be learnt, together with
invariants in the data. We review the properties of these different phases, of
the transition separating them and some open questions. Our treatment
emphasizes analogies with physical systems, scaling arguments and the
development of numerical observables to quantitatively test these results
empirically.
