  Learning is a distinctive feature of intelligent behaviour. High-throughput
experimental data and Big Data promise to open new windows on complex systems
such as cells, the brain or our societies. Yet, the puzzling success of
Artificial Intelligence and Machine Learning shows that we still have a poor
conceptual understanding of learning. These applications push statistical
inference into uncharted territories where data is high-dimensional and scarce,
and prior information on "true" models is scant if not totally absent. Here we
review recent progress on understanding learning, based on the notion of
"relevance". The relevance, as we define it here, quantifies the amount of
information that a dataset or the internal representation of a learning machine
contains on the generative model of the data. This allows us to define
maximally informative samples, on one hand, and optimal learning machines on
the other. These are ideal limits of samples and of machines, that contain the
maximal amount of information about the unknown generative process, at a given
resolution (or level of compression). Both ideal limits exhibit critical
features in the statistical sense: Maximally informative samples are
characterised by a power-law frequency distribution (statistical criticality)
and optimal learning machines by an anomalously large susceptibility. The
trade-off between resolution (i.e. compression) and relevance distinguishes the
regime of noisy representations from that of lossy compression. These are
separated by a special point characterised by Zipf's law statistics. This
identifies samples obeying Zipf's law as the most compressed loss-less
representations that are optimal in the sense of maximal relevance. Criticality
in optimal learning machines manifests in an exponential degeneracy of energy
levels, that leads to unusual thermodynamic properties.
