  Inference capabilities of machine learning (ML) systems skyrocketed in recent
years, now playing a pivotal role in various aspect of society. The goal in
statistical learning is to use data to obtain simple algorithms for predicting
a random variable $Y$ from a correlated observation $X$. Since the dimension of
$X$ is typically huge, computationally feasible solutions should summarize it
into a lower-dimensional feature vector $T$, from which $Y$ is predicted. The
algorithm will successfully make the prediction if $T$ is a good proxy of $Y$,
despite the said dimensionality-reduction. A myriad of ML algorithms (mostly
employing deep learning (DL)) for finding such representations $T$ based on
real-world data are now available. While these methods are often effective in
practice, their success is hindered by the lack of a comprehensive theory to
explain it. The information bottleneck (IB) theory recently emerged as a bold
information-theoretic paradigm for analyzing DL systems. Adopting mutual
information as the figure of merit, it suggests that the best representation
$T$ should be maximally informative about $Y$ while minimizing the mutual
information with $X$. In this tutorial we survey the information-theoretic
origins of this abstract principle, and its recent impact on DL. For the
latter, we cover implications of the IB problem on DL theory, as well as
practical algorithms inspired by it. Our goal is to provide a unified and
cohesive description. A clear view of current knowledge is particularly
important for further leveraging IB and other information-theoretic ideas to
study DL models.
