  Identifying the relevant coarse-grained degrees of freedom in a complex
physical system is a key stage in developing powerful effective theories in and
out of equilibrium. The celebrated renormalization group provides a framework
for this task, but its practical execution in unfamiliar systems is fraught
with ad hoc choices, whereas machine learning approaches, though promising,
often lack formal interpretability. Recently, the optimal coarse-graining in a
statistical system was shown to exist, based on a universal, but
computationally difficult information-theoretic variational principle. This
limited its applicability to but the simplest systems; moreover, the relation
to standard formalism of field theory was unclear. Here we present an algorithm
employing state-of-art results in machine-learning-based estimation of
information-theoretic quantities, overcoming these challenges. We use this
advance to develop a new paradigm in identifying the most relevant field theory
operators describing properties of the system, going beyond the existing
approaches to real-space renormalization. We evidence its power on an
interacting model, where the emergent degrees of freedom are qualitatively
different from the microscopic building blocks of the theory. Our results push
the boundary of formally interpretable applications of machine learning,
conceptually paving the way towards automated theory building.
