  We extend the theory of PAC learning in a way which allows to model a rich
variety of learning tasks where the data satisfy special properties that ease
the learning process. For example, tasks where the distance of the data from
the decision boundary is bounded away from zero. The basic and simple idea is
to consider partial concepts: these are functions that can be undefined on
certain parts of the space. When learning a partial concept, we assume that the
source distribution is supported only on points where the partial concept is
defined.
  This way, one can naturally express assumptions on the data such as lying on
a lower dimensional surface or margin conditions. In contrast, it is not at all
clear that such assumptions can be expressed by the traditional PAC theory. In
fact we exhibit easy-to-learn partial concept classes which provably cannot be
captured by the traditional PAC theory. This also resolves a question posed by
Attias, Kontorovich, and Mansour 2019.
  We characterize PAC learnability of partial concept classes and reveal an
algorithmic landscape which is fundamentally different than the classical one.
For example, in the classical PAC model, learning boils down to Empirical Risk
Minimization (ERM). In stark contrast, we show that the ERM principle fails in
explaining learnability of partial concept classes. In fact, we demonstrate
classes that are incredibly easy to learn, but such that any algorithm that
learns them must use an hypothesis space with unbounded VC dimension. We also
find that the sample compression conjecture fails in this setting.
  Thus, this theory features problems that cannot be represented nor solved in
the traditional way. We view this as evidence that it might provide insights on
the nature of learnability in realistic scenarios which the classical theory
fails to explain.
