  This work studies distributed learning in the spirit of Yao's model of
communication complexity: consider a two-party setting, where each of the
players gets a list of labelled examples and they communicate in order to
jointly perform some learning task. To naturally fit into the framework of
learning theory, the players can send each other examples (as well as bits)
where each example/bit costs one unit of communication. This enables a uniform
treatment of infinite classes such as half-spaces in $\mathbb{R}^d$, which are
ubiquitous in machine learning.
  We study several fundamental questions in this model. For example, we provide
combinatorial characterizations of the classes that can be learned with
efficient communication in the proper-case as well as in the improper-case.
These findings imply unconditional separations between various learning
contexts, e.g.\ realizable versus agnostic learning, proper versus improper
learning, etc.
  The derivation of these results hinges on a type of decision problems we term
"{\it realizability problems}" where the goal is deciding whether a distributed
input sample is consistent with an hypothesis from a pre-specified class.
  From a technical perspective, the protocols we use are based on ideas from
machine learning theory and the impossibility results are based on ideas from
communication complexity theory.
