  We study algorithmic learning of algebraic structures. In our framework, a
learner receives larger and larger pieces of an arbitrary copy of a computable
structure and, at each stage, is required to output a conjecture about the
isomorphism type of such a structure. The learning is successful if the
conjectures eventually stabilize to a correct guess. We prove that a family of
structures is learnable if and only if its learning domain is continuously
reducible to the relation $E_0$ of eventual agreement on reals. This motivates
a novel research program, that is, using descriptive set theoretic tools to
calibrate the (learning) complexity of nonlearnable families. Here, we focus on
the learning power of well-known benchmark Borel equivalence relations (i.e.,
$E_1$, $E_2$, $E_3$, $Z_0$, and $E_{set}$).
