  We revisit known constructions of efficient learning algorithms from various
notions of constructive circuit lower bounds such as distinguishers breaking
pseudorandom generators or efficient witnessing algorithms which find errors of
small circuits attempting to compute hard functions. As our main result we
prove that if it is possible to find efficiently, in a particular interactive
way, errors of many p-size circuits attempting to solve hard problems, then
p-size circuits can be PAC learned over the uniform distribution with
membership queries by circuits of subexponential size. The opposite implication
holds as well. This provides a new characterisation of learning algorithms and
extends the natural proofs barrier of Razborov and Rudich. The proof is based
on a method of exploiting Nisan-Wigderson generators introduced by
Kraj\'{i}\v{c}ek (2010) and used to analyze complexity of circuit lower bounds
in bounded arithmetic.
  An interesting consequence of known constructions of learning algorithms from
circuit lower bounds is a learning speedup of Oliveira and Santhanam (2016). We
present an alternative proof of this phenomenon and discuss its potential to
advance the program of hardness magnification.
