  Heuristic tools from statistical physics have been used in the past to locate
the phase transitions and compute the optimal learning and generalization
errors in the teacher-student scenario in multi-layer neural networks. In this
contribution, we provide a rigorous justification of these approaches for a
two-layers neural network model called the committee machine. We also introduce
a version of the approximate message passing (AMP) algorithm for the committee
machine that allows to perform optimal learning in polynomial time for a large
set of parameters. We find that there are regimes in which a low generalization
error is information-theoretically achievable while the AMP algorithm fails to
deliver it, strongly suggesting that no efficient algorithm exists for those
cases, and unveiling a large computational gap.
