  We consider the Ising perceptron model with N spins and M = N*alpha patterns,
with a general activation function U that is bounded above. For U bounded away
from zero, or U a one-sided threshold function, it was shown by Talagrand
(2000, 2011) that for small densities alpha, the free energy of the model
converges in the large-N limit to the replica symmetric formula conjectured in
the physics literature (Krauth--Mezard 1989, see also Gardner--Derrida 1988).
We give a new proof of this result, which covers the more general class of all
functions U that are bounded above and satisfy a certain variance bound. The
proof uses the (first and second) moment method conditional on the approximate
message passing iterates of the model. In order to deduce our main theorem, we
also prove a new concentration result for the perceptron model in the case
where U is not bounded away from zero.
