  We consider the generalization problem for a perceptron with binary synapses,
implementing the Stochastic Belief-Propagation-Inspired (SBPI) learning
algorithm which we proposed earlier, and perform a mean-field calculation to
obtain a differential equation which describes the behaviour of the device in
the limit of a large number of synapses N. We show that the solving time of
SBPI is of order N*sqrt(log(N)), while the similar, well-known clipped
perceptron (CP) algorithm does not converge to a solution at all in the time
frame we considered. The analysis gives some insight into the ongoing process
and shows that, in this context, the SBPI algorithm is equivalent to a new,
simpler algorithm, which only differs from the CP algorithm by the addition of
a stochastic, unsupervised meta-plastic reinforcement process, whose rate of
application must be less than sqrt(2/(\pi * N)) for the learning to be achieved
effectively. The analytical results are confirmed by simulations.
