Training a perceptron in a discrete weight space.

Learning in a perceptron having a discrete weight space, where each weight can take 2L+1 different values, is examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined, among them on-line learning with discrete and continuous transfer functions. The generalization error of the clipped weights decays asymptotically as exp(-Kalpha(2)) in the case of on-line learning with binary activation functions and exp(-e(|lambda|alpha)) in the case of on-line learning with continuous one, where alpha is the number of examples divided by N, the size of the input vector and K is a positive constant. For finite N and L, perfect agreement between the discrete student and the teacher is obtained for alpha~Lsqrt[ln(NL)]. A crossover to the generalization error approximately 1/alpha, characterizing continuous weights with binary output, is obtained for synaptic depth L>O(sqrt[N]).