Generalization ability and information gain of clock-model perceptrons

We study the generalization abilityg_QofQ-state Clock-model perceptrons for (i) Hebbian and for certain Non-Hebbian learning procedures, namely (ii) learning with maximal stability, (iii) zero stability and (iv) optimal generalization, for the case of random training sets. Among other results we find thatg_Qbehaves quite different in the Hebbian and in the Non-Hebbian cases in the limitQ. E.g. in the Hebbian case for finite ,g_Qvanishes always 1/Q, whereas in the Non-Hebbian cases considered,g_Qconverges forQ to a non-trivial continuous functiong(), which vanishes for <2, but increases rapidly for >2. This means that for (ii), (iii) and (iv), as a function of atQ=, there is a 2nd-order phase transition from a non-generalizing phase for 2 to a generalizing phase for >2. Different behaviour of the Hebbian and Non-Hebbian cases, respectively, is also observed for the information gain obtained through learning. For the particular case of AdaTron Learning, which is identical to case (ii), we find a geometrical formulation forg_Q(), which is applicable to more general models.