Generalization and capacity of extensively large two-layered perceptrons

The generalization ability and storage capacity of a treelike two-layered neural network with a number of hidden units scaling as the input dimension is examined. The mapping from the input to the hidden layer is via Boolean functions; the mapping from the hidden layer to the output is done by a perceptron. The analysis is within the replica framework where an order parameter characterizing the overlap between two networks in the combined space of Boolean functions and hidden-to-output couplings is introduced. The maximal capacity of such networks is found to scale linearly with the logarithm of the number of Boolean functions per hidden unit. The generalization process exhibits a first-order phase transition from poor to perfect learning for the case of discrete hidden-to-output couplings. The critical number of examples per input dimension, ${\alpha }_c,$ at which the transition occurs, again scales linearly with the logarithm of the number of Boolean functions. In the case of continuous hidden-to-output couplings, the generalization error decreases according to the same power law as for the perceptron, with the prefactor being different.