Tilinglike learning in the parity machine

An algorithm for the training of multilayered feedforward neural networks is presented. The strategy is very similar to the well-known tiling algorithm, yet the resulting architecture is completely different. New hidden units are added to one layer only in order to correct the errors of the previous ones; standard perceptron learning can be applied. The output of the network is given by the product of these k (±1) neurons (parity machine). In a special case with two hidden units, the capacity ${\mathrm{\alpha }}_{\mathit{c}}$ and stability of the network can be derived exactly by means of a replica-symmetric calculation. Correlations between the two sets of couplings vanish exactly. For the case of arbitrary k, estimates of ${\mathrm{\alpha }}_{\mathit{c}}$ are given. The asymptotic capacity per input neuron of a network trained according to the proposed algorithm is found to be ${\mathrm{\alpha }}_{\mathit{c}}$∼k lnk for k→∞ in the estimation. This is in agreement with recent analytic results for the algorithm-independent capacity of a parity machine.