Phase transitions in the generalization behaviour of multilayer perceptrons: II. The influence of noise

We extend our study of phase transitions in the generalization behaviour of multilayer perceptrons with non-overlapping receptive fields to the problem of the influence of noise concerning, for example, the input units and/or the couplings between the input units and the hidden units of the second layer (= 'input noise') or the final output unit (= 'output noise'). Without output noise, the output itself is given by a general, permutation-invariant Boolean function of the outputs of the hidden units. As a result we find that the phase transitions which we found in the deterministic case, mostly persist in the presence of noise. The influence of the noise on the position of the phase transition, as well as on the behaviour in other regimes of the loading parameter alpha, can often be described by a simple rescaling of alpha depending on the strength and type of the noise. We then consider the problem of the optimal noise level for Gibbsian and Bayesian learning, looking at replica symmetry breaking as well. Finally, we consider the question of why learning with errors is useful at all.