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We analyze the learning of noisy teacher-generated examples by nonlinear anddifferentiable student perceptrons using the cavity method. The genericactivation of an example is a function of the cavity activation of the example,which is its activation in the perceptron that learns without the example. Meanfield equations for the macroscopic parameters and the stability conditionyield results consistent with the replica method. When a single value of thecavity activation maps to multiple values of the generic activation, there is acompetition in learning strategy between preferentially learning an example andsacrificing it in favor of the background adjustment. We find parameter regimesin which examples are learned preferentially or sacrificially, leading to a gapin the activation distribution. Full phase diagrams of this complex system arepresented, and the theory predicts the existence of a phase transition frompoor to good generalization states in the system. Simulation results confirmthe theoretical predictions.Comment: 13 figures, one tabl

Cavity approach to noisy learning in nonlinear perceptrons