Phase transitions in soft-committee machines

Equilibrium statistical physics is applied to the off-line training of layered neural networks with differentiable activation functions. A first analysis of soft-committee machines with an arbitrary number (K) of hidden units and continuous weights learning a perfectly matching rule is performed. Our results are exact in the limit of high training temperatures (beta --> 0). For K = 2 we find a second-order phase transition from unspecialized to specialized student configurations at a critical size P of the training set, whereas for K greater than or equal to 3 the transition is first order. The limit K --> infinity can be performed analytically, the transition occurs after presenting on the order of NK/beta examples. However, an unspecialized metastable state persists up tu P proportional to NK2/beta.