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 $(\beta \to 0)$. 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 $K\geq 3$ the transition is first order. The limit $K\to\infty$ can be performed analytically, the transition occurs after presenting on the order of $ N K/\beta$ examples. However, an unspecialized metastable state persists up to $P \propto N K^2/\beta$