Phase transitions in soft-committee machines

PACS. 87.10+e { General, theoretical, and mathematical biophysics (including logic of biosys-tems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics, and bionics). PACS. 07.05Mh { Neural networks, fuzzy logic, articial intelligence. PACS. 05.90+m { Other topics in statistical physics and themodynamics. Abstract. { Equilibrium statistical physics is applied to the o-line training of layered neural networks with dierentiable activation functions. A rst 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 ( ! 0). For K = 2 we nd a second-order phase transition from unspecialized to specialized student congurations at a critical size P of the training set, whereas for K 3 the transition is rst order. The limit K! 1 can be performed analytically, the transition occurs after presenting on the order of NK = examples. However, an unspecialized metastable state persists up to P / NK2 =. Statistical physics provides tools for the investigation of learning processes in adaptiv