Phase transitions in soft-committee machines

TL;DR

This paper applies equilibrium statistical physics to analyze phase transitions in soft-committee neural networks, revealing different transition types depending on the number of hidden units and providing insights into learning dynamics.

Contribution

It provides an exact analysis of off-line learning in soft-committee machines with finite hidden units, identifying phase transition types and their dependence on network size and training set.

Findings

Second order phase transition for K=2 at a critical training set size

First order transition for K>2

Metastable unspecialized states persist at large training set sizes

Abstract

Equilibrium statistical physics is applied to layered neural networks with differentiable activation functions. A first analysis of off-line learning in soft-committee machines with a finite number (K) of hidden units learning a perfectly matching rule is performed. Our results are exact in the limit of high training temperatures. 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 > 2 the transition is first order. Monte Carlo simulations indicate that our results are also valid for moderately low temperatures qualitatively. The limit K to infinity can be performed analytically, the transition occurs after presenting on the order of N K examples. However, an unspecialized metastable state persists up to P= O (N K^2).