On-Line Learning Theory of Soft Committee Machines with Correlated Hidden Units –Steepest Gradient Descent and Natural Gradient Descent–

The permutation symmetry of the hidden units in multilayer perceptrons causes the saddle structure and plateaus of the learning dynamics in gradient learning methods. The correlation of the weight vectors of hidden units in a teacher network is thought to affect this saddle structure, resulting in a prolonged learning time, but this mechanism is still unclear. In this paper, we discuss it with regard to soft committee machines and on-line learning using statistical mechanics. Conventional gradient descent needs more time to break the symmetry as the correlation of the teacher weight vectors rises. On the other hand, no plateaus occur with natural gradient descent regardless of the correlation for the limit of a low learning rate. Analytical results support these dynamics around the saddle point.