  In order to elucidate the plateau phenomena caused by vanishing gradient, we
herein analyse stability of stochastic gradient descent near degenerated
subspaces in a multi-layer perceptron. In stochastic gradient descent for
Fukumizu-Amari model, which is the minimal multi-layer perceptron showing
non-trivial plateau phenomena, we show that (1) attracting regions exist in
multiply degenerated subspaces, (2) a strong plateau phenomenon emerges as a
noise-induced synchronisation, which is not observed in deterministic gradient
descent, (3) an optimal fluctuation exists to minimise the escape time from the
degenerated subspace. The noise-induced degeneration observed herein is
expected to be found in a broad class of machine learning via neural networks.
