Non-Deterministic Learning Dynamics in Large Neural Networks due to Structural Data Bias

We study the dynamics of on-line learning in large perceptrons, for the case of training sets with a structural bias of the input vectors, by deriving exact and closed macroscopic dynamical laws using non-equilibrium statistical mechanical tools. In sharp contrast to the more conventional theories developed for homogeneously distributed or only weakly biased data, these laws are found to describe a non-trivial and persistently non-deterministic macroscopic evolution, and a generalisation error which retains both stochastic and sample-to-sample fluctuations, even for infinitely large networks. Furthermore, for the standard error-correcting microscopic algorithms (such as the perceptron learning rule) one obtains learning curves with distinct bias-induced phases. Our theoretical predictions find excellent confirmation in numerical simulations.