Variational foundations of online backpropagation

On-line Backpropagation has become very popular and it has been the subject of in-depth theoretical analyses and massive experimentation. Yet, after almost three decades from its publication, it is still surprisingly the source of tough theoretical questions and of experimental results that are somewhat shrouded in mystery. Although seriously plagued by local minima, the batch-mode version of the algorithm is clearly posed as an optimization problem while, in spite of its effectiveness, in many real-world problems the on-line mode version has not been given a clean formulation, yet. Using variational arguments, in this paper, the on-line formulation is proposed as the minimization of a classic functional that is inspired by the principle of minimal action in analytic mechanics. The proposed approach clashes sharply with common interpretations of on-line learning as an approximation of batch-mode, and it suggests that processing data all at once might be just an artificial formulation of learning that is hopeless in difficult real-world problems. © 2013 Springer-Verlag Berlin Heidelberg.