H^∞ Optimal Training Algorithms and their Relation to Backpropagation

We derive global H^∞ optimal training algorithms for neural networks. These algorithms guarantee the smallest possible prediction error energy over all possible disturbances of fixed energy, and are therefore robust with respect to model uncertainties and lack of statistical information on the exogenous signals. The ensuing estimators are infinite-dimensional, in the sense that updating the weight vector estimate requires knowledge of all previous weight esimates. A certain finite-dimensional approximation to these estimators is the backpropagation algorithm. This explains the local H6∞ optimality of backpropagation that has been previously demonstrated