H 1 Optimality Criteria for Lms and Backpropagation

We have recently shown that the widely known LMS algorithm is an H 1 optimal estimator. The H 1 criterion has been introduced, initially in the control theory literature, as a means to ensure robust performance in the face of model uncertainties and lack of statistical information on the exogenous signals. We extend here our analysis to the nonlinear setting often encountered in neural networks, and show that the backpropagation algorithm is locally H 1 optimal. This fact provides a theoretical justiication of the widely observed excellent robustness properties of the LMS and backpropagation algorithms. We further discuss some implications of these results.