Relationships Between the A Priori and A Posteriori Errors in Nonlinear Adaptive Neural Filters

The lower bounds for the a posteriori prediction error of a nonlinear predictor realized as a neural network are provided. These are obtained for a priori adaptation and a posteriori error networks with sigmoid nonlinearities trained by gradient-descent learning algorithms. A contractivity condition is imposed on a nonlinear activation function of a neuron so that the a posteriori prediction error is smaller in magnitude than the corresponding a priori one. Furthermore, an upper bound is imposed on the learning rate η so that the approach is feasible. The analysis is undertaken for both feedforward and recurrent nonlinear predictors realized as neural networks.