Signal-flow-graph derivation of on-line gradient learning algorithms

In this paper, making use of the Signal-Flow-Graph (SFG) representation and its known properties, we derive a new general method for backward gradient computation of a system output or cost function with respect to past (or present) system parameters. The system can be any causal, in general non-linear and time-variant, dynamic system represented by a SFG, in particular any feed forward or recurrent neural network. In this work we use discrete time notation, but the same theory holds for the continuos time case. The gradient is obtained by the analysis of two SFGs, the original one and its adjoint. This method can be used both for on-line and off-line learning. In the latter case using the Mean Square Error cost function, our approach particularises to E. Wan's method that is not suited for online training of recurrent networks. Computer simulations of non-linear dynamic systems identification will also be presented to assess the performance of the algorithm resulting from the application of the proposed method in the case of locally recurrent neural networks.