A signal-flow-graph approach to on-line gradient calculation

A large class of nonlinear dynamic adaptive systems such as dynamic recurre nt neural networks can be effectively represented by signal flow graphs (SF Gs). By this method, complex systems are described as a general connection of many simple components, each of them implementing a simple one-input, on e-output transformation, as in an electrical circuit. Even if graph represe ntations are popular in the neural network community, they are often used f or qualitative description rather than for rigorous representation and comp utational purposes. In this article, a method for both on-line and batch-ba ckward gradient computation of a system output or cost function with respec t to system parameters is derived by the SFG representation theory and its known properties. The system can be any causal, in general nonlinear and ti me-variant, dynamic system represented by an SFG, in particular any feedfor ward, time-delay, or recurrent neural network. In this work, we use discret e-time notation, but the same theory holds for the continuous-time case. Th e gradient is obtained in a straightforward way by the analysis of two SFGs , the original one and its adjoint (obtained from the first by simple trans formations), without the complex chain rule expansions of derivatives usual ly employed. This method can be used for sensitivity analysis and for learning both off- line and on-line. On-line learning is particularly important since it is re quired by many real applications, such as digital signal processing, system identification and control, channel equalization, and predistortion.