When does online BP training converge?

The backpropogation (BP) neural networks have been widely applied in scientific research and engineering. The success of the application, however, relies upon the convergence of the training procedure involved in the neural network learning. We settle down the convergence analysis issue through proving two fundamental theorems on the convergence of the online BP training procedure. One theorem claims that under mild conditions, the gradient sequence of the error function will converge to zero (the weak convergence), and another theorem concludes the convergence of the weight sequence defined by the procedure to a fixed value at which the error function attains its minimum (the strong convergence). The weak convergence theorem sharpens and generalizes the existing convergence analysis conducted before, while the strong convergence theorem provides new analysis results on convergence of the online BP training procedure. The results obtained reveal that with any analytic sigmoid activation function, the online BP training procedure is always convergent, which then underlies successful application of the BP neural networks.