Abstract
This paper deals with deriving the properties of updated neural network model that is exploited to identify an unknown nonlinear system via the standard gradient learning algorithm. The convergence of this algorithm for online training the three-layer neural networks in stochastic environment is studied. A special case where an unknown nonlinearity can exactly be approximated by some neural network with a nonlinear activation function for its output layer is considered. To analyze the asymptotic behavior of the learning processes, the so-called Lyapunov-like approach is utilized. As the Lyapunov function, the expected value of the square of approximation error depending on network parameters is chosen. Within this approach, sufficient conditions guaranteeing the convergence of learning algorithm with probability 1 are derived. Simulation results are presented to support the theoretical analysis.
Highlights
Design of mathematical models for technical, economic, social and other systems with uncertainties is the important problem from both theoretical and practical points of view
A special case where an unknown nonlinearity can exactly be approximated by some neural network with a nonlinear activation function for its output layer is considered
The proposed approach to deriving these convergence results is based on utilizing the Lyapunov methodology [36]. They make it possible to reveal some new features of the multilayer neural networks with nonlinear activation function in output layer which use the online gradient-type training algorithms having a constant learning rate
Summary
Design of mathematical models for technical, economic, social and other systems with uncertainties is the important problem from both theoretical and practical points of view. This work has been motivated by the fact that the standard gradient algorithm is widely exploited for online updating the neural network weights in accordance with the gradient-descent principle whereas the following important question related to its ultimate properties remained in part open as yet: when does the sequential procedure based on this algorithm converge if the learning rate is. The proposed approach to deriving these convergence results is based on utilizing the Lyapunov methodology [36] They make it possible to reveal some new features of the multilayer neural networks with nonlinear activation function in output layer which use the online gradient-type training algorithms having a constant learning rate
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