Boundedness and Convergence Analysis of a Pi-Sigma Neural Network Based on Online Gradient Method and Sparse Optimization

High order neural networks have strong nonlinear mapping ability, but the network structure is more complex, which restricts the efficiency of the network, and the relevant theoretical analysis is still not perfect up to now. To solve these problems, an online gradient learning algorithm model of Pi-Sigma neural network with a smooth set lasso regular term is proposed. Since the original lasso regular term contains absolute values and is not differentiable at the origin, it causes experiment oscillations and poses a great challenge to the convergence analysis of the algorithm. We use grinding technology to overcome this deficiency. The main contribution of this paper lies in the adoption of online learning algorithm, which effectively improves the efficiency of the algorithm. At the same time, strict theoretical proofs are presented, including strong convergence and weak convergence. Finally, the effectiveness of the algorithm and the correctness of the theoretical results are verified by numerical experiments.