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Convergence Analysis of Online Gradient Method for High-Order Neural Networks and Their Sparse Optimization.

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    This study introduces a smoothing technique for sigma-pi-sigma neural networks (SPSNNs) to improve network sparseness and generalization. The method enhances online gradient descent by optimizing network structure and controlling redundancy, supported by theoretical and experimental results.

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    Area of Science:

    • Artificial Intelligence
    • Machine Learning
    • Neural Networks

    Background:

    • Traditional group regularization in neural networks can lead to non-convex and non-smooth error functions, causing oscillations.
    • Sigma-pi-sigma neural networks (SPSNNs) require methods to enhance sparseness and generalization ability.

    Purpose of the Study:

    • To investigate the boundedness and convergence of an online gradient method using smoothing group regularization for SPSNNs.
    • To address the limitations of non-smooth error functions in original group regularization techniques.

    Main Methods:

    • Developed a novel smoothing technique to overcome the deficiencies of non-smooth error functions in group regularization.
    • Applied the online gradient method with the proposed smoothing group regularization to SPSNNs.
    • Analyzed the boundedness of weights and convergence properties (strong and weak) of the method.

    Main Results:

    • The smoothing technique effectively eliminates oscillations caused by non-smooth error functions.
    • The proposed method optimizes network structure by driving redundant hidden nodes and weights towards zero.
    • Demonstrated strong and weak convergence, and boundedness of weights for the online gradient method with smoothing group regularization.

    Conclusions:

    • The smoothing group regularization is an effective technique for enhancing SPSNN sparseness and generalization.
    • Experimental results validate the theoretical findings, confirming the method's capability and redundancy control effectiveness.