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This chapter explores second-order optimization techniques for neural network training, focusing on methods that leverage curvature information beyond first-order gradients. It begins with Newton’s method, detailing both basic and global variants that utilize Hessian matrices for faster convergence. The Levenberg-Marquardt (LM) algorithm is extensively covered as a hybrid approach combining gradient descent and Gauss-Newton methods, particularly effective for nonlinear least squares problems. The conjugate gradient method is presented as an efficient iterative solver for symmetric positive definite systems. The chapter emphasizes practical implementation through Python code examples, including a complete LM algorithm implementation and comparative analysis using SciPy’s optimization tools. Key insights highlight the computational trade-offs between providing analytical Jacobians versus numerical approximations, demonstrating how second-order methods can achieve superior convergence characteristics despite their increased computational complexity.

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Neural Networks Training Based on Second-Order Optimization Technique

  • 章节

  • 发表日期：

    2026年

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  • Chunwei Zhang,
  • Tianpeng Li,
  • Ying Dai,
  • Li Sun,
  • Ardashir Mohammadzadeh

摘要

This chapter explores second-order optimization techniques for neural network training, focusing on methods that leverage curvature information beyond first-order gradients. It begins with Newton’s method, detailing both basic and global variants that utilize Hessian matrices for faster convergence. The Levenberg-Marquardt (LM) algorithm is extensively covered as a hybrid approach combining gradient descent and Gauss-Newton methods, particularly effective for nonlinear least squares problems. The conjugate gradient method is presented as an efficient iterative solver for symmetric positive definite systems. The chapter emphasizes practical implementation through Python code examples, including a complete LM algorithm implementation and comparative analysis using SciPy’s optimization tools. Key insights highlight the computational trade-offs between providing analytical Jacobians versus numerical approximations, demonstrating how second-order methods can achieve superior convergence characteristics despite their increased computational complexity.