Feedforward Neural Networks: Algorithms and Optimization Approaches

The backpropagation neural network algorithm stands as one of the pioneering optimization solutions in machine learning. This paper focuses on the most extensively implemented feedforward neural networks and their weight learning methodologies. The error backpropagation algorithm (commonly abbreviated as BP algorithm) constitutes the most influential component in this domain, typically implemented through gradient descent optimization where weights are updated using partial derivatives of the loss function with respect to network parameters. However, the BP algorithm presents several computational limitations, including tendency to converge to local minima and slow convergence rates particularly in deep network architectures. To mitigate these issues, the Levenberg-Marquardt algorithm derived from optimization theory offers improved performance, though it approximates the Hessian matrix by neglecting second-order terms in its formulation. Therefore, this paper investigates scenarios where error functions are non-zero or nonlinear, making the second-order term S(W) non-negligible. We develop methodologies for approximate Hessian matrix computation under these conditions, employing techniques like finite difference approximations or Quasi-Newton methods where exact Hessian calculation becomes computationally prohibitive. The enhanced approach facilitates more robust network training through better curvature information utilization in optimization steps, implemented through modified weight update rules that incorporate second-order information while maintaining computational feasibility.