Levenberg-Marquardt Optimization Algorithm (trainlm) and Bayesian Regularization Algorithm (trainbr)

This text discusses two prominent optimization algorithms used in neural network training: the Levenberg-Marquardt optimization algorithm (trainlm) and the Bayesian regularization algorithm (trainbr). The Levenberg-Marquardt algorithm (trainlm) combines gradient descent and Gauss-Newton methods to achieve fast convergence for medium-sized networks. In MATLAB implementation, trainlm calculates the Jacobian matrix instead of the Hessian, making it computationally efficient for networks with up to several hundred parameters. The algorithm dynamically adjusts the damping parameter μ - increasing it for gradient descent behavior when far from solutions, and decreasing it for Gauss-Newton convergence when approaching minima. Bayesian regularization (trainbr) modifies the typical performance function F = E_D (mean squared errors) to F = βE_D + αE_W, where E_W represents the sum of squared weights. This automatically penalizes large weights during training, effectively preventing overfitting. The algorithm uses Bayesian inference to statistically determine optimal regularization parameters α and β, enhancing network generalization without requiring a separate validation set. When implementing these algorithms, developers should consider: trainlm for fast convergence where computational resources allow Jacobian calculations, and trainbr for improved generalization particularly with limited training data. The choice depends on specific requirements - trainlm for rapid training of moderate-sized networks, and trainbr for robust performance where overfitting is a concern. Proper algorithm selection significantly impacts neural network performance and application success.