Artificial Neural Networks: Understanding the Levenberg-Marquardt algorithm

Hi guys,

I´m working with the matlab toolbox for neural networks. My target is a nn for function approximation.

I understand the backpropagation algorithm to update the weights and biases after each round of training. For understanding I used Hagan, M.T., H.B. Demuth, and M.H. Beale, Neural Network Design, Boston, MA: PWS Publishing, 1996 which matlab advise. For the backpropagation for multilayer networks this is working with the sensitive and this is all fine. I understand that the algorithm has some problems to find every time fast the best solution. So the algorithm is modified. Now I´m trying to understand the Levenberg-Marquardt algorithm.

My understanding is, that steepest gradient descent is moving slowly in the “deepest valley” but I don´t understand, what the Gauss-Newton algorithm is doing.

Another problem is, that I don´t understand what are the Jacobian and Hessian matrix and why do we need them? I the previous calculation with the sensitive we don`t need them and now they confuse me.

I hope someone can help me.

Michael

Answer 1

[Prasanna]

Hi,

It is my understanding that you want to know more regarding the Levenberg-Marquardt algorithm.

Firstly, The Gauss-Newton algorithm is an optimization technique that improves upon gradient descent for least squares problems, often encountered in function approximation tasks. It enhances convergence by leveraging curvature (second derivative) information through an approximation of the Hessian matrix using first derivatives. This approach, focusing on curvature rather than just the gradient, allows for more direct and efficient progress toward the error minimum.

The Jacobian and Hessian matrices are used for optimization algorithms to make better informed updates to the weights and biases aiming for faster convergence and better solutions.

The Levenberg-Marquardt algorithm merges the advantages of gradient descent and Gauss-Newton methods, using the Jacobian to approximate the Hessian for curvature insights and introducing a damping factor to modulate updates for stability and convergence on complex error surfaces. This factor is dynamically adjusted: decreased after successful steps for faster convergence (akin to Gauss-Newton) and increased following error increments for more cautious updates (like gradient descent).

In summary, the transition from basic backpropagation to the Levenberg-Marquardt algorithm involves moving from simple gradient-based updates to more sophisticated updates that consider the curvature of the error surface. This shift necessitates the introduction of the Jacobian and Hessian matrices to better inform the training process, aiming for faster convergence and improved handling of complex error surfaces encountered in neural network training for function approximation tasks.

Refer the following link to know more about the Levenberg-Marquardt and Gauss-Newton algorithm: https://sites.cs.ucsb.edu/~yfwang/courses/cs290i_mvg/pdf/LMA.pdf

Hope this helps.