Conjugated gradient method. What is an A-matrix in case of neural networks
The traditional conjugate gradient descent is an increment on the gradient descent that just takes a direction that is fully orthogonal to the previous descent direction. There is no $A$ matrix in that case.

There are different rules (you can check some in my old optimization toolbox at https://github.com/mbrucher/scikit-optimization/blob/master/scikits/optimization/step/conjugate_gradient_step.py). If I remember properly FR combined with strong Wolfe-Powell line search rule give one of the best answer. The issue is that it requires more computation, which is why line search is never used in neural networks optimization.