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In neural nets, the back-propagation algorithm is 'based on' gradient descent, optimizing over a cost function $C(w,b)$ of the weights and biases of the network.

Conjugate gradient descent has way faster convergence than standard gradient descent (and stochastic gradient descent). Is there a similar back-propagation algorithm based off of conjugate gradient descent?

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Yes,

see the article Optimization in an error backpropagation neural network environment with a performance test on a spectral pattern classification problem

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