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My question is at the bottom. (Most of the descriptive words come from Chris. Bishop's Neural Networks for Pattern Recognition)

let $w$ be the weight vector of the neural network and $E$ the error function.

According to the Robbins-Monro algorithm, this $$w_{kj}^{(r+1)}=w_{kj}^{(r)}-\eta\left.\frac{\partial E}{\partial w_{kj}}\right|_{w^{(r)}}$$ will converge to where $$\frac{\partial E}{\partial w_{kj}}=0$$

In general the error function is given by a sum of terms each of which is calculated using one of the patterns from the training set, so that $$E=\sum_nE^n(w)$$ And in applications we just update the weight vector using one pattern at a time $$w_{kj}^{(r+1)}=w_{kj}^{(r)}-\eta\frac{\partial E^n}{\partial w_{kj}}$$

My question is: Why the algorithm will converge using the last formula? Once we use it to update the $w$, the value of $w$ is changed, I can't prove the convergence using $$\frac{\partial E}{\partial w_{kj}}=\sum_n \frac{\partial E^n}{\partial w_{kj}}$$

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I don't know anything about neural networks, but this looks a multidimensional Newton's method, see en.wikipedia.org/wiki/Kantorovich_theorem . Two common strategies to prove convergence of methods like this are 1) show the iteration is a contraction mapping, or 2) Find some quantity that decreases by at least a fixed fraction every iteration that is zero at the true solution. It will be difficult to help much more without knowing more properties of E. –  Nick Alger Apr 8 '12 at 4:20

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