Updating the gradients

With the back propagate error for each $l=L-1,L-2,\dots,1$ as

$$\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}sigma'(z_j^l),$$

we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,1$ and update the weights and biases according to the rules

$$w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1},$$ $$b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,$$

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