# Explanation of the partial derivatives in back-propogation algorithm

I understand that this is the method for conducting back-propogation:

• With a three layer network (input layer, one hidden layer, one output layer); start with input $$I_i$$ as an exemplar input.
• Calculate the output for this input (by multiplying by weights, checking against thresholds, and propogating through the network to the output); output = $$O_k$$
• $$y_k$$ is the target output for $$O_k$$

To update the weights leading to this output in the final layer (i.e. $$w_{jk}$$; between the one hidden layer and the output layer):

1. Calculate $$\frac{\delta E}{\delta X_k} = y_k(1-y_k)(y_k-O_k)$$
2. Calculate $$\frac{\delta E}{\delta w_{jk}} = \frac{\delta E}{\delta X_k}y_j$$
3. For all j,k; $$w_{jk} := w_{jk} - C\frac{\delta E}{\delta w_{jk}}$$

Then for the $$w_{ij}$$ weights (i.e. the weights between the input layer and the hidden layer):

1. Calculate all the $$\frac{\delta E}{\delta X_j} = y_j(1-y_j) \sum_k \frac{\delta E}{\delta X_k} w_{jk}$$
2. Calculate all the $$\frac{\delta E}{\delta w_{ij}} = \frac{\delta E}{\delta x_j}I_i$$
3. For all i,j: $$w_{ij} = w_{ij} - C\frac{\delta E}{\delta w_{jk}}$$

What I understand:

1. I understand what step 3 and 6 is doing. You have the weight, a learning rate ($$C$$) and the error you've calculated to be associated with that weight, and you're updating the weight to reflect the error.

2. For step 1,2,4 and 5, I understand the left hand side of each equation: 1 is measuring the change in total error for that input X, 2 is measuring total error for the weight $$w_{jk}$$, 5 is measuring the total error for the weight $$w_{ij}$$ and 4 is measuring the error for that X value.

What I don't understand is, in plain basic english, what the right hand side of equations 1,2,4 and 5 are doing. Can someone else with this?

• I would suggest that, in this post, you focus on a specific equation, maybe the first one that you need to understand before understanding the others, so that people can focus on a single equation at a time. So, I would actually suggest that you split this post into multiple ones. And try to put a more specific question in the title.
– nbro
Jan 4 at 22:38