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How Backpropagation Algorithm works?

The backpropagation algorithm is used in layered feed-forward ANNs.

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Artificial neurons are organized in layers, and send their signals ?forward?, and then the errors are propagated backwards. The network receives inputs by neurons in the input layer, and the output of the network is given by the neurons on an output layer. There may be one or more intermediate hidden layers.
The backpropagation algorithm uses supervised learning, which means that we provide the algorithm with examples of the inputs and outputs we want the network to compute, and then the error  is calculated. The idea of the backpropagation algorithm is to reduce this error, until the ANN learns the training data. The training begins with random weights, and the goal is to adjust them so that the error will be minimal.
The activation function of the artificial neurons in ANNs implementing the backpropagation algorithm is a weighted sum :

If the output function would be the identity (output=activation), then the neuron would be called linear. But these have severe limitations. The most common output function is the sigmoidal function:
The sigmoidal function is very close to one for large positive numbers, 0.5 at zero, and very close to zero for large negative numbers. This allows a smooth transition between the low and high output of the neuron (close to zero or close to one). We can see that the output depends only in the activation, which in turn depends on the values of the inputs and their respective weights.
Now need to adjust the weights in order to minimize the error. We can define the error function for the output of each neuron:
 The error of the network will simply be the sum of the errors of all the neurons in the output layer:

The backpropagation algorithm now calculates how the error depends on the output, inputs, and weights. After we find this, we can adjust the weights using the method of gradient descendent:
Now we need to calculate how much the error depends on the output, which is the derivative of E in respect j to O (from (3)). 

How much the output depends on the activation, which in turn depends on the weights (from (1) and (2)):

 From (6) and (7)):

 The adjustment to each weight will be (from (5) and (8)):

We can use (9) as it is for training an ANN with two layers. Now, for training the network with one more layer we need to make some considerations. If we want to adjust ik the weights (let?s call them v ) of a previous layer, we need first to calculate how the error depends not on the weight, but in the input from the previous layer. This is easy, we would i ji just need to change x with w in (7), (8), and (9). But we also need to see how the error of ik the network depends on the adjustment of v . So:

 Assuming that there k ik are inputs u into the neuron with v (from (7)):
If we want to add yet another layer, we can do the same, calculating how the error depends on the inputs and weights of the first layer. We should just be careful with the indexes, since each layer can have a different number of neurons, and we should not confuse them. For practical reasons, ANNs implementing the backpropagation algorithm do not have too many layers, since the time for training the networks grows exponentially. Also, there are refinements to the backpropagation algorithm which allow a faster learning. 

Backpropagation Algorithm
Tech writer at newsandstory