Neural network: Backpropagation algorithm

Backpropagation is an algorithm used to train artificial neural networks.
It is a supervised learning algorithm used to update the weights of the network
in order to minimize the error between the predicted outputs and the actual outputs.

The backpropagation algorithm works as follows

Step 1: Feedforward: The input data is passed through the network and
the activations of each layer are calculated.
The final output is compared to the actual target.

Step 2: Calculate the error: The error between the predicted output
and the actual target is calculated using a loss function
such as mean squared error.

Step 3: Backpropagate the error: The error is then backpropagated
through the network by computing the gradient of the loss
with respect to the weights.
This is done using the chain rule of differentiation.

Step 4: Update the weights: The weights are updated using an optimization algorithm
such as gradient descent, using the gradients calculated in the previous step.

Repeat: Steps 1-4 are repeated until the network reaches convergence
or the maximum number of iterations is reached.

Backpropagation is a powerful algorithm that allows neural networks
to learn complex relationships in the data.
It is the foundation of training deep neural networks
and is used in many applications including image classification,
natural language processing, and reinforcement learning.