An Artificial Neural Network (ANN) is created and trained on the pima-indians-diabetes dataset using R. Rather than doing backpropagation one instance at a time, the weight updates are calculated for all training instances at once, and summing these to calculate the net change.
For a given layer in the ANN, a matrix W is used to hold the values of all weights, where W(i,j) is the weight of the edge going from input node i to output node j.

The following functions are implemented in the code:

• The function sigmoid calculates the sigmoid [f(x) = 1/(1+e^-x)] of an input vector x. The function sigmoid_derivative calculates the derivative of the sigmoid function with respect to a matrix x.
• The calculate_loss function takes the predicted values and actual values as input and computes the loss of predictions.
• The calculate_activations function is used to calculate the activation vector for a given node, given the input matrix of activations from the prior layer, and the weight matrix. The input is a matrix, where each row corresponds to one training instance and each column corresponds to the activation of a node in the previous layer. The output is a vector, where each item in the vector represents the outputs for one training instance.
• There are two kinds of gradients calculated to perform back propagation, besides the gradient for the activation function that's already calculated in sigmoid_derivative(). In calculate_dCdw(), the derivative of the cost with respect to a weight matrix is calculated. In calculate_dCdf(), the derivative of the cost with respect to activation values, calculated by calculate_activations() is computed.
• The function neuralnet utilizes all the above mentioned function to train the neural network.