Write Back Propagation algorithm, and showcase its execution on a neural network of your choice (make suitable assumptions if any)

Backpropagation is a fundamental algorithm used to train artificial neural networks. It is based on the gradient descent optimization technique, and its goal is to minimize the error in the network's predictions by adjusting the weights and biases of the network. The backpropagation algorithm works by calculating the gradient of the loss function with respect to each weight by the chain rule, and then updating the weights in the direction that minimizes the error.

Backpropagation Algorithm

The backpropagation algorithm can be divided into two main phases: the forward pass and the backward pass.

1. Forward Pass: In this phase, input data is passed through the network, layer by layer, to compute the predicted output. This involves multiplying the inputs by the weights, applying an activation function at each layer, and propagating the outputs through to the final layer.

   • Input Layer: Each node in the input layer receives an input feature from the dataset.

   • Hidden Layers: The weighted sum of the inputs is computed for each node in the hidden layers, and an activation function (e.g., sigmoid, ReLU) is applied to introduce non-linearity.

   • Output Layer: The output layer computes the final output, which is compared to the true target values using a loss function (e.g., Mean Squared Error, Cross-Entropy).

2. Backward Pass: Once the network has made a prediction, the error (loss) is computed, and this error is propagated backward through the network to update the weights and biases.

   The backward pass consists of three steps:

   • Step 1: Compute the Loss Function The loss function measures how well the network’s prediction matches the true value. A common loss function used for regression tasks is the Mean Squared Error (MSE), and for classification tasks, it is Cross-Entropy. The error (or loss) is calculated as:

     L=12∑(ytrue−ypred)2L = \frac{1}{2} \sum (y_{true} - y_{pred})^2

     where ytruey_{true} is the actual target value and ypredy_{pred} is the predicted value.

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