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Backpropagation Algorithm in MLP Training

The Backpropagation Algorithm is a fundamental method for training Multilayer Perceptrons (MLPs) by adjusting weights to minimize error. It efficiently computes gradients using the chain rule and updates weights through gradient descent.

Steps of Backpropagation

1. Forward Propagation

   • Input data passes through the network.

   • Each neuron computes a weighted sum and applies an activation function (e.g., ReLU, Sigmoid).

   • The output layer generates predictions.

2. Compute Loss

   • The difference between the predicted and actual output is measured using a loss function (e.g., Mean Squared Error for regression, Cross-Entropy for classification).

3. Backward Propagation (Gradient Computation)

   • Error Propagation: The loss is propagated backward through the network to compute gradients for each weight.

   • Uses the chain rule to compute partial derivatives of the loss function with respect to each weight.

   • For a weight $w$, the gradient is computed as:

     $$\frac{\partial L}{\partial w} = \frac{\partial L}{\partial y} \cdot \frac{\partial y}{\partial w}$$

     where $L$ is the loss, $y$ is the neuron's output.

4. Weight Update

   • Weights are updated using Gradient Descent (or its variants like Adam, RMSprop):

     $$w = w - \eta \cdot \frac{\partial L}{\partial w}$$

     where $\eta$ is the learning rate.

Why Backpropagation?

• Efficiently updates weights using gradients.

• Reduces error by optimizing the loss function.

• Enables deep networks to learn complex patterns.

Backpropagation is the foundation of modern deep learning, making MLPs and neural networks highly effective in various AI applications.