Q18Machine Learning
Question
Explain the Backpropagation algorithm in Artificial Neural Networks with a detailed step-by-step mathematical derivation.
Answer
Backpropagation systematically updates neural network weights by calculating the gradient of the loss function using the chain rule.
### Introduction to Backpropagation Backpropagation is the backbone of neural network training. It propagates the error backwards from the output layer to the input layer to update weights using Gradient Descent.
### Step-by-Step Derivation
1. Forward Pass: The input X is fed through the network. Pre-activations z and activations a are computed for each layer.
2. Loss Calculation: The total error is calculated using a loss function like Mean Squared Error or Cross-Entropy.
3. Backward Pass: We compute the gradient of the error with respect to each weight using the chain rule.
4. Weight Update: Weights are updated proportional to the negative gradient, controlled by the learning rate.
By repeatedly applying these steps over multiple epochs, the network converges to a state with minimal loss.
