Forward and Backpropagation - Rishan Solutions

Forward and Backpropagation in Neural Networks: Detailed Explanation

Forward and backward propagation are two key steps in the training process of a neural network. These steps are fundamental to how a neural network learns from data. Forward propagation helps make predictions, while backpropagation helps to update the weights of the network to minimize errors. Together, they allow neural networks to learn and improve through training, enabling them to handle tasks like classification, regression, and pattern recognition.

Let’s break down each step in detail:

Forward Propagation

Forward propagation is the process where input data is passed through the network to generate an output prediction. During this process, data flows through the network from the input layer to the output layer, passing through hidden layers along the way.

Here’s how forward propagation works step by step:

Step 1: Input Layer

The process starts with the input layer, where the features of the data are passed into the network. Each neuron in the input layer corresponds to one feature from the input dataset.
Example: For a neural network tasked with classifying images, the input layer would receive pixel values of the image.

Step 2: Weighted Sum

Each neuron in the input layer sends its output (the input feature) to the neurons in the next layer. Before this output is sent, each input value is multiplied by a weight.
Each connection between neurons has a weight that indicates the strength of the relationship between two neurons.
- If the input to a neuron in a hidden layer is xx and the weight is ww, then the weighted sum is calculated as:
z=w×x+bz = w \times x + b Where:
- zz is the weighted sum
- bb is the bias term
- xx is the input

Step 3: Activation Function

The weighted sum zz is then passed through an activation function. The activation function introduces non-linearity to the model, allowing the neural network to learn complex patterns.
Common activation functions include:
- Sigmoid: Output values between 0 and 1.
- ReLU (Rectified Linear Unit): Outputs the input value if it’s positive, otherwise zero.
- Tanh (Hyperbolic Tangent): Outputs values between -1 and 1.
- Softmax: Used in the output layer of a multi-class classification network to convert the raw output into a probability distribution.
The function transforms the weighted sum into a value that will be passed as the output to the next layer. The output of the neuron can be denoted as: a=f(z)a = f(z) Where:
- aa is the activation output.
- f(z)f(z) represents the activation function.

Step 4: Propagation Through Layers

The process of calculating weighted sums and passing them through activation functions continues from the input layer through one or more hidden layers to the output layer.
In each hidden layer, the same operation is repeated: a weighted sum of the inputs is calculated, passed through an activation function, and then sent to the next layer.

Step 5: Output Layer

The final layer in the network is the output layer, where the network produces its prediction. For example:
- In a binary classification task, the output layer might have a single neuron with a sigmoid activation function to output a probability between 0 and 1.
- In a multi-class classification task, the output layer may have multiple neurons, with a softmax activation function applied to produce a probability distribution across different classes.
The predicted output from the network is the result of forward propagation.

Backpropagation

Backpropagation is the process used to update the weights of the neural network by minimizing the error (or loss) between the network’s predictions and the true target values. It uses the concept of gradient descent to propagate the error backward through the network and adjust the weights accordingly. The goal of backpropagation is to optimize the weights so that the error is minimized.

Here’s how backpropagation works step by step:

Step 1: Calculate the Loss

After forward propagation, the network generates a prediction. The next step is to calculate the loss (or error) between the predicted output and the actual target output. The loss function is used to compute this error. For example:
- Mean Squared Error (MSE) for regression tasks.
- Cross-Entropy Loss for classification tasks.
The loss is computed as: Loss=1N∑i=1N(ytrue−ypredicted)2\text{Loss} = \frac{1}{N} \sum_{i=1}^{N} (y_{\text{true}} – y_{\text{predicted}})^2 Where:
- ytruey_{\text{true}} is the true output (target).
- ypredictedy_{\text{predicted}} is the predicted output from the neural network.
- NN is the number of samples.

Step 2: Calculate Gradients

Once the loss is calculated, backpropagation begins. The goal is to compute the gradient (partial derivative) of the loss function with respect to each weight in the network.
This is done by applying the chain rule of calculus, which allows the error to be propagated backward through each layer of the network. The gradient for a weight ww in a neuron is computed as: ∂Loss∂w=∂Loss∂a×∂a∂z×∂z∂w\frac{\partial \text{Loss}}{\partial w} = \frac{\partial \text{Loss}}{\partial a} \times \frac{\partial a}{\partial z} \times \frac{\partial z}{\partial w} Where:
- ∂Loss∂a\frac{\partial \text{Loss}}{\partial a} is the derivative of the loss with respect to the activation.
- ∂a∂z\frac{\partial a}{\partial z} is the derivative of the activation function with respect to the weighted sum.
- ∂z∂w\frac{\partial z}{\partial w} is the derivative of the weighted sum with respect to the weight.

Step 3: Backpropagate the Error

The gradient is propagated backward from the output layer to the input layer. This involves calculating the gradients for the weights in the output layer first, then the hidden layers, and finally the input layer.
The gradients represent how much the error will change if the weights are adjusted. Larger gradients indicate that the weights need to be adjusted more significantly.

Step 4: Update the Weights

After calculating the gradients, the weights are updated to minimize the loss function using an optimization algorithm such as gradient descent. The weight update is computed as: wnew=wold−η×∂Loss∂ww_{\text{new}} = w_{\text{old}} – \eta \times \frac{\partial \text{Loss}}{\partial w} Where:
- wneww_{\text{new}} is the updated weight.
- woldw_{\text{old}} is the current weight.
- η\eta is the learning rate, which determines the size of the weight update.
- ∂Loss∂w\frac{\partial \text{Loss}}{\partial w} is the gradient of the loss with respect to the weight.

Step 5: Repeat the Process

This process of forward propagation, loss calculation, backpropagation, and weight updates is repeated for multiple epochs (iterations over the training dataset) until the network converges, i.e., until the weights are optimized to minimize the loss.

Key Differences between Forward and Backpropagation

Aspect	Forward Propagation	Backpropagation
Purpose	Generate the prediction/output from the input data.	Minimize the loss by adjusting the weights based on the error.
Direction	Data moves from input to output layer.	Error moves from output to input layer.
Process	Involves calculating weighted sums and passing them through activation functions.	Involves calculating gradients and updating weights based on error.
Output	Produces the predicted output from the network.	Updates the network’s weights to reduce error.
Computation	Involves simple arithmetic operations like addition and multiplication.	Involves differentiation and application of the chain rule.
Iteration	Occurs once during the forward pass of the training phase.	Occurs after forward propagation during the backpropagation phase.