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Chapter 1: Building a Neural Network from Scratch

Introduction

In this chapter, we will walk through the process of building a simple Feedforward Neural Network (FNN) from scratch using Python and NumPy. Neural networks are the foundation of deep learning models, and understanding how they work at the core level is essential for mastering more complex models like Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs).

We will implement a basic neural network with one hidden layer. The network will be trained using backpropagation and gradient descent to minimize the error between predicted and actual outputs.

This tutorial will cover the following:

1. Understanding the architecture of a neural network.
2. Implementing forward propagation.
3. Implementing the backpropagation algorithm.
4. Training the network using gradient descent.
5. Testing the model on a simple dataset.
6. Visualizing the training process.

By the end of this chapter, you will have a deep understanding of how neural networks work and how they are trained.

1. Neural Network Architecture

A neural network consists of layers of neurons, each of which performs a mathematical operation on the input data. The architecture of a neural network typically consists of three types of layers:

• Input Layer: Receives the input features.
• Hidden Layer(s): Performs computations on the input data using weights and activation functions.
• Output Layer: Produces the final output (predicted value or classification).

In this example, we will create a network with:

• One input layer with n neurons (equal to the number of features in the dataset).
• One hidden layer with h neurons.
• One output layer with a single neuron for binary classification.

2. Mathematical Representation of Neural Networks

The neural network operates through forward propagation and backpropagation.

1. Forward Propagation:
• Each layer receives an input, applies a weighted sum, and passes it through an activation function to generate the output.
• In a fully connected network, the output of a layer a^(l) is computed as:

Where:

• W^(l) is the weight matrix of layer l,
• b^(l) is the bias vector of layer l,
• σ is the activation function (e.g., ReLU, sigmoid).
1. Backpropagation:
• Backpropagation is used to update the weights of the network by minimizing the cost function (e.g., mean squared error or cross-entropy loss) using gradient descent.

3. Implementing Forward Propagation

3.1 Initializing the Weights and Biases

Before the forward pass, we need to initialize the weights and biases. The weights are usually initialized randomly, and the biases are initialized to zero or small values.

Code Sample:

import numpy as np

# Initialize the weights and biases

def initialize_parameters(input_size, hidden_size, output_size):

    W1 = np.random.randn(hidden_size, input_size) * 0.01  # Weight matrix for hidden layer

    b1 = np.zeros((hidden_size, 1))  # Bias for hidden layer

    W2 = np.random.randn(output_size, hidden_size) * 0.01  # Weight matrix for output layer

    b2 = np.zeros((output_size, 1))  # Bias for output layer

    parameters = {

        'W1': W1,

        'b1': b1,

        'W2': W2,

        'b2': b2

    }

    return parameters

Explanation:

• W1, b1 are the weights and biases for the hidden layer.
• W2, b2 are the weights and biases for the output layer.
• The weights are initialized randomly, and the biases are initialized to zero.

3.2 Forward Propagation

The forward propagation step involves computing the activations for each layer. For the hidden layer, we compute the weighted sum of the inputs, pass it through the ReLU activation function, and then repeat the same for the output layer with a sigmoid activation function for binary classification.

Code Sample:

def forward_propagation(X, parameters):

    W1 = parameters['W1']

    b1 = parameters['b1']

    W2 = parameters['W2']

    b2 = parameters['b2']

    # Compute activations for hidden layer

    Z1 = np.dot(W1, X) + b1

    A1 = np.maximum(0, Z1)  # ReLU activation function

    # Compute activations for output layer

    Z2 = np.dot(W2, A1) + b2

    A2 = 1 / (1 + np.exp(-Z2))  # Sigmoid activation function

    cache = (Z1, A1, Z2, A2)

    return A2, cache

Explanation:

• We calculate the activations for the hidden layer using ReLU and for the output layer using the sigmoid function.
• The activations are stored in the cache, which will be used during backpropagation.

4. Implementing Backpropagation

4.1 Calculating the Cost Function

The cost function measures the difference between the predicted outputs and the true labels. For binary classification, we use the cross-entropy loss:

Where:

• y is the true label.
• ŷ​ is the predicted probability (output of the sigmoid function).

Code Sample:

def compute_cost(A2, Y):

    m = Y.shape[1]  # Number of examples

    cost = - (1/m) * np.sum(Y * np.log(A2) + (1 - Y) * np.log(1 - A2))

    return cost

Explanation:

• The cost is computed by summing the cross-entropy loss for all examples.

4.2 Backpropagation to Compute Gradients

Backpropagation computes the gradients of the cost function with respect to each parameter (weights and biases). We will use these gradients to update the weights using gradient descent.

Code Sample:

def backpropagation(X, Y, parameters, cache):

    m = X.shape[1]  # Number of examples

    W1 = parameters['W1']

    b1 = parameters['b1']

    W2 = parameters['W2']

    b2 = parameters['b2']

    # Retrieve values from the cache

    Z1, A1, Z2, A2 = cache

    # Compute gradients for output layer

    dZ2 = A2 - Y

    dW2 = (1/m) * np.dot(dZ2, A1.T)

    db2 = (1/m) * np.sum(dZ2, axis=1, keepdims=True)

    # Compute gradients for hidden layer

    dZ1 = np.dot(W2.T, dZ2) * (A1 > 0)  # Derivative of ReLU

    dW1 = (1/m) * np.dot(dZ1, X.T)

    db1 = (1/m) * np.sum(dZ1, axis=1, keepdims=True)

    gradients = {

        'dW1': dW1,

        'db1': db1,

        'dW2': dW2,

        'db2': db2

    }

    return gradients

Explanation:

• We compute the gradient of the cost function with respect to each weight and bias in the network.
• For the hidden layer, we use the derivative of the ReLU activation function to compute the gradients.

4.3 Gradient Descent to Update Parameters

After calculating the gradients, we update the weights and biases using gradient descent:

Where α is the learning rate.

Code Sample:

def gradient_descent(parameters, gradients, learning_rate=0.01):

    W1 = parameters['W1']

    b1 = parameters['b1']

    W2 = parameters['W2']

    b2 = parameters['b2']

    # Update parameters

    W1 -= learning_rate * gradients['dW1']

    b1 -= learning_rate * gradients['db1']

    W2 -= learning_rate * gradients['dW2']

    b2 -= learning_rate * gradients['db2']

    updated_parameters = {

        'W1': W1,

        'b1': b1,

        'W2': W2,

        'b2': b2

    }

    return updated_parameters

Explanation:

• The weights and biases are updated using the calculated gradients and the learning rate.

5. Putting It All Together: Training the Neural Network

We can now train the neural network by performing forward propagation, calculating the cost, backpropagating the gradients, and updating the parameters over multiple iterations (epochs).

Code Sample:

def train_neural_network(X_train, Y_train, input_size, hidden_size, output_size, epochs=1000, learning_rate=0.01):

    # Initialize parameters

    parameters = initialize_parameters(input_size, hidden_size, output_size)

    for epoch in range(epochs):

        # Forward propagation

        A2, cache = forward_propagation(X_train, parameters)

        # Compute cost

        cost = compute_cost(A2, Y_train)

        # Backpropagation

        gradients = backpropagation(X_train, Y_train, parameters, cache)

        # Update parameters

        parameters = gradient_descent(parameters, gradients, learning_rate)

        if epoch % 100 == 0:

            print(f"Epoch {epoch}, Cost: {cost}")

    return parameters

Explanation:

• We initialize the parameters and then train the model over multiple epochs, updating the weights and biases after each forward and backward pass.
• Every 100 epochs, we print the cost to monitor the training process.

6. Conclusion

In this chapter, we built a Feedforward Neural Network (FNN) from scratch using Python and NumPy. We covered:

1. Neural network architecture and how forward and backpropagation work.
2. Forward propagation with ReLU and sigmoid activation functions.
3. Backpropagation to compute gradients and update the weights using gradient descent.
4. Training the network over multiple epochs and monitoring the cost.

This tutorial provided a hands-on approach to understanding how neural networks work and how they are trained. While deep learning frameworks like TensorFlow and PyTorch handle these steps automatically, implementing a neural network from scratch gives you a deeper understanding of the inner workings of neural networks.