Chapter 6: Neural Networks

The Limit of Known Patterns: The Artificial Brain

In dermatology, Ethan recalls Hazel’s ImageNet moment and the team turns to neural networks to flag risky lesions while specialists stay in the loop.

This chapter demystifies neural networks: the perceptron as a building block, how gradients flow via backpropagation, and how architecture and hyperparameters shape learning.

1. 6.1 The Perceptron: The Artificial Neuron: Train a Perceptron on a simple 2D dataset and see how a linear decision rule emerges.

2. 6.2 Backpropagation: The Engine of Learning: You will visualize the backpropagation algorithm, the engine of learning in neural networks. You will understand how errors propagate backward through the network, allowing each neuron to adjust its connections and improve overall model performance.

3. 6.2 Interactive Tutorial: Backpropagation Step by Step: A guided, hands-on trainer where you compute the forward pass, calculate output error, propagate gradients backward, and update weights — step by step.

4. 6.2 Classic Paper: "Learning representations by back-propagating errors": A concise note on Rumelhart, Hinton & Williams (1986), the paper that popularized backpropagation and demonstrated its practical power.

5. 6.3 Neural Network Playground: Experimenting with Deep Learning: Experience hands-on experimentation with TensorFlow Playground. Build, train, and visualize neural networks in your browser, exploring how architecture, hyperparameters, and data affect learning in real time.

6. 6.4 CNN Digit Lab: Draw, Train, and Predict: Build intuition for convolutional neural networks by recognizing handwritten digits. Draw in the canvas, compare predictions, train a compact CNN from scratch, or jump directly to a pretrained snapshot.

Algorithm Pseudocode

• 📝 Perceptron Training Pseudocode: The perceptron learning rule, convergence theorem, Pocket algorithm, and Voted Perceptron variants.
• 📝 Backpropagation Pseudocode: Forward pass, backward pass, chain rule derivations, and full training loop with mini-batches.

Mathematical Foundations

• 📐 Mathematical Proof: The XOR Problem: Explore the formal mathematical proof that single-layer perceptrons cannot solve the XOR problem, the insight that sparked the rise of multilayer networks.
• Perceptron Convergence Theorem: Rosenblatt–Novikoff mistake bounds proving the perceptron converges on linearly separable datasets.
• Backpropagation via the Chain Rule: Jacobian-based derivation of the gradient recursions.

Bibliography and Additional Resources

• 📚 Neural Networks and Perceptron: Verified resources and references on neural networks, Rosenblatt's perceptron and the backpropagation algorithm.