Backpropagation in Neural Networks

Overview

• Core algorithm for neural network learning.
• Intuitive walkthrough before diving into the math.
• Key concepts: neural networks, gradient descent, cost functions.

Neural Network Structure

• Input Layer: 784 neurons for pixel values of handwritten digits.
• Hidden Layers: 2 layers, each with 16 neurons.
• Output Layer: 10 neurons representing digits (0-9).

Key Concepts

• Gradient Descent: Method to minimize cost function by adjusting weights and biases.
• Cost Function: Measures difference between network output and desired output.
  • Calculate cost for a single example: sum of squared differences.
  • Average cost across all training examples for total cost.
• Negative Gradient: Indicates how to change weights/biases to decrease cost efficiently.

Understanding Backpropagation

• Backpropagation computes the gradient of the cost function.
• Think of gradient sensitivity:
  • Higher sensitivity in cost function leads to bigger adjustments in weights/biases.

Example with Digit Recognition

• Using an example of recognizing the digit "2":
  • Initial randomly activated outputs (e.g., 0.5, 0.8, 0.2).
  • Adjustments needed:
    • Increase activation for target neuron (digit 2).
    • Decrease activations for other neurons.
• Weight Adjustment:
  • Weights with brighter activations contribute more to the output.
  • Hebbian theory: "neurons that fire together, wire together" – connections strengthen between active neurons.

Propagation of Changes

• Changes propagate backwards through layers:
  • Desired changes for one neuron influence the preceding layers.
  • Aggregate changes from all output neurons to inform adjustments in previous layers.

Averaging Changes for Training Examples

• Compute adjustments for a single example, but need to consider all examples.
• Average desired changes for weights/biases across all training examples to find approximate gradient.

Stochastic Gradient Descent (SGD)

• Instead of using all training data, use mini-batches (e.g., 100 examples).
• Faster computation: gets close enough to the gradient without calculating for all data.
• Results in a quicker, albeit less precise, optimization path.

Summary of Backpropagation

• Backpropagation determines how to adjust weights/biases based on training data.
• Efficiently combines changes from mini-batches to converge towards minimum cost function.
• Importance of large training datasets, illustrated by the MNIST database for digit recognition.

Future Learning

• Next video will cover underlying calculus of backpropagation.