Lecture Notes: Backpropagation in Machine Learning

Introduction

• Commonality in ML Systems: Nearly all machine learning systems (e.g., GPT, Mid-Journey, AlphaFold) utilize a common algorithm called Backpropagation.
• Purpose: Understand backpropagation, how it works, and its differences from biological learning.

Backpropagation

Historical Context

• Leibniz (17th Century): Early concepts trace back.
• Seppu Linanma (1970): First modern formulation in a thesis.
• Rumelhart, Hinton, Williams (1986): Demonstrated backpropagation in multilayer perceptrons, showing it enabled learning.

Basic Concept

• Curve Fitting Problem:
  • Fit a curve (polynomial of degree 5) to a set of data points.
  • Define a loss function to measure fit quality (square distance between data points and curve).
• Loss Function: Minimize this to find the best curve.

Optimization Process

Random Perturbation Method

• Use a machine (CurveFitter 6000) with adjustable knobs to test different settings.
• Inefficient due to randomness in adjustments.

Gradient Descent

• Differentiability: Key property allowing efficient optimization.
• Upgrade the machine to indicate optimal direction for adjustments using the derivative.

Mathematical Foundation

Derivative Basics

• Definition: Rate of change or steepness at a point.
• Visual Representation: Slope of a tangent line on a graph.

Partial Derivatives and Gradient

• Partial Derivative: Change in output per change in one parameter, holding others constant.
• Gradient Vector: Packs partial derivatives, pointing in direction of steepest ascent for multivariable functions.
• Gradient Descent: Iteratively adjust parameters in the opposite direction of gradient.

Backpropagation Algorithm

Chain Rule

• Purpose: Calculate derivatives of complex functions using simple, known derivatives.
• Process: Use chain rule in sequence for functions composed of simpler functions.

Computational Graph

• Forward Step: Calculate loss using operations like addition, multiplication.
• Backward Step (Backpropagation): Calculate derivatives by applying chain rule backward along the graph.

Training Process

• Iterations: Forward pass, backward pass, adjust parameters, repeat.
• Modern ML systems: Use this iterative process with differentiable models.

Biological Learning

• Future Discussion: Next video will explore synaptic plasticity and biological learning mechanisms, questioning backpropagation’s biological relevance.

Conclusion

• Backpropagation's Importance: Fundamental to ML but distinct from how biological brains learn.
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• Recommendation: Stay tuned for next video on biological learning and synaptic plasticity.