Lecture Notes: Understanding Gradient Descent in Linear Regression

Introduction

• Continuation of a video series on machine learning.
• Focus on linear regression with gradient descent.
• Video is supplementary, providing background math.
• References to 3Blue1Brown for deeper calculus understanding.

Key Concepts

Linear Regression Basics

• 2D Data Points & Line:
  • Formula: ( y = mx + b )
  • Prediction involves input ( x ) and known ( y ).
  • Error: Difference between prediction and actual ( y ).

Cost Function

• Definition: Evaluates performance of the model.
• Formula: ( \text{Cost} = \sum (y_i - \text{guess}_i)^2 )
• Goal: Minimize the error (cost).

Function Minimization

• Example: ( y = x^2 ) to illustrate minimizing functions.
• Aim: Find ( x ) producing lowest ( y ), analogous to minimizing loss in models.

Gradient Descent

• Purpose: Minimizing the cost function.
• Derivative: Key concept to determine the slope and direction for minimizing.
  • Allows determination of step direction and size.

Gradient Descent Mechanism

• Derivative Calculation:
  • Find slope indicating which direction (and how much) to change ( m ) and ( b ).
  • Use calculus: derivative as slope of tangent.

Application

• Adjust ( m ) and ( b ) to minimize error.
• Gradient Descent Algorithm: Adjusts these values iteratively.

Mathematical Explanation

Derivatives

• Power Rule: ( f(x) = x^n \rightarrow n \times x^{n-1} )
• Chain Rule: Used when function is composed of other functions.

Gradient Descent Formula

• For ( m ): Derivative of error relative to ( m )

  • Formula: ( 2 \times \text{error} \times x )
  • Simplifies to: ( \text{error} \times x )

• For ( b ): Derivative of error relative to ( b )

  • Formula: ( 2 \times \text{error} \times 1 )
  • Simplifies to: ( \text{error} )

Adjusting with Learning Rate

• Learning rate determines step size.
• ( \Delta m = \text{error} \times x )
• ( \Delta b = \text{error} )

Conclusion

• Insight into why gradient descent formulas are applied in linear regression.
• Prepares for further machine learning modeling, such as neural networks.
• Encouragement to explore other resources for deeper understanding of calculus and derivatives.

Additional Resources

• Suggested viewing of 3Blue1Brown's calculus series for comprehensive understanding.
• Links in video description for further learning.

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Note: This lecture aims to provide the groundwork for understanding how linear regression functions are adjusted through a gradient descent approach, using calculus fundamentals involving derivatives.