Linear Regression and Gradient Descent

Introduction

• Host: JP
• Topic: Linear Regression Gradient Descent
• Purpose: To explain how linear regression makes predictions using gradient descent.

Linear Regression Overview

• Linear Regression: Predicts by plotting a straight line that best fits the data set.
• Cost Function: Measures how well predictions match actual values.
  • Represents the error between actual and predicted values.
  • Formula: ( J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{y}_i)^2 )
  • ( m ): Total number of data points
  • ( y ): Actual values
  • ( \hat{y} ): Predictions
• Goal: Minimize the cost function to best fit the straight line.

Gradient Descent Algorithm

• Used to minimize the cost function value.
• Graph Representation:
  • Cost vs. Theta graph resembles a parabola.
  • The goal is to reach the minimum point (minima) in the graph.
• Step Process:
  • Start at a point on the parabola (cost value).
  • Take steps towards the minima:
    • If cost decreases, continue moving.
    • If cost exceeds, backtrack.
    • Oscillation around the minima is expected.

Mathematical Representation

• The iterative step is represented as:
  • ( \theta = \theta - \alpha \frac{\partial J(\theta)}{\partial \theta} )
  • ( \alpha ): Positive constant (learning rate).

Understanding Derivatives

• Slope of the cost function informs the adjustment of ( \theta ):
  • If the slope is negative, ( \theta ) increases.
  • If the slope is positive, ( \theta ) decreases.

Cost Function Derivation

• Data set: ( X ) with ( m ) data points and ( n' ) features.
• Adding a column of ones increases dimensions to ( n' + 1 ).
• Predictions are given by:
  • ( \hat{y} = \theta^T X )
• Cost function in matrix form:
  • ( J(\theta) = \frac{1}{m} \sum (Y - \hat{Y})^2 )

Gradient Descent Implementation

• Initialize ( \theta ) as zeros.
• Repeat the following for a set number of iterations (e.g., 1000 times):
  1. Calculate predictions: ( \hat{Y} = X \cdot \theta )
  2. Calculate cost function value.
  3. Calculate derivative:
     • ( \frac{\partial J}{\partial \theta} = \frac{1}{m} X^T (Y - \hat{Y}) )
  4. Update ( \theta ):
     • ( \theta = \theta - \alpha \cdot \frac{\partial J}{\partial \theta} )

Learning Rate (( \alpha ))

• Determines step size during updates:
  • Large ( \alpha ): Risk of overshooting minima; may diverge.
  • Small ( \alpha ): Smaller steps towards minima.

Conclusion

• Implement the linear regression model to fully understand the concepts.
• Suggested further learning: Complete implementation of the linear regression model using Python.

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