Gradient Descent

Introduction

• Presented by: Josh Starmer of StatQuest
• Key Topic: Gradient Descent Algorithm
• Assumptions: Basic understanding of least squares and linear regression
• Fields of Application: Statistics, Machine Learning, Data Science

Optimization Examples

• Linear Regression: Optimize intercept and slope
• Logistic Regression: Optimize a squiggle
• t-SNE: Optimize clusters

Importance of Gradient Descent

• Can be used to optimize diverse functions in statistics and data science
• Once learned, it applies to various optimization problems

Example Dataset

• X-axis: Weight
• Y-axis: Height
• Objective: Fit a line to predict height from weight

Initial Setup

• Start with: Least Squares estimate for slope (0.64)
• Goal: Use Gradient Descent to find optimal intercept

Gradient Descent Steps

1. Initial Guess: Pick a random value for the intercept (e.g., 0)
2. Loss Function: Sum of squared residuals
3. Calculate Residuals: Difference between observed and predicted heights
4. Plot Sum of Squared Residuals: Against different intercept values

Efficiency of Gradient Descent

• Big Steps: When far from optimal value
• Small Steps: When close to optimal value
• Adaptive Step Size Calculation: Related to slope and learning rate

In-Depth Calculation

1. Compute Derivative: Of the sum of squared residuals w.r.t intercept
2. Update Intercept: Based on step size
3. Iterate: Until step size is close to zero or maximum iterations are reached

Stopping Criteria

• When step size is very close to zero
• Maximum number of steps (e.g., 1000)

Review

1. Use sum of squared residuals as the loss function
2. Compute derivatives to find slopes
3. Iteratively update intercept values

Extending to Multiple Parameters

• Derive derivatives w.r.t. all parameters including slope
• Iterate with updated parameters to minimize loss function

Gradient Descent with More Parameters

• Treat intercept and slope as constants depending on the derivative
• Calculate step sizes and update both intercept and slope
• Same approach generalizes to more than two parameters

Different Loss Functions

• Gradient Descent also applies to other loss functions beyond sum of squared residuals
• Common Steps: Derive gradients, pick initial parameters, iterative optimization

Stochastic Gradient Descent

• Uses subsets of the data to reduce computation time
• Effective for large datasets

Closing Remarks

• Gradient Descent is versatile for optimization problems in data science
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