Week 5: Regression Methods: From Linear to Logistic
Overview
This week progresses from linear regression through logistic regression, exploring both direct and iterative solutions, and extending to multiclass classification.
Learning Objectives
- Master Ordinary Least Squares (OLS)
- Understand iterative optimization methods
- Implement binary logistic regression
- Extend to multiclass classification
- Apply gradient descent variations
1. Linear Regression Solutions
- Ordinary Least Squares
- Direct solution method
- Matrix formulation
- Computational considerations
- Limitations
- Iterative Approach
- Gradient descent formulation
- Step size selection
- Convergence criteria
- Advantages over direct solution
2. Binary Logistic Regression
- Problem Formulation
- From linear to logistic
- Sigmoid function
- Probability interpretation
- Decision boundaries
- Gradient Descent Solution
- Loss function
- Gradient computation
- Parameter updates
- Optimization process
3. Multiclass Extension
- One-vs-All Approach
- Multiple binary classifiers
- Decision boundaries
- Implementation considerations
- Softmax Regression
- Multinomial logistic regression
- Cross-entropy loss
- Gradient computation
- Class probabilities
4. Optimization Methods
- Gradient Descent Variations
- Batch gradient descent
- Stochastic gradient descent
- Mini-batch approach
- Implementation Details
- Learning rate selection
- Batch size considerations
- Convergence monitoring
- Performance trade-offs
Key Takeaways
- OLS provides direct solution for linear regression
- Iterative methods enable more complex models
- Logistic regression handles classification
- Multiple approaches for multiclass problems
Practical Exercises
- Implement OLS solution
- Build iterative optimizer
- Create binary classifier
- Extend to multiclass problems