Lecture Notes: Ridge Regression and Regularization

Introduction

• Regularization: Technique to reduce model complexity and prevent overfitting by adding a penalty term to the loss function.
• Ridge Regression: A type of linear regression that includes a regularization term.
  • Part of a series on regularization techniques.
  • Assumes knowledge of regularization, bias & variance, linear models, and cross-validation.

Main Goals

1. Understanding Ridge Regression: Introduction to concepts and how it works.
2. Details of Ridge Regression: Mathematical formulation and functioning.
3. Applications: How it works in various scenarios.
4. Capabilities: Solving high-dimensional problems.

Ridge Regression Basics

• Model weight and size of mice using linear regression (least squares).
• Least Squares: Minimizes sum of squared residuals.
  • Works well with large datasets.
  • Risk of overfitting with small datasets.

Overfitting and Variance

• Overfitting occurs when a model fits training data too closely, leading to high variance on testing data.
• Ridge Regression introduces bias to reduce variance.
  • Better long-term predictions by starting with a slightly worse fit.

Mathematical Formulation

• Ridge Regression Equation: Minimizes sum of squared residuals plus lambda times slope squared.
• Lambda (λ): Regularization parameter controlling penalty severity.
  • When λ = 0, ridge regression is equivalent to least squares.
  • Increasing λ reduces slope, decreasing sensitivity of predictions to changes in independent variables.

Effect of Ridge Regression

• Slope penalty reduces sensitivity of predictions to changes in independent variables.
• Ridge regression produces less sensitive predictions, reducing overfitting.

Choosing Lambda (λ)

• Determine optimal λ using cross-validation (e.g., 10-fold cross-validation).

Ridge Regression with Discrete Variables

• Also applies when predictors are discrete (e.g., diet types).
  • Penalizes differences between group means.
• Effective for small datasets, improving prediction accuracy.

Ridge Regression in Logistic Regression

• Applies to binary outcomes (e.g., obesity prediction).
• Optimizes likelihoods rather than squared residuals.

High-Dimensional Problems

• Effective in situations with more parameters than data points.
  • E.g., gene expression data.
• Solves for parameters in underdetermined systems using penalty term.

Summary

• Ridge regression adds a penalty term to reduce variance and improve predictions on new data.
• Useful for small sample sizes and high-dimensional datasets.
• Cross-validation used to determine optimal λ.
• Allows estimation of parameters when traditional least squares cannot.

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