Ridge & Lasso Regression
Regularization techniques to prevent overfitting in linear models
What are Ridge and Lasso Regression?
Regularization techniques that extend linear regression to prevent overfitting
Ridge and Lasso regression are regularization techniques that extend linear regression to address overfitting, particularly when dealing with many features or multicollinearity. Both methods add a penalty term to the linear regression cost function, but they differ in the type of penalty applied.
Ridge Regression (L2 Regularization)
Adds a penalty equal to the square of the magnitude of coefficients.
Cost = RSS + α * Σ(β_j²)
Ridge regression shrinks coefficients toward zero but rarely eliminates them completely.
Lasso Regression (L1 Regularization)
Adds a penalty equal to the absolute value of coefficients.
Cost = RSS + α * Σ|β_j|
Lasso regression can reduce coefficients exactly to zero, effectively performing feature selection.
Key Concepts in Regularized Regression
- Regularization Parameter (α): Controls the strength of the penalty; higher values of α apply stronger regularization
- Feature Selection: Lasso can completely eliminate less important features by setting their coefficients to zero
- Bias-Variance Tradeoff: Regularization increases bias but reduces variance, which can lead to better generalization
- Elastic Net: A hybrid approach that combines L1 and L2 penalties, offering a middle ground between Ridge and Lasso
When to Use Regularized Regression
- When dealing with datasets with many features relative to the number of observations
- When there is multicollinearity among features
- To prevent overfitting in complex models
- Use Ridge when you believe most features contribute to the outcome
- Use Lasso when you suspect only a subset of features are relevant
- Use Elastic Net when you have groups of correlated features
Advantages and Limitations
Advantages
- Reduces overfitting in high-dimensional data
- Handles multicollinearity effectively
- Lasso provides built-in feature selection
- Ridge works well when all features are relevant
- Improves model generalization
Limitations
- Requires tuning of the regularization parameter
- May introduce bias in coefficient estimates
- Lasso may be unstable with highly correlated features
- Ridge doesn't perform feature selection
- Performance depends on proper feature scaling