What is Lasso Regression?

Lasso Regression is an extension of linear regression designed to reduce overfitting and improve model simplicity. It’s especially helpful when dealing with high-dimensional data by automatically eliminating irrelevant features, making the model more interpretable and efficient.


Concept of Lasso Regression

Lasso stands for Least Absolute Shrinkage and Selection Operator. It is a linear regression model that includes an L1 regularization term to penalize large coefficients.

Standard Linear Regression Equation:


y = β₀ + β₁x₁ + β₂x₂ + ... + βₙxₙ + ε

Lasso Regression Loss Function:


Loss = RSS + λ \* Σ|βᵢ|

  • RSS: Residual Sum of Squares
  • λ: Regularization strength (higher λ means more shrinkage)
  • Σ|βᵢ|: Sum of the absolute values of the coefficients (L1 norm)

The L1 penalty causes some coefficients to shrink to exactly zero, which effectively removes those variables from the model.


Real-World Example: House Price Prediction

Imagine you’re building a model to predict house prices with features like:

  • Size (sqft)
  • Number of rooms
  • Elevator availability
  • Year built
  • Complex name
  • Maintenance fee

Some of these variables may not significantly impact the price. Lasso Regression automatically eliminates less important features by assigning their coefficients to zero, leaving only the most relevant predictors.


When to Use Lasso?

  1. High-dimensional datasets
    • Especially useful when the number of features is large or features are correlated.
  2. Need for feature selection
    • Lasso automatically performs variable selection.
  3. Interpretability
    • It produces simpler, more interpretable models by reducing the number of active features.

Limitations of Lasso

  1. Unstable when features > samples
    • It may not select the correct variables in such cases.
  2. Biased estimation
    • Coefficients are biased due to the shrinkage effect.
  3. Sensitivity to correlated variables
    • Among highly correlated features, Lasso tends to select one and ignore the others.

Comparison: Lasso vs Ridge vs ElasticNet

Model Regularization Feature Elimination Key Feature
Ridge L2 (squared) Shrinks all coefficients
Lasso L1 (absolute) Eliminates some coefficients to 0
ElasticNet L1 + L2 Hybrid of Ridge and Lasso

Scikit-Learn Example

from sklearn.linear_model import Lasso
from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error

# Load data
X, y = load_boston(return_X_y=True)

# Split train/test
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Train model
lasso = Lasso(alpha=1.0)
lasso.fit(X_train, y_train)

# Predict and evaluate
y_pred = lasso.predict(X_test)
print("MSE:", mean_squared_error(y_test, y_pred))
print("Coefficients:", lasso.coef_)

Summary

Lasso Regression is a powerful technique that enhances linear regression by simplifying models and reducing overfitting through L1 regularization. Its ability to perform automatic feature selection makes it particularly useful in real-world applications involving many variables.

For interpretable, efficient, and generalizable models, Lasso Regression is a go-to method in the machine learning toolbox.