Ridge vs Lasso Regression, Visualized!!!

Summary

In this StatQuest video, Josh Stormer explains the differences between Ridge and Lasso regression, using a simple dataset of weight and height measurements. The video visually demonstrates how Ridge (L2 penalty) shrinks the optimal slope towards zero without fully eliminating it, while Lasso (L1 penalty) forces the optimal slope to exactly zero as the penalty increases. The video highlights how the penalties affect the sum of squared residuals, with Ridge resulting in a smoother decrease and Lasso producing a sharp kink at zero, illustrating their distinct impact on model fitting and variable selection.

Takeaways

• 😀 Ridge and Lasso regression are both techniques used for regularization in linear models, but they differ in the way they penalize coefficients.
• 😀 Ridge regression uses the L2 penalty (squared penalty), which shrinks coefficients towards zero but never forces them to be exactly zero.
• 😀 Lasso regression uses the L1 penalty (absolute value penalty), which can shrink coefficients to zero, effectively performing variable selection.
• 😀 When lambda (penalty) is zero, both Ridge and Lasso regressions behave the same as standard linear regression without any regularization.
• 😀 As lambda increases, Ridge regression gradually shrinks the slope values, keeping the coefficients close to zero but never exactly at zero.
• 😀 In contrast, Lasso regression results in a sharp kink in the curve, where the optimal slope becomes exactly zero when lambda is sufficiently large.
• 😀 With Ridge regression, the optimal slope values shift towards zero but remain positive, even with very high lambda values.
• 😀 Lasso regression's optimal slope can reach zero when the penalty is large enough, essentially removing certain variables from the model.
• 😀 The Ridge regression curve is smooth and parabolic, while the Lasso regression curve shows a sharp change at the point where the coefficient becomes zero.
• 😀 Increasing lambda in Ridge regression results in a smooth, continuous shrinkage of the coefficients, while Lasso leads to a more discrete variable selection process.

Q & A

What is the main difference between Ridge regression and Lasso regression?

The main difference between Ridge and Lasso regression lies in their penalty functions. Ridge regression uses the L2-norm penalty, which shrinks the coefficients towards zero without eliminating them. Lasso regression uses the L1-norm penalty, which can shrink some coefficients all the way to zero, effectively eliminating certain features from the model.

Why is the L2-norm penalty called the squared penalty?

The L2-norm penalty is called the squared penalty because it involves squaring the coefficients when calculating the penalty. This results in a penalty proportional to the square of the coefficients, which encourages smaller values but never reduces them to zero.

What happens to the optimal slope as the lambda value increases in Ridge regression?

As the lambda value increases in Ridge regression, the optimal slope moves closer to zero. This happens because the penalty term increases, shrinking the coefficients, but the slope will never reach zero, even with a very large lambda.

What is the effect of increasing lambda in Lasso regression?

In Lasso regression, increasing lambda not only shrinks the coefficients but can also reduce some of them to exactly zero. This results in the elimination of some features from the model, making Lasso useful for feature selection.

What is the significance of the 'kink' in the Lasso regression penalty curve?

The 'kink' in the Lasso regression penalty curve represents a point where the coefficient becomes exactly zero. As lambda increases, the curve shows this kink more clearly, and beyond a certain point, the optimal slope becomes zero, meaning that the variable is no longer used in the model.

How does Ridge regression prevent overfitting?

Ridge regression prevents overfitting by adding a penalty term to the sum of squared residuals. This penalty discourages large coefficients, which can lead to overfitting by reducing the model's complexity and making it more generalizable.

Why might you choose Ridge regression over Lasso?

You might choose Ridge regression over Lasso if you believe that all features should be included in the model, but you want to reduce their influence. Ridge is better when you have many small, correlated features that you don't want to exclude from the model.

What happens when lambda is set to a very large value in Ridge regression?

When lambda is set to a very large value in Ridge regression, the penalty becomes very strong, causing the coefficients to shrink further. However, the slope will never reach zero, even with an extremely large lambda, which means all features remain in the model, albeit with very small coefficients.

How can Lasso regression help with feature selection?

Lasso regression helps with feature selection by shrinking some coefficients to zero as lambda increases. This effectively removes certain variables from the model, allowing it to focus only on the most important features for prediction.

In the example, how does the penalty affect the best-fitting line in Ridge regression?

In the Ridge regression example, as the penalty (lambda) increases, the best-fitting line's slope decreases. The optimal slope gets closer to zero, but it never reaches zero, meaning the feature is still retained, though its influence is reduced.