Explain the contrast between L1 and L2 regularization methods used in regression analysis, and when one would be favored over the other.

Question Analysis

This question tests your understanding of regularization techniques in machine learning, specifically L1 and L2 regularization. Regularization is used to prevent overfitting by adding a penalty term to the loss function. You need to explain the differences between these two types of regularization and provide scenarios where one might be preferred over the other. This involves discussing how each method influences model complexity, feature selection, and computational efficiency.

Answer

L1 and L2 Regularization: An Overview

• L1 Regularization (Lasso)

  • Penalty Term: Adds the absolute value of the magnitude of coefficients as a penalty term to the loss function.
  • Equation: ( \text{Loss} = \text{RSS} + \lambda \sum |w_i| )
  • Effect on Model: Encourages sparsity, meaning it tends to drive some coefficients to zero, effectively performing feature selection.
  • When to Use:
    • When you suspect that only a few features are important.
    • When you want to produce a model that is easy to interpret.
    • In situations where you need to perform feature selection.

• L2 Regularization (Ridge)

  • Penalty Term: Adds the square of the magnitude of coefficients as a penalty term to the loss function.
  • Equation: ( \text{Loss} = \text{RSS} + \lambda \sum w_i^2 )
  • Effect on Model: Tends to shrink coefficients but doesn't necessarily zero them out. This helps in distributing the weight more evenly.
  • When to Use:
    • When you have many features that contribute to the output.
    • When you expect that all features contribute to the output to some extent.
    • In cases where multicollinearity is a concern, as L2 can handle correlated features better.

Choosing Between L1 and L2

• L1 Regularization is favored when model interpretability is crucial and when you expect that only a subset of features will be significant.
• L2 Regularization is preferred when you want to maintain all features and ensure that they contribute to the model, especially when dealing with multicollinearity.

Both methods can be used in conjunction (Elastic Net) to leverage the benefits of both L1 and L2 regularization when neither method alone suffices.