Ahmed Dawoud

While Lasso regression helps in variable selection by shrinking some coefficients to zero, it does not guarantee that it will select the exact set of true predictors. This limitation is especially pronounced in situations where predictors are highly correlated or when the true model does not exhibit strong sparsity.

Mathematical Explanation

Let’s consider the linear regression model:

where:

• Y is the response variable.
• X is a vector of p predictors.
• β is a vector of coefficients.
• ϵ is the error term.

In Lasso regression, we solve the following optimization problem:

where:

• The first term is the sum of squared errors.
• The second term is the penalty term, proportional to the sum of the absolute values of the coefficients bj​.
• λ is the penalty parameter controlling the strength of regularization.

Why Lasso May Not Select the Correct Variables

1. Correlated Predictors: When predictors are highly correlated, Lasso may arbitrarily select one predictor over another, even if both are important.
2. Small but Non-Zero Coefficients: Lasso tends to set coefficients with small but non-zero true values to zero, potentially missing important predictors.
3. Approximate Sparsity: Lasso is designed to work well under the assumption of approximate sparsity, but if the true model deviates significantly from this assumption, Lasso may not perform well.

Consider a scenario where we aim to predict a response variable Y using three predictors X1, X2, X3​. Suppose the true model is:

Scenario 1: Correlated Predictors

If X1​ and X2​ are highly correlated, Lasso may arbitrarily choose one over the other. For instance:

• If X1​ and X2​ are highly correlated (cor(X1, X2)≈1), Lasso might set β1​ to 0 and retain β2​, or vice versa.

Scenario 2: Small but Non-Zero Coefficients

If β3​ is small but non-zero, Lasso might set it to zero, potentially missing an important predictor

Mathematical Illustration:

• Suppose λ is chosen such that Lasso sets β3​ to 0. The model becomes: Y=0.5X1 + 0.5X2 + 0 +ϵ
• The resulting model ignores X3​, which contributes to the response variable, albeit with a small coefficient.

Practical Importance

1. Interpretation of Selected Variables: Practitioners should be cautious when interpreting the variables selected by Lasso. The absence of a variable in the Lasso model does not necessarily imply that it has no effect on the response variable.
2. Model Stability: In high-dimensional settings with correlated predictors, the Lasso solution can be unstable. Small changes in the data can lead to different sets of selected variables. This instability can be problematic for understanding the underlying data-generating process.
3. Alternative Methods:
   • Other methods, such as Elastic Net, which combines the Lasso (L1) and Ridge (L2) penalties, can help mitigate some of the limitations of Lasso, especially in the presence of correlated predictors.
   • Stability selection, a technique that involves repeatedly applying Lasso to subsamples of the data, can provide more stable variable selection results.

Summary

Lasso regression is a powerful tool for variable selection in high-dimensional settings, but it does not guarantee the selection of the correct set of predictors. This limitation arises primarily due to correlated predictors and small but non-zero coefficients. Understanding these limitations is crucial for practitioners to make informed decisions about model interpretation and selection. Employing alternative methods or techniques can help address some of these challenges and lead to more robust and reliable models.