Regression Shrinkage and Selection via the Lasso

Rob Tibshirani 1996

Robert Tibshirani’s 1995 paper introduces a new method for regression called the Lasso. The Lasso is designed to improve the prediction accuracy and interpretability of statistical models by imposing a constraint that results in simpler models.

Key Concepts

• Linear Regression: This is a basic statistical method used to model the relationship between a dependent variable and one or more independent variables.

• Overfitting: This occurs when a model is too complex and captures the noise in the data rather than the underlying pattern. Overfitted models perform well on training data but poorly on new, unseen data.

• Shrinkage Methods: These are techniques used to address overfitting by introducing a penalty that shrinks the regression coefficients towards zero. Ridge regression is an example, where the penalty is the sum of the squared coefficients.

The Lasso

The Lasso modifies the linear regression by adding a penalty equal to the sum of the absolute values of the coefficients. Mathematically, it solves the following optimization problem:

\[\min_{\beta} \left( \sum_{i=1}^{n} (y_i - \sum_{j=1}^{p} x_{ij} \beta_j)^2 + \lambda \sum_{j=1}^{p} |\beta_j| \right)\]

• \(\beta\) represents the regression coefficients.

• \(y_i\) is the observed value.

• \(x_{ij}\) are the predictor variables.

• \(\lambda\) is a parameter that controls the amount of shrinkage. Larger values of \lambda result in more coefficients being shrunk to zero.

Benefits of the Lasso

• Variable Selection: Unlike ridge regression, the Lasso can set some coefficients exactly to zero, effectively selecting a simpler model with fewer predictors.

• Interpretability: By producing simpler models with fewer predictors, the Lasso makes it easier to understand the relationship between the variables.

• Prediction Accuracy: The Lasso can improve the accuracy of predictions on new data by reducing overfitting.

• The Lasso is particularly useful when dealing with datasets that have a large number of predictors.

• It is a powerful tool for both improving prediction performance and simplifying models, making it easier to communicate results to non-experts.

Tibshirani’s introduction of the Lasso provides a valuable addition to the set of tools available for regression analysis. By combining the benefits of variable selection and shrinkage, the Lasso helps statisticians build models that are both accurate and interpretable.

Conversation with Rob Tibshirani

1995 Paper

If you are unable to view the interview, please watch it here. If you are unable to view any of the documents above, please download the paper.