Regression analysis is a powerful statistical technique to model the relationship between a dependent variable and one or more independent variables. It finds applications in various fields, including economics, finance, biology, and machine learning. One of the critical considerations in regression analysis is the choice of regularization techniques, with L1 and L2 regression being two of the most widely used methods.
● L1 Regularization (Lasso): L1 regularization adds the absolute values of the coefficients as a penalty term to the linear regression’s loss function. It encourages some coefficients to be exactly zero, effectively performing feature selection.
● L2 Regularization (Ridge): L2 regularization adds the squared values of the coefficients as a penalty term to the linear regression’s loss function. It penalizes large coefficient values and encourages all coefficients to be small but non-zero.
● The primary purpose of L1 and L2 regularization in linear regression is to prevent overfitting. They achieve this by adding a penalty term to the loss function that discourages the model from fitting the data too closely, which can result in complex and unstable models. L1 encourages sparsity and feature selection, while L2 encourages smaller, more evenly balanced coefficients.
● L1 regularization tends to drive some of the coefficients to exactly zero, effectively performing feature selection by eliminating certain features from the model. L2 regularization, on the other hand, shrinks the coefficients toward zero but rarely makes them exactly zero. It encourages all features to contribute to the prediction but with smaller values.
● For L1 regularization (Lasso), the hyperparameter is usually denoted as “alpha” or “lambda,” which controls the strength of the penalty. Higher values of alpha result in more coefficients being exactly zero.
● For L2 regularization (Ridge), the hyperparameter is also typically “alpha” or “lambda.” Increasing alpha leads to smaller coefficient values.
● You might choose L1 regularization (Lasso) when you suspect that only a subset of the features is relevant for the task, and you want to perform feature selection. L2 regularization (Ridge) is a good choice when you want all features to contribute to the prediction, but you want to prevent any one feature from having a disproportionately large effect.
● Elastic Net regularization combines both L1 and L2 regularization. It adds a linear combination of the L1 and L2 penalties to the loss function. This allows it to address the limitations of both L1 and L2 regularization and find a balance between feature selection and coefficient shrinking.
● The optimal alpha value is typically found through techniques like cross-validation. By trying a range of alpha values and measuring the model’s performance on a validation set, you can determine which value results in the best trade-off between bias and variance.