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Penalized regression is a general term that refers to the practice of adding a penalty term to the standard linear regression objective function. This penalty term is used to discourage large coefficients in the model. It's a broad concept that encompasses various types of regularization, including Lasso and Ridge. Elastic Net is a penalized regression technique. It combines the L1 regularization (Lasso) and L2 regularization (Ridge) techniques into a single model, providing a compromise between the two.

Here's why Elastic Net is considered a penalized regression technique:

1. L1 Regularization (Lasso Penalty): Elastic Net includes the L1 regularization term, which adds the absolute values of the coefficients as a penalty to the linear regression loss function. This penalty encourages sparsity in the model, meaning it helps drive some of the coefficient estimates to exactly zero. In other words, it can automatically select a subset of the most important features for the model while reducing the impact of less important ones. This is a form of regularization because it prevents overfitting by discouraging the model from assigning overly large coefficients to any single feature.

2. L2 Regularization (Ridge Penalty): Elastic Net also includes the L2 regularization term, which adds the squares of the coefficients as a penalty to the loss function. This penalty discourages coefficients from becoming too large and helps in preventing multicollinearity (correlation among predictor variables). L2 regularization can improve the numerical stability of the model by preventing very large coefficient values.

By combining both L1 and L2 regularization terms, Elastic Net offers a flexible way to control the bias-variance trade-off in regression modeling. The L1 penalty helps with feature selection and sparsity, while the L2 penalty helps with coefficient shrinkage and multicollinearity control. The mixing parameter in Elastic Net allows you to adjust the balance between these two penalties, providing a versatile tool for regression tasks, especially when dealing with high-dimensional datasets with potentially correlated features.

Note that penalized regression, Lasso, and Ridge regression are related concepts in the field of linear regression, but they are not exactly the same as L1 and L2 regularization.

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