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Concept# Elastic net regularization

Summary

In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods.
The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on
Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part () to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization).
The estimates from the elastic net method are defined by
The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where or . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by .
Examples of where the elastic net method has been applied are:
Support vector machine
Metric learning
Portfolio optimization
Cancer prognosis
In late 2014, it was proven that the elastic net can be reduced to the linear support vector machine.
A similar reduction was previously proven for the LASSO in 2014.

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