Elastic Net Regression : The Best of L1 and L2 Norm Penalties

Elastic net represents a merging of the well-known regularized versions of linear regression, namely ridge and lasso. While ridge employs an L2 penalty and lasso utilizes an L1 penalty, elastic net incorporates both penalties. The advantage of elastic net is that you aren’t constrained to choosing between these two models, as it seamlessly integrates both L2 and L1 penalties. In practical applications, opting for elastic net is generally preferred over selecting ridge or lasso individually. This article aims to equip you with comprehensive knowledge to successfully employ elastic net in your analyses.

Introduction:
You may be familiar with linear regression, and perhaps you’ve encountered ridge and lasso regression in your studies. These two variations of linear regression aim to enhance its robustness. In modern applications, it’s unusual to use traditional linear regression without employing variations like ridge or lasso. In previous articles, we explored how ridge and lasso work, their differences, strengths, weaknesses, and practical implementation. Now, the question arises: Should you use Ridge or Lasso? The good news is that with elastic net, you don’t have to choose. This article, part of a series on ridge and lasso regression, delves into elastic net and how to use both ridge and lasso penalties simultaneously.

Prerequisites:
Before delving into elastic net, it’s recommended to read articles on ridge and lasso as this article builds upon that knowledge. Elastic net is based on ridge and lasso, making it essential to understand these models first.

The Problem:
Why do we need variations like ridge and lasso when linear regression exists? This question is explored in previous articles. To recap, consider a dataset of figure prices with features like age and price. Linear regression, when applied to predict figure prices based on age, may lead to overfitting. Ridge regression was introduced as a remedy, addressing overfitting caused by large model parameters. The loss function was modified to penalize large parameters, either by squaring or taking absolute values.

We then split our dataset into a train set and a test set, and trained our linear regression (OLS regression) model on the training data. Here’s how that looked like:

Elastic Net:
Elastic net combines the penalties of ridge and lasso, resulting in a comprehensive loss function. The elastic net loss function includes both L1 and L2 penalties, controlled by two parameters, α₁ and α₂. If α₁ is 0, elastic net becomes ridge regression; if α₂ is 0, it becomes lasso regression. Alternatively, a single α parameter and an L1-ratio parameter can be used. Cross-validation helps determine the best ratio between L1 and L2 penalty strength. Elastic net is often recommended over lasso or ridge, especially when feature importance is uncertain.

However there is one problem with this loss function. Since the model parameters can be negative, adding them might decrease the loss instead of increasing it. In order to circumvent this, we can either square our model parameters or take their absolute values:

Solving Elastic Net:
Solving elastic net involves different approaches based on the L1-ratio parameter. If L1-ratio is 0, treating it as ridge regression allows using the normal equation or gradient descent. For L1-ratio of 1 (lasso regression), methods like subgradient descent or coordinate descent are employed due to the presence of absolute values. When both L1 and L2 penalties are applied, similar techniques as used for lasso regression can be employed.

In conclusion, elastic net offers a flexible regularization approach by combining ridge and lasso penalties. This provides a nuanced solution, especially when uncertainty exists regarding the importance of features. Cross-validation aids in finding the optimal L1 and L2 penalty strengths. Understanding the prerequisites of ridge and lasso regression is crucial for effectively implementing elastic net.