Elastic Net Regularization: Balancing Between L1 and L2 Penalties
Published: 2024-06-07 18:46:40

Elastic Net regularization stands out by combining the strengths of both L1(lasso) and L2(Ridge) regularization methods. This article will explore the theoretical, mathematical and practical aspects of the Elastic Net regularization. 


## Lasso vs. Ridge Regression

- **Lasso Regression:** Adding L1 norm penalty, promoting sparsity by driving some coefficients to zero. This can lead to feature selection. However, Lasso can struggle with highly correlated features. 
- **Ridge Regression:** Adding L2 norm penalty, shrinking all coefficients towards zero but not necessarily driving them to zero. This avoids sparsity but can be less effective in feature selection. 


## Elastic Net Regularization

Elastic Net regularization is a combined approach that blends L1 and L2 regularization penalties. Elastic Net addresses some limitations of Lasso and Ridge, particularly in scenarios with highly correlated features. 

## Mathematical Formulation

The Elastic Net regularization adds both L1 and L2 penalties to the loss function. The penalty term is: 

![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/r58p0vbpndbqvl1d97ua.png)

## Understanding the impact: 

- The L1 penalty from Lasso encourages sparsity, potentially driving some coefficients to zero(feature selection)
- The L2 penalty from ridge regression shrinks all coefficients towards zero, promoting smoother coefficient shrinkage and potentially better handling of correlated features. 

By adjusting the values of lambda1 and lambda2, we can control the relative influence of the L1 and L2 penalties. A higher lambda1, encourages more sparsity, while a lower lambda2 smother coefficients shrinkage. 

## Benefits of Elastic Net: 

- **Overfitting:** Elastic net helps prevents overfitting by penalizing overly complex models. 
- **Feature Selection:** The L1 component can drive coefficients to zero, potentially performing feature selection. 
- **Handles Correlated Features:** Elastic net can be more robust to highly correlated features. 

## Choosing the Right value:

Finding the optimal values for λ₁ and λ₂ is crucial for optimal performance. Techniques like cross-validation are employed to identify the combination of λ₁ and λ₂ that minimizes the validation error while maintaining a desirable sparsity level.

## When to use 
- When the dataset is quite large
- input columns have multicollinearity

## Practical Implementation

```
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import ElasticNet
from sklearn.datasets import make_regression

X, y = make_regression(n_samples=100, n_features=10, noise=0.1, random_state=42)

elastic_net = ElasticNet(alpha=0.1, l1_ratio=0.5)  # alpha controls L1 & L2, l1_ratio controls L1 vs L2 ratio
elastic_net.fit(X, y)

plt.figure(figsize=(12, 6))
plt.plot(range(X.shape[1]), elastic_net.coef_, marker='o', linestyle='none')
plt.xlabel('Feature Index')
plt.ylabel('Coefficient Value')
plt.title('Elastic Net Coefficients')
plt.xticks(range(X.shape[1]))
plt.grid(True)
plt.show()
```
## Conclusion
In conclusion, Elastic Net regularization is a versatile and effective technique for improving the performance and interpretability of linear regression models. By leveraging both L1 and L2 penalties, it offers a comprehensive solution that can be fine-tuned to suit a variety of datasets and modelling challenges.