Understanding Regularization in Supervised Learning

Regularization is an important technique used in supervised learning. It helps solve the problem of overfitting. Overfitting happens when a model learns too much from the training data but doesn’t perform well with new, unseen data.

There are two common types of regularization: L1 Regularization and L2 Regularization. Each has its own benefits and uses.

L1 Regularization (Lasso)

L1 regularization, also called Lasso, adds a penalty based on the size of the model's coefficients. Coefficients are the numbers used by the model to make predictions.

The formula looks like this:

$$J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{y}_i)^2 + \lambda \sum_{j=1}^{n} |\theta_j|$$

Here’s what the symbols mean:

• $J(\theta)$: This is the cost function, which helps measure how well the model is performing.
• $m$: This is how many training examples we have.
• $y_i$: This is the actual output or the true value.
• $\hat{y}_i$: This is the predicted output from the model.
• $\theta_j$: These are the model’s parameters or coefficients.
• $\lambda$: This controls the strength of the penalty.

Benefits of L1 Regularization:

1. Feature Selection: L1 helps pick important features for the model. It can reduce the number of features used, making the model easier to understand.

2. Managing Multicollinearity: When features are closely related, it can cause problems. L1 helps reduce this issue by keeping coefficients smaller and the model more stable.

L2 Regularization (Ridge)

L2 regularization is also known as Ridge regression. This method adds a penalty based on the square of the size of the coefficients. The formula here is:

$$J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{y}_i)^2 + \lambda \sum_{j=1}^{n} \theta_j^2$$

Benefits of L2 Regularization:

1. Weight Distribution: L2 keeps all the coefficients small but doesn't make any of them zero. This is great when there are fewer features than samples.

2. Better Generalization: By keeping coefficients from getting too large, L2 often helps the model work better with new data, reducing overfitting.

Statistical Impact

Regularization can be measured statistically. Research shows that models using L1 or L2 can be 10% to 30% more accurate than those without regularization, especially with complex data. Additionally, regularization can reduce the variance of the predictions while keeping the bias low.

Practical Considerations

• Choosing Between L1 and L2: Which method to use depends on the problem. If there are many unnecessary features, L1 might be better. If the model needs to be smooth, L2 could be the way to go.

• Tuning Hyperparameters: The $\lambda$ parameter that controls the strength of regularization needs careful tuning. This can be done using methods like grid search or Bayesian optimization.

In summary, L1 and L2 regularization are vital in preventing overfitting in supervised learning. They each have unique strengths that help with different data situations.