Scikit-learn 26: Understanding Ridge in Supervised Learning

In this tutorial, we will be discussing the intuition behind Ridge regression, a widely used technique in supervised learning for dealing with multicollinearity in linear regression models.

Before diving into Ridge regression, let’s briefly discuss linear regression. Linear regression is a simple yet powerful technique used to model the relationship between a dependent variable and one or more independent variables. The goal of linear regression is to find the best-fit line that minimizes the sum of squared errors between the predicted values and the actual values in the data set.

However, in practice, linear regression can suffer from multicollinearity, which occurs when two or more independent variables in the model are highly correlated. This can lead to unstable coefficient estimates and inflated variances, making the model less reliable.

This is where Ridge regression comes into play. Ridge regression is a regularized version of linear regression that adds a penalty term to the cost function, which helps to shrink the coefficient estimates towards zero and reduce multicollinearity.

The mathematical formulation of Ridge regression can be represented as follows:

minimize ||Y - Xw||^2 + alpha * ||w||^2

Where:

• Y is the vector of observed target values
• X is the matrix of feature values
• w is the vector of coefficient estimates
• alpha is the regularization parameter

The first term in the cost function represents the squared error between the predicted values and the actual values, while the second term represents the L2 norm of the coefficient estimates.

By introducing the penalty term, Ridge regression encourages smaller coefficient values, which helps to reduce the impact of multicollinearity and improve the stability of the model.

Now, let’s discuss the intuition behind Ridge regression. The penalty term in Ridge regression serves as a deterrent against large coefficient values. When the penalty term is large (i.e., when alpha is large), the model will prioritize smaller coefficient values in order to minimize the total cost. This can help to prevent overfitting and improve the generalization performance of the model.

On the other hand, when the penalty term is small (i.e., when alpha is small), the model will place less emphasis on shrinking the coefficient values, allowing them to take on larger values. In this case, the model may be more susceptible to multicollinearity issues.

In conclusion, Ridge regression is a powerful technique for dealing with multicollinearity in linear regression models. By introducing a penalty term to shrink the coefficient estimates towards zero, Ridge regression helps to improve the stability and generalization performance of the model. It is important to tune the regularization parameter alpha carefully to achieve the best balance between bias and variance in the model.