Ridge Regression in Machine Learning

Ridge Regression is a type of linear regression that addresses the problem of multicollinearity and overfitting by adding a regularization term to the cost function. It is particularly useful when the dataset has highly correlated features or when there are more predictors than observations.

The key idea of ridge regression is to shrink the regression coefficients toward zero by penalizing their size. This prevents the model from becoming overly complex and helps improve its generalization ability.

How Does Ridge Regression Work?

Step 1: Linear Combination of Inputs

Like standard linear regression, ridge regression starts by calculating a linear combination of the input features:

\( y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_nx_n \)

Here:

\( y \): Predicted value of the target variable
\( x_1, x_2, \dots, x_n \): Input features (independent variables)
\( \beta_0 \): Intercept term (bias)
\( \beta_1, \beta_2, \dots, \beta_n \): Coefficients or weights of the features

The primary difference lies in how the model is trained, as ridge regression modifies the cost function to include a regularization term.

Step 2: Regularized Cost Function

Ridge regression minimizes the Mean Squared Error (MSE) with an additional penalty term based on the L2 norm of the coefficients:

\( J(\beta) = \frac{1}{n} \sum_{i=1}^n (y_i – \hat{y}_i)^2 + \lambda \sum_{j=1}^n \beta_j^2 \)

Here:

\( J(\beta) \): Ridge regression cost function
\( y_i \): Actual target value for the \(i^{th}\) data point
\( \hat{y}_i \): Predicted target value for the \(i^{th}\) data point
\( \lambda \): Regularization parameter (controls the strength of the penalty)
\( \beta_j \): Coefficient of the \(j^{th}\) feature

The regularization parameter \( \lambda \) determines the trade-off between minimizing the error and shrinking the coefficients. A higher \( \lambda \) results in greater shrinkage, reducing the model’s complexity.

Step 3: Model Training

During training, ridge regression solves for the coefficients (\( \beta_0, \beta_1, \dots, \beta_n \)) using the regularized cost function. The optimal coefficients can be obtained by solving the following equation:

\( \beta = (X^TX + \lambda I)^{-1}X^Ty \)

Here:

\( X \): Feature matrix
\( y \): Target variable vector
\( I \): Identity matrix (used for regularization)
\( \lambda \): Regularization parameter

By adding \( \lambda I \), ridge regression ensures that the feature matrix \( X \) is invertible, even when the features are highly correlated or when there are more features than observations.

Step 4: Prediction

Once trained, the model predicts the target values for new data points using the learned coefficients:

\( \hat{y} = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_nx_n \)

Here, \( \hat{y} \) is the predicted value, and \( x_1, x_2, \dots, x_n \) are the feature values of the new data point.

Key Characteristics of Ridge Regression

Addresses Multicollinearity: Ridge regression stabilizes the solution when features are highly correlated.
Prevents Overfitting: The regularization term penalizes large coefficients, reducing model complexity.
Bias-Variance Trade-off: Introduces a small bias to reduce variance and improve generalization.

Advantages of Ridge Regression

Handles Correlated Features: Works well when features are collinear, unlike ordinary linear regression.
Improves Generalization: Reduces overfitting, making the model more robust to new data.
Works with High-Dimensional Data: Effective when the number of features exceeds the number of observations.

Limitations of Ridge Regression

Does Not Perform Feature Selection: Unlike Lasso regression, ridge regression does not shrink coefficients to zero, retaining all features.
Sensitive to Scaling: Features need to be standardized or normalized for effective regularization.
Choice of Regularization Parameter: The performance depends on selecting an appropriate \( \lambda \), which often requires cross-validation.

TutorialKart

Ridge Regression in Machine Learning