What is Ridge Regression?

Ridge regression is a type of linear regression that includes a regularization term in its cost function. The purpose of this regularization is to prevent overfitting by discouraging large coefficients in the model. The model adds a penalty to the sum of the squared coefficients, which helps control the complexity of the model.

Ridge regression is also known as L2 regularization because it penalizes the sum of the squares of the coefficients.

Key Concepts in Ridge Regression:

1. Linear Regression:
   • In standard linear regression, the model predicts the target variable (y) using a linear combination of input features (X):

y=β0+β1X1+β2X2+…+βnXn

1. • where β0​ is the intercept and β1​,β2​,…,βn​ are the coefficients of the features.
2. Ridge Regression Objective:
   • In ridge regression, we modify the linear regression objective function to include a penalty term based on the sum of the squared values of the coefficients:

Cost Function=MSE + α ∑i=1,n​ βi^2​ where:

1. • MSE (Mean Squared Error) is the original loss function of linear regression.
   • α is a hyperparameter that controls the amount of regularization. A larger $\alpha$ puts more emphasis on the penalty term.
   • βi^2 are the squared coefficients of the model.
2. Effect of Regularization:
   • The regularization term α ∑i=1,n​ βi^2​ discourages large values of the coefficients, which can help reduce overfitting.
   • When α=0, ridge regression is equivalent to standard linear regression (no regularization).
   • As $\alpha$ increases, the coefficients get shrunk towards zero, leading to simpler models that may generalize better on new data.
3. Advantages:
   • Prevents Overfitting: By penalizing large coefficients, ridge regression can prevent the model from fitting noise or irrelevant features in the data.
   • Improves Model Stability: Regularization helps make the model more stable when there is multicollinearity (correlated features).
   • Computationally Efficient: Ridge regression can be solved analytically and is computationally efficient, even for high-dimensional datasets.
4. Disadvantages:
   • Does Not Perform Feature Selection: Unlike Lasso regression (L1 regularization), ridge regression doesn’t produce sparse models (i.e., it doesn’t set some coefficients to exactly zero).
   • Requires Tuning of $\alpha$: The hyperparameter $\alpha$ needs to be chosen carefully using techniques like cross-validation.

Ridge Regression Formula:

The cost function for ridge regression becomes:

$$N$$

Where:

• $N$ is the number of data points.
• yi​ is the actual value for the i-th data point.
• yi​b is the predicted value for the i-th data point.
• βj​ are the model coefficients.
• α is the regularization parameter.

Choosing the Regularization Parameter (α:

• $\alpha =0$: The model behaves like regular linear regression (no regularization).
• α > 0: The model applies regularization, and as α increases, the penalty for large coefficients becomes more significant.
• The optimal value of α is typically chosen via cross-validation.

How Ridge Regression Helps in Practice:

1. Handling Multicollinearity: If the input features are highly correlated, ridge regression can help prevent large fluctuations in the model coefficients, which can be a problem in standard linear regression.
2. Improving Generalization: By controlling the magnitude of the coefficients, ridge regression can help reduce the risk of overfitting, leading to better performance on unseen data.
3. Model Complexity: Ridge regression helps to balance model complexity and performance by adding the regularization term, effectively controlling how much the model can “fit” the training data.

Summary:

• Ridge regression is a regularized version of linear regression that penalizes large coefficients using an L2 penalty term.
• It helps to prevent overfitting, especially when there are many features or multicollinearity.
• The main parameter to tune is α, which controls the amount of regularization.

Ridge regression is widely used in scenarios where you want to build a linear model but avoid overfitting, especially with datasets that have many features.