In ridge regression, a form of regularized linear regression, the bias term is typically not subjected to regularization. Understanding why the bias term is left unregularized involves delving into the mechanics of ridge regression and the role that the bias plays in model training.

Ridge Regression Overview

Ridge regression, also known as $L2$ regularization, is used to prevent overfitting in linear regression models by adding a penalty to the loss function. The standard linear regression model minimizes the residual sum of squares between observed outcomes and predicted values. Ridge regression modifies this objective by adding a penalty equal to the square of the magnitude of coefficients, multiplied by a regularization parameter $\lambda$.

The ridge regression loss function is given by:

\L(\beta) = \sum\_{i=1}^{n} (y\_i - \beta\_0 - \sum\_{j=1}^{p} \beta\_j x\_{ij})^2 + \lambda \sum\_{j=1}^{p} \beta\_j^2\

where $\beta_0$ is the bias (intercept) term, $\lambda$ is the regularization strength, $\beta_j$ are the coefficients, $x_{ij}$ are the feature values, and $y_i$ are the target values.

Noticeably, the bias term $\beta_0$ is not included in the regularization term $\sum_{j=1}^{p} \beta_j^2$.

Reason for Not Regularizing the Bias Term

1. Interpretation of the Bias Term: • The bias term, also known as the intercept, captures the average value of the dependent variable when all independent variables are zero. Regularizing the bias would shift this baseline level, potentially leading to a model that does not align well with the true underlying data distribution.
2. Ensuring Model Flexibility: • Regularization is intended to shrink only the coefficients of the feature variables to prevent overfitting. Including the bias term in this penalty would constrain the model unnecessarily, potentially leading to underfitting.
3. Normalization Considerations: • When regularizing, features are typically standardized (or normalized) to zero mean and unit variance. The bias term accommodates any overall shift in the data, and regularization would interfere with its ability to do so effectively.
4. Geometric Perspective: • From a geometric viewpoint, regularizing the bias would change the orientation of the regression hyperplane in a way that could degrade the fit's accuracy. It would translate the hyperplane regardless of the data alignment.
5. Empirical Performance: • Practically, experiments and empirical results demonstrate less reliable performance when the bias term is regularized. Models tend to perform better with only the feature coefficients being regularized.

Example

Consider a dataset where the feature values are standardized but the target values have a non-zero mean. Regularizing the bias term in such a scenario could cause the model to fit poorly as it’s unable to fully capture the mean shift present in the target distribution.

Summary Table

Additional Considerations

• Alternative Approaches: If concerns about the bias remain, other regularization methods or feature engineering techniques can be explored to balance the model's complexity and performance. Elastic Net regularization, for example, combines $L1$ and $L2$ penalties but also excludes the bias term unless otherwise modified. • Software Implementations: Popular machine learning libraries such as scikit-learn implement ridge regression with an unregularized bias term, reflecting best practices based on theoretical considerations and empirical findings.

In conclusion, leaving the bias term unregularized allows ridge regression to maintain a necessary degree of flexibility to accurately model data while controlling for overfitting through regularization of only the feature coefficients.