/rɪdʒ rɪˈɡrɛʃ.ən/
noun … “OLS with a leash on wild coefficients.”
Ridge Regression is a regularized variant of Ordinary Least Squares used in Linear Regression to prevent overfitting when predictors are highly correlated or when the number of features is large relative to observations. By adding a penalty term proportional to the square of the magnitude of coefficients, Ridge Regression shrinks estimates toward zero without eliminating variables, balancing bias and Variance to improve predictive performance and numerical stability.
Mathematically, Ridge Regression minimizes the objective function:
β̂ = argmin ||Y - Xβ||² + λ||β||²Here, Y is the response vector, X is the predictor matrix, β is the coefficient vector, ||·||² denotes the squared Euclidean norm, and λ ≥ 0 is the regularization parameter controlling the strength of shrinkage. When λ = 0, Ridge reduces to standard OLS; as λ increases, coefficients are pulled closer to zero, reducing sensitivity to multicollinearity and extreme values.
Ridge Regression is widely used in high-dimensional data, including genomics, finance, and machine learning pipelines, where feature count can exceed sample size. It works hand-in-hand with concepts such as Covariance Matrix analysis, Expectation Values, and residual variance to ensure stable and interpretable models. It is also a foundation for other regularization techniques like Lasso and Elastic Net.
Example conceptual workflow for Ridge Regression:
collect dataset with predictors and response
standardize features to ensure comparable scaling
choose a range of λ values to control regularization
fit Ridge Regression for each λ
evaluate model performance using cross-validation
select λ minimizing prediction error and assess coefficientsIntuitively, Ridge Regression is like putting a leash on OLS coefficients: it allows them to move and respond to data but prevents them from swinging wildly due to correlated predictors or small sample noise. The result is a more disciplined, reliable model that balances fit and generalization, taming complexity without discarding valuable information.