Regression Isn’t Regularization: A Simple Guide to Understanding Both

Regression and regularization are both important concepts in machine learning and statistics, but they solve different problems. Regression is primarily used to model relationships and make predictions. Regularization is used to improve a model's ability to generalize by controlling its complexity. Regression This is a statistical and machine learning technique used to predict a continuous numerical outcome based on one or more input variables. For example, we might want to predict: A house's price based on its size and location A student's exam score based on study hours A company's sales based on advertising spending Simple Linear Regression In simple linear regression, we model the relationship between an input variable (x) and an output (y): $$ y = \beta_0 + \beta_1x + \epsilon $$ Where: (y) is the predicted outcome (\beta_0) is the intercept (\beta_1) is the coefficient or slope (x) is the input variable (\epsilon) represents the error The model learns values for (\beta_0) and (\beta_1) that make its predictions as close as possible to the actual values. Multiple Linear Regression In multiple linear regression, several predictors are used: $$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \cdots + \beta_px_p + \epsilon $$ The goal is typically to minimize the sum of squared errors (SSE) : $$ \text{SSE} = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 $$ This approach is known as Ordinary Least Squares (OLS) . Regularization Regularization is a technique used to prevent a machine learning model from becoming too complex. A model can perform extremely well on training data but poorly on new, unseen data. This problem is called overfitting . Regularization addresses overfitting by adding a penalty for large model coefficients to the model's objective function. Instead of minimizing only the prediction error, the model minimizes: $$ \text{Prediction Error} + \text{Complexity Penalty} $$ The penalty discourages the model from relying too heavily on individual features. The Main Types o