# Regression Isn’t Regularization: A Simple Guide to Understanding Both > Source: > Published: 2026-07-27 06:44:11+00:00 Regression and regularization are both important concepts in machine learning and statistics, but they solve different problems. 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: In simple linear regression, we model the relationship between an input variable (x) and an output (y): $$ y = \beta_0 + \beta_1x + \epsilon $$ Where: 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 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. Ridge regression adds a penalty based on the squared values of the coefficients: $$ \sum_{i=1}^{n}(y_i - \hat{y}*i)^2 + \lambda\sum*{j=1}^{p}\beta_j^2 Lasso regression uses the absolute values of the coefficients: $$ \sum_{i=1}^{n}(y_i - \hat{y}*i)^2 + \lambda\sum*{j=1}^{p}|\beta_j| This means Lasso can perform a type of **feature selection** by effectively removing less important variables from the model. Elastic Net combines both L1 and L2 regularization: $$ \text{SSE} + \lambda_1\sum_j|\beta_j| + \lambda_2\sum_j\beta_j^2 $$ It combines: The simplest way to understand the difference is: Regression builds a model to explain or predict an outcome. Regularization modifies the learning process to control the model's complexity. For example, ordinary linear regression might optimize the following objective: $$ \min_{\beta} \text{SSE} $$ Ridge regression changes it to: $$ \min_{\beta} \left( \text{SSE} + \lambda\sum_j\beta_j^2 \right) $$ The underlying task is still regression. Regularization simply adds a constraint or penalty to make the model less likely to overfit. Imagine you are predicting house prices using 100 features, including: An ordinary regression model may fit the training data very closely. However, if there are too many features or strong correlations between them, the model may overfit. Regularization can help: Here is the corrected and clearly formatted Markdown table: | Feature | Ordinary Regression | Regularized Regression | |---|---|---| Main purpose | Model relationships and make predictions | Reduce overfitting and control complexity | Objective | Minimize prediction error | Minimize prediction error plus a penalty | Coefficients | Can become very large | Penalized and typically smaller | Feature selection | Usually no | Lasso can select features | Handles multicollinearity | Can be sensitive | Ridge handles it well | Generalization | May overfit | Often improves performance on unseen data | Regularization works by introducing a small amount of **bias** in exchange for reducing **variance**. The goal is to find the right balance. The regularization parameter (\lambda) is commonly selected using **cross-validation**. Regularization is sensitive to the scale of features. For example: Because regularization penalizes coefficient sizes, variables with different scales can be treated unfairly. Therefore, features are often standardized before applying **Ridge, Lasso, or Elastic Net**. Regression and regularization are not competing concepts. **Regression** answers: How can we model the relationship between inputs and a numerical outcome? Regularizationanswers: How can we prevent that model from becoming unnecessarily complex and overfitting the training data? Ordinary linear regression focuses on minimizing prediction error. Regularized regression adds a penalty that discourages overly large coefficients. Ridge, Lasso, and Elastic Net are common examples of regularized regression techniques. A useful way to remember the distinction is: Regression learns the relationship. Regularization controls the complexity of what is learned.