Lasso Regression in Machine Learning
Lasso Regression (Least Absolute Shrinkage and Selection Operator) is a type of linear regression that incorporates L1 regularization. It adds a penalty equal to the absolute value of the magnitude of coefficients to the cost function, encouraging sparsity in the model by shrinking some coefficients to exactly zero. This makes Lasso particularly useful for feature selection in high-dimensional datasets.
By introducing L1 regularization, Lasso Regression addresses overfitting and multicollinearity while providing a simpler, interpretable model by selecting only the most important features.
How Does Lasso Regression Work?
Step 1: Linear Combination of Inputs
Lasso regression starts with the linear combination of input features, similar to ordinary linear regression:
\( y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_nx_n \)
Here:
- \( y \): Predicted value of the target variable
- \( x_1, x_2, \dots, x_n \): Input features (independent variables)
- \( \beta_0 \): Intercept term (bias)
- \( \beta_1, \beta_2, \dots, \beta_n \): Coefficients or weights of the features
Lasso Regression modifies the training process by introducing an L1 penalty to the cost function.
Step 2: Regularized Cost Function
The cost function for Lasso Regression includes the Mean Squared Error (MSE) with an L1 regularization term:
\( J(\beta) = \frac{1}{n} \sum_{i=1}^n (y_i – \hat{y}_i)^2 + \lambda \sum_{j=1}^n |\beta_j| \)
Here:
- \( J(\beta) \): Lasso regression cost function
- \( y_i \): Actual target value for the \(i^{th}\) data point
- \( \hat{y}_i \): Predicted target value for the \(i^{th}\) data point
- \( \lambda \): Regularization parameter (controls the strength of the penalty)
- \( |\beta_j| \): Absolute value of the \(j^{th}\) coefficient
The L1 penalty forces some coefficients to become exactly zero, effectively removing those features from the model. The regularization parameter \( \lambda \) controls the degree of sparsity.
Step 3: Model Training
During training, Lasso Regression solves for the coefficients (\( \beta_0, \beta_1, \dots, \beta_n \)) by minimizing the regularized cost function. The optimization process uses techniques like coordinate descent or gradient-based methods to handle the L1 penalty.
The result is a sparse model where some coefficients are exactly zero, leaving only the most significant predictors in the model.
Step 4: Prediction
After training, the model predicts the target values for new data points using the learned coefficients:
\( \hat{y} = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_nx_n \)
Here, \( \hat{y} \) is the predicted value, and \( x_1, x_2, \dots, x_n \) are the feature values of the new data point.
Key Characteristics of Lasso Regression
- Feature Selection: Automatically selects the most important features by shrinking irrelevant coefficients to zero.
- Prevents Overfitting: The L1 penalty reduces model complexity, improving generalization on unseen data.
- Sparse Solutions: Results in a simpler, more interpretable model by retaining only the most significant predictors.
Advantages of Lasso Regression
- Feature Selection: Automatically eliminates irrelevant features, simplifying the model.
- Improves Generalization: Reduces overfitting by regularizing the coefficients.
- Interpretable Model: Produces a sparse solution, making it easier to understand the relationship between features and the target variable.
Limitations of Lasso Regression
- Handles Multicollinearity Poorly: When features are highly correlated, Lasso may arbitrarily select one and ignore others.
- Sensitive to Data Scaling: Requires features to be standardized or normalized for effective regularization.
- Choice of Regularization Parameter: Performance depends on selecting an appropriate \( \lambda \), which often requires cross-validation.