Today, in data science and machine learning, predictive modeling mostly deals with high-dimensional datasets that have numerous features. The issue is that not all features contribute equally to predictions, and these redundant features can reduce model accuracy. Lasso regression is a statistical technique that aims to remove the contribution of redundant features through variable selection and regularization.
In this article, we will understand Lasso regression in detail, explain its working, and highlight how it differs from other regression techniques like Ridge regression. We will also discuss the advantages and disadvantages of using these regularization tasks, and finally, end the article with a simple implementation in Python.
What is Lasso Regression?
Lasso regression is actually short for Least Absolute Shrinkage and Selection Operator. It is theoretically an extension of linear regression that differs by including a regularization term. This method is useful when you work with datasets where some features are not relevant and have close to no impact on the target variable.
The main objectives or principles that are followed by Lasso regression are:
- It aims to reduce overfitting by discouraging large coefficient values by adding a penalty to them while training the model.
- It attempts to simplify the models by eliminating unimportant or redundant features.
- It also tries to make the model easier to understand by keeping only the most important features.
By decreasing the effects by shrinking the coefficients of less important features to zero, Lasso actually performs automatic feature selection. This nature makes it a preferred choice for high-dimensional datasets, which are common in fields like finance, healthcare, and genomics.
Now that we are familiar with the meaning of Lasso regression and what it does, let us see “how it does” it.
Mathematics Behind Lasso Regression
Normally, linear regression estimates the relationship between input features and a target variable by minimizing the residual sum of squares (RSS). The formula used to do that is:
Where:
- yi is the actual observed value of the target variable for the i-th observation.
- yi hat is the value predicted by the model for the i-th observation
By minimizing RSS, the model finds the best-fitting line (or hyperplane) that reduces the difference between actual and predicted values. This is how linear regression works.
On the other hand, Lasso modifies the approach followed in linear regression by adding an L1 regularization term to the RSS term. The formula is noted below:
Here:
- Bj (beta) represents the coefficient of the j-th feature (not the datapoint but the feature). And,
- λ is the regularization parameter that controls the strength or the impact of the penalty term (or the regularization term).
Now, let us understand this impact on the regularization term by understanding what happens at various values of λ.
- If λ = 0, it means that lasso regression has been reduced to standard linear regression because the penalty terms become zero.
- As λ increases, more coefficients are shrunk toward zero, which simplifies the model one feature at a time. Lasso can actually reduce coefficients exactly to zero. This property allows it to eliminate irrelevant features automatically, improving model interpretability.
This was a technique to decrease the impact of a feature by adjusting its penalty. But there is also the concept of bias and variance that is of importance in machine learning models. Let us look at how Lasso regression manages that in the next section.
Bias-Variance Trade-off in Lasso Regression
Like we stated earlier, in any predictive model, balancing bias and variance is very important:
- Bias refers to the errors that occur due to overly simplistic assumptions in the model. It is like underfitting, where the model is too simple to capture the true patterns in the data. And,
- Variance refers to errors caused by the model when it is too sensitive to small changes in the training data. It is like overfitting, where the model follows noise in the training data instead of the true trend.
Lasso regression influences the variance issue by shrinking the coefficients of less important features. This again reduces model complexity and prevents overfitting on noisy data. However, increasing too much may introduce bias, resulting in underfitting.
The goal is to choose an optimal, balanced value that achieves a balance between bias and variance, which is often done through cross-validation techniques.
How Lasso Regression Works?
Before we move on to the implementation of Lasso, let us understand how it works with its internal mechanism. This will help you apply it effectively and make things intuitive:
1. Start with Linear Regression
Since lasso regression is an extension of linear regression, we will naturally start by writing the linear regression formula. This models the target variable as a linear combination of input features, being the error term that is added to the model training:
2. Adding the L1 Penalty
Now, Lasso regression uniquely introduces an L1 penalty term, which is actually the sum of the absolute values of the coefficients. This penalty shrinks the coefficients of less relevant features, sometimes to zero.
3. Shrinking Coefficients
- Features that have a weak influence on the target variable, Lasso reduces their coefficient to zero.
- Because of this, only the important predictors or features retain non-zero coefficients, making the model easier to interpret.
4. Selecting the Optimal Lambda
This is actually a very important step. The effectiveness and accuracy of your lasso regression depend on this term, and choosing the right one is critical. A few common approaches to finding the optimal value are:
- K-fold Cross-Validation: In this technique, we divide the data into K subsets and evaluate the model’s performance across them by alternating between testing and training subsets.
- Grid Search: Here, we test multiple values to find the best tradeoff between model complexity and accuracy.
- Analytical Approaches: You can also use information criteria like AIC or BIC to select the best model.
An optimal lambda will give you the sparest model (Models that focus on key predictors only, that are also few in number) without sacrificing accuracy.
This is the complete working of L1 regularization or Lasso. Now, it will be easy for you to follow through with the implementation.
Difference Between Ridge and Lasso Regression
Both Ridge and Lasso are regularization techniques that are used to prevent overfitting, but they differ in the kind of penalty they apply. Lasso is preferred for simplified models and automatic feature selection, while Ridge is suitable when features are highly correlated with each other (multicollinearity exists), and all features are important.
| Feature | Ridge Regression | Lasso Regression |
| Penalty Type | L2 (squared sum of coefficients) | L1 (absolute sum of coefficients) |
| Coefficient Shrinkage | Reduces magnitude but is rarely zero | Can shrink coefficients to zero |
| Feature Selection | No automatic feature elimination | Performs automatic feature selection |
| Use Case | When all predictors are relevant but need regularization | When many predictors may be irrelevant |
| Interpretation | Coefficients shrink uniformly | Produces sparse models with fewer features |
Implementing Lasso Regression in Practice
This Python example demonstrates how to apply Lasso regression using scikit-learn. We will train the model on a random sample of data, make predictions, and evaluate performance using Mean Squared Error while showing which features are selected.
Python Example Using scikit-learn:
Output:
Explanation: The Mean Squared Error (0.055) shows the model’s predictions are reasonably close to actual values. Lasso shrinks many feature weights to zero, keeping only important predictors. Here, all coefficients are zero, so no features were strongly predictive for this random dataset.
Applications of Lasso Regression
Till now, we have discussed all the technical details of Lasso regression. It is actually widely used in scenarios involving high-dimensional data. Some of the common applications include:
- Healthcare: Lasso regression is used in the healthcare domain to identify key factors influencing disease progression from genomic data.
- Finance: In finance, it helps the developers in selecting the most important indicators for stock price prediction.
- Marketing: In this domain, it helps in reducing the number of predictors in customer behavior analysis.
- Image Processing: It helps in feature selection in high-dimensional pixel data.
- Natural Language Processing (NLP): Here, lasso is used in selecting important words or n-grams in text classification.
Advantages of Lasso Regression
The main aim of Lasso regression is to simplify the models so that they improve interpretability and decrease the computational load. Let us look at some more advantages of using Lasso:
- Automatic Feature Selection: This regression technique eliminates the need for manual feature elimination, and once you find the optimal value, you can also increase the efficiency of the model.
- Reduces Overfitting: L1 regularization, which is used by Lasso, shrinks coefficients, which lowers the model variance.
- Improved Interpretability: Sparse models, as we have discussed earlier in the article, make it easier to identify influential features.
- Handles High-Dimensional Data: The biggest advantage of this technique is that it’s effective even when predictors outnumber observations.
Disadvantages of Lasso Regression
While lasso regression has many benefits, there are some disadvantages that you should be aware of when developing a model:
- Selection Bias: Sometimes, it randomly selects one feature from a group of highly correlated features. This introduces a bias to your trained model, decreasing its accuracy.
- Sensitive to Feature Scale: Before using Lasso, it’s important to scale or normalize the features so they are on the same range. This ensures that the penalty treats all features fairly, instead of unfairly shrinking features with larger values. This increases the computational overhead.
- Affected by Outliers: Lasso can be influenced by extreme values in the data. Very large or small values can distort the model, causing it to shrink some coefficients too much or too little.
- Complexity in Lambda Selection: The penalty strength in Lasso is controlled by λ. Choosing the right value is important, as too high or too low can hurt model performance.
Conclusion
Lasso regression is an extremely practical and powerful regression method that tackles two very important problems of predictive modeling, overfitting and feature selection. The application of an L1 penalty shrinks less important coefficients toward zero to produce a model that is sparse and interpretable. Knowledge of the bias-variance tradeoff, the selection of an optimal, and how it’s different from Ridge regression is critical in proper application.
For data scientists and analysts dealing with high-dimensional data, knowing how to apply lasso regression is an important skill to have in order to make model development more accurate and interpretable.
Understanding Lasso Regression – FAQs
Q1. What is the main advantage of lasso regression?
It performs automatic feature selection by shrinking irrelevant coefficients to zero, simplifying the model and reducing overfitting.
Q2. Can lasso regression handle multicollinearity?
Yes, but it may arbitrarily select one feature among correlated predictors, unlike Ridge regression, which shrinks all correlated coefficients.
Q3. How is lambda selected in Lasso regression?
Lambda is typically selected using cross-validation or grid search to balance bias and variance for optimal model performance.
Q4. When should I prefer Lasso over Ridge regression?
Prefer lasso when you expect many features to be irrelevant and want automatic feature selection. Use Ridge when all features are potentially important.
Q5. Does Lasso regression work for non-linear relationships?
Standard lasso regression is linear. For non-linear relationships, features may be transformed (e.g., polynomial features) before applying lasso.
Q6. Can Lasso regression handle categorical variables?
Yes, but categorical variables must be one-hot encoded to convert them into a numerical format before applying lasso regression.
Q7. What happens if lambda is too large?
A very large lambda may shrink most coefficients to zero, resulting in underfitting and high bias.
Q8. Is feature scaling necessary for Lasso regression?
Yes, standardizing features ensures the penalty affects all coefficients equally, improving model stability.
Q9. How does Lasso compare to Elastic Net?
Elastic Net combines L1 and L2 penalties, offering better performance when features are highly correlated, while Lasso uses only the L1 penalty.
Q10. Is lasso regression computationally expensive?
Lasso is more computationally intensive than standard linear regression due to iterative optimization, but it is feasible for most datasets with modern libraries.