What is Least Angle Regression (LARS)?

Least Angle Regression (LARS) is a regression algorithm introduced by Bradley Efron and colleagues in 2004. Designed to address challenges in high-dimensional linear regression models, LARS provides a computationally efficient way to perform feature selection while estimating regression coefficients. The algorithm is particularly useful when the number of predictors (features) is large compared to the number of observations.

LARS bridges the gap between traditional forward selection methods and shrinkage-based methods like Lasso. It constructs a piecewise linear solution path that can be interpreted and computed efficiently, offering insights into how model complexity evolves with added predictors.

Mathematical Foundations of LARS

LARS operates under the framework of linear regression. Consider a response variable $y \in \mathbb{R}^n$ and a predictor matrix $X \in \mathbb{R}^{n \times p}$. The goal is to find a coefficient vector $\beta \in \mathbb{R}^p$ that minimizes the residual sum of squares:

\[\min_\beta \|y - X\beta\|_2^2\]

However, in cases where $p \gg n$ or there exists multicollinearity among predictors, standard least squares becomes unstable or unidentifiable. LARS addresses this by choosing predictors incrementally, adjusting the coefficient vector in the direction that is most correlated with the current residual.

At each iteration, instead of making a full step as in standard forward selection, LARS takes a small step in the direction of the predictor most correlated with the residual, gradually incorporating more predictors as needed.

Comparison with Lasso and Forward Stepwise Regression

LARS, Lasso, and Forward Stepwise Regression all share a goal of model simplicity and interpretability. However, they differ significantly in methodology and outcomes.

Forward Stepwise Regression adds one variable at a time to the model, based on which variable reduces the residual error the most. Once added, variables are never removed. This approach can be greedy and may not yield the optimal subset of predictors.

Lasso Regression adds a regularization term to the objective function:

\[\min_\beta \left\{ \|y - X\beta\|_2^2 + \lambda \|\beta\|_1 \right\}\]

This penalizes the absolute size of the coefficients and tends to produce sparse solutions, where many coefficients are exactly zero.

LARS behaves like Forward Stepwise Regression in its stepwise variable inclusion, but it adjusts the coefficients less aggressively. Interestingly, when used with an appropriate modification, LARS can produce the exact solution path of the Lasso without the need to tune the penalty parameter explicitly at each step.

How LARS Works: Step-by-Step Process

The LARS algorithm follows these main steps:

  1. Initialization: Set all coefficients to zero, and compute the correlation of each predictor with the response vector $y$.

  2. Select Most Correlated Predictor: Identify the predictor most correlated with the current residuals and start moving the coefficient in that direction.

  3. Move Along Equiangular Direction: Instead of fully fitting this predictor, LARS takes a step in a direction that is equiangular between all active predictors, i.e., those currently in the model.

  4. Add Next Predictor When Correlation Matches: As the algorithm proceeds, it adds a new predictor into the active set when its correlation with the residuals equals that of the current active predictors.

  5. Repeat Until All Predictors Are Included or Stopping Criterion Met.

This stepwise, piecewise linear path allows practitioners to examine how model fit evolves as complexity increases.

Advantages and Limitations

One of the main strengths of Least Angle Regression is its efficiency. Unlike traditional subset selection methods that can be computationally expensive, LARS has a cost comparable to fitting a single least squares model. For $p$ predictors, LARS typically requires only $O(p^2)$ operations.

It is also interpretably sparse. Because predictors enter the model incrementally, the algorithm naturally produces a series of increasingly complex models, making it easy to identify a preferred balance between simplicity and accuracy.

Another advantage lies in its relationship with Lasso. With a slight modification, LARS can exactly trace out the Lasso path, making it a valuable tool for understanding how Lasso solutions evolve with varying regularization.

However, LARS has some limitations:

  • It is sensitive to noise and outliers due to its reliance on correlation.
  • Like most linear methods, it assumes a linear relationship between predictors and the response.
  • It can struggle when predictors are highly collinear, as the decision of which predictor to enter next can become unstable.
  • Moreover, LARS is designed for linear models only and does not generalize to non-linear or non-parametric settings without substantial changes.

Applications in High-Dimensional Data Analysis

LARS is particularly effective in high-dimensional settings, such as genomics, image analysis, and signal processing, where the number of variables can exceed the number of observations. In these cases, ordinary least squares becomes impractical or ill-posed due to overfitting.

In genomics, for example, LARS can identify a small subset of genes most relevant to predicting disease risk or drug response, enabling biologically interpretable and statistically sound models.

In machine learning pipelines, LARS is often used as a feature selection step before applying more complex models like support vector machines or ensemble methods. By reducing the dimensionality of the data, LARS can improve computational efficiency and reduce overfitting in downstream models.

LARS has also found applications in compressed sensing and sparse signal recovery, where its ability to produce sparse solutions is especially valuable.

Final Thoughts and Future Directions

Least Angle Regression occupies an elegant middle ground between computational efficiency and statistical rigor. By building models incrementally, it provides both transparency and adaptability. Its close relationship with Lasso and its ability to handle high-dimensional data make it a tool of lasting relevance in modern statistical learning.

Going forward, LARS continues to inspire variations and improvements, including hybrid methods that incorporate Bayesian priors or non-linear transformations. Additionally, integrating LARS into deep learning architectures or extending it to generalized linear models are active areas of research.

As machine learning and statistics continue to evolve in tandem, algorithms like LARS remind us that simplicity and insight often go hand-in-hand.

Appendix: Python Example of Least Angle Regression (LARS)

import numpy as np
from sklearn import datasets
from sklearn.linear_model import Lars
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, r2_score
import matplotlib.pyplot as plt

# Load a high-dimensional dataset (e.g., diabetes or synthetic)
X, y = datasets.make_regression(n_samples=100, n_features=50, n_informative=10, noise=0.1, random_state=42)

# Split the dataset into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

# Initialize and fit the LARS model
lars = Lars(n_nonzero_coefs=10)  # Limit to 10 predictors for sparsity
lars.fit(X_train, y_train)

# Predict on test data
y_pred = lars.predict(X_test)

# Evaluate model performance
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)

print("Mean Squared Error:", mse)
print("R^2 Score:", r2)
print("Selected Coefficients:", lars.coef_)

# Plot coefficient progression (coefficient paths)
_, _, coefs = lars.path(X_train, y_train)

plt.figure(figsize=(10, 6))
for coef_path in coefs.T:
    plt.plot(coef_path)
plt.title("LARS Coefficient Paths")
plt.xlabel("Step")
plt.ylabel("Coefficient Value")
plt.grid(True)
plt.show()

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