Modern machine learning models often deal with high-dimensional data where the number of features can be large compared to the number of observations. In such settings, regularisation becomes essential to control overfitting and improve model interpretability. Techniques like Ridge and Lasso regression are widely used, but their computational cost can grow when models must be trained repeatedly for different regularisation strengths. This is where the concept of the regularisation path and the Least Angle Regression (LARS) algorithm become particularly relevant. For learners exploring advanced regression methods in data science classes in Pune, understanding LARS offers valuable insight into how efficient algorithms can scale to complex problems.
Regularisation Paths in Linear Models
A regularisation path represents how model coefficients change as the regularisation parameter varies. In Ridge regression, coefficients shrink smoothly as the penalty increases, while in Lasso regression, coefficients can become exactly zero, enabling feature selection. Typically, computing these paths requires solving the optimisation problem multiple times for different values of the regularisation parameter.
Standard solvers for Lasso or Ridge often rely on iterative optimisation techniques such as coordinate descent or gradient-based methods. While effective, these approaches may require many iterations for convergence, especially when the feature space is large. The idea behind LARS is to compute the entire regularisation path more efficiently, without repeatedly solving the problem from scratch.
The Least Angle Regression (LARS) Algorithm
Least Angle Regression is an algorithm designed to build linear models in a stepwise and computationally efficient manner. Conceptually, LARS begins with all coefficients set to zero. At each step, it identifies the predictor most correlated with the current residuals and moves the coefficient of that predictor in the direction that most reduces the error.
Unlike traditional forward selection, LARS does not fully commit to one predictor at a time. Instead, it adjusts coefficients gradually, allowing multiple predictors to enter the model as soon as they become equally correlated with the residual. This “least angle” movement ensures that the algorithm progresses in a balanced way across features.
A key advantage of LARS is that it can be modified to produce the exact Lasso solution. By incorporating a simple constraint that forces coefficients to drop to zero when their sign changes, the LARS-Lasso variant traces the same regularisation path as Lasso regression. This makes LARS not just an approximation, but an exact and elegant solution method.
Computational Efficiency Compared to Lasso and Ridge Solvers
The computational strength of LARS lies in its ability to compute the full regularisation path in roughly the same cost as fitting a single ordinary least squares model, under certain assumptions. This is particularly beneficial when analysts need insights across many levels of regularisation.
In contrast, standard Lasso solvers typically compute solutions for individual penalty values. Even with warm starts, the cumulative cost of solving multiple optimisation problems can be significant. Ridge regression, while having a closed-form solution, still requires matrix inversions that scale poorly with dimensionality.
LARS avoids repeated optimisation by updating coefficients analytically at each step. For datasets where the number of features is large but sparsity is expected, this leads to noticeable performance gains. These efficiency considerations are often highlighted in advanced discussions within data science classes in Pune, especially when covering scalable machine learning algorithms.
Practical Implications and Use Cases
In practice, LARS is most useful when feature selection and interpretability are priorities. Since it reveals the order in which features enter the model, it provides insights into variable importance that are not always obvious from standard solvers. This is valuable in domains such as finance, bioinformatics, and marketing analytics.
However, LARS does have limitations. Its computational advantages are most pronounced when the number of observations exceeds the number of features and when predictors are reasonably well-conditioned. For extremely large datasets or non-linear models, other optimisation techniques may be more suitable.
From a learning perspective, mastering LARS helps practitioners understand the geometry of regression problems and the relationship between different regularisation techniques. This theoretical clarity complements practical skills gained through hands-on projects in data science classes in Pune, where learners often experiment with multiple regression approaches.
Conclusion
Least Angle Regression offers a clear and efficient way to compute regularisation paths for linear models, particularly for Lasso regression. By incrementally adjusting coefficients based on correlations with residuals, LARS achieves computational efficiency that traditional solvers may struggle to match. While it is not a universal replacement for all optimisation methods, it remains a powerful tool for understanding and implementing regularised regression. For anyone aiming to deepen their grasp of algorithmic efficiency and model interpretability, especially those enrolled in data science classes in Pune, LARS represents an important concept in the modern data science toolkit.
