Lasso Regression is a Laplace Prior

2026-04-11 · Source: DataMListic · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, quick

Summary

Lasso Regression, or Least Absolute Shrinkage and Selection Operator, addresses the challenge of feature selection in models with numerous variables by employing an L1 penalty. Unlike Ridge Regression, which uses an L2 penalty to shrink coefficients towards zero without reaching it, Lasso adds lambda times the sum of the absolute values of coefficients to the ordinary least squares objective. Geometrically, this L1 penalty constrains coefficients within a diamond shape in parameter space, whose corners often align with coordinate axes. This alignment causes the optimal solution to frequently land on an axis, effectively setting one or more coefficients to exactly zero. As the regularization parameter lambda increases, coefficients for weaker features progressively shrink to zero and are eliminated, facilitating automatic feature selection. This method also has a Bayesian interpretation, corresponding to the Maximum A Posteriori (MAP) estimate when a Laplace prior is placed on the coefficients, contrasting with Ridge's Gaussian prior.

Key takeaway

For data scientists and machine learning engineers building models with many features, Lasso regression offers a powerful alternative to Ridge regression. If your goal is not just to shrink coefficients but to perform automatic feature selection by eliminating irrelevant variables, Lasso's L1 penalty will drive those coefficients to exactly zero. This simplifies model interpretation and can improve generalization by focusing on the most impactful predictors.

Key insights

Lasso regression uses an L1 penalty to achieve automatic feature selection by driving less important coefficients to zero.

Principles

• L1 penalty promotes sparsity.
• Geometric shape of constraint matters.
• Bayesian priors explain regularization.

Method

Lasso regression modifies the OLS objective by adding an L1 penalty (sum of absolute coefficients) scaled by lambda, which forces less impactful feature coefficients to exactly zero, enabling automatic feature selection.

In practice

• Use Lasso for high-dimensional datasets.
• Apply Lasso when feature sparsity is desired.
• Compare L1 vs. L2 for different outcomes.

Topics

• Lasso Regression
• L1 Regularization
• Feature Selection
• Ridge Regression
• L2 Regularization

Best for: Machine Learning Engineer, Data Scientist, AI Student

Related on AIssential

• Lasso Regression: Why the Solution Lives on a Diamond — Lasso Regression uses vector projections and coefficient constraints to prevent overfitting by shrinking feature coefficients.
• Elastic Net - Explained — Elastic Net combines L1 and L2 regularization for stable feature selection and grouping of correlated variables.
• L2 Regularization Is Secretly Bayesian — L2 regularization is mathematically equivalent to applying a Gaussian prior in Bayesian maximum a posteriori estimation.
• Adaptive Forward Stepwise: A Method for High Sparsity Regression — Adaptive Forward Stepwise Regression offers sparser models with shrinkage, outperforming LASSO and Forward Stepwise.
• Machine Learning | Ridge Regression-L2 — Ridge Regression uses L2 regularization to stabilize coefficients, reduce overfitting, and manage multicollinearity in linear models.
• A Regularization-Sharpness Tradeoff for Linear Interpolators — A regularization-sharpness tradeoff replaces bias-variance in overparameterized linear models, decomposing model selection into alignment and local perturbation effects.

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Editorial summary, takeaway, and curation by AIssential. Original article published by DataMListic.