Lasso (statistics)

In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso, LASSO or L1 regularization) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. The lasso method assumes that the coefficients of the linear model are sparse, meaning that few of them are non-zero. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term. Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator. These include its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do not need to be unique if covariates are collinear. Though originally defined for linear regression, lasso regularization is easily extended to other statistical models including generalized linear models, generalized estimating equations, proportional hazards models, and M-estimators. Lasso's ability to perform subset selection relies on the form of the constraint and has a variety of interpretations including in terms of geometry, Bayesian statistics and convex analysis. The LASSO is closely related to basis pursuit denoising.

History Lasso was introduced in order to improve the prediction accuracy and interpretability of regression models. It selects a reduced set of the known covariates for use in a model. Lasso was developed independently in geophysics literature in 1986, based on prior work that used the ℓ 1 {\displaystyle \ell ^{1}} penalty for both fitting and penalization of the coefficients. Statistician Robert Tibshirani independently rediscovered and popularized it in 1996, based on Breiman's nonnegative garrote. Prior to lasso, the most widely used method for choosing covariates was stepwise selection. That approach only improves prediction accuracy in certain cases, such as when only a few covariates have a strong relationship with the outcome. However, in other cases, it can increase prediction error. At the time, ridge regression was the most popular technique for improving prediction accuracy. Ridge regression improves prediction error by shrinking the sum of the squares of the regression coefficients to be less than a fixed value in order to reduce overfitting, but it does not perform covariate selection and therefore does not help to make the model more interpretable. Lasso achieves both of these goals by forcing the sum of the absolute value of the regression coefficients to be less than a fixed value, which forces certain coefficients to zero, excluding them from impacting prediction. This idea is similar to ridge regression, which also shrinks the size of the coefficients; however, ridge regression does not set coefficients to zero (and, thus, does not perform variable selection).

Basic form

Least squares Consider a sample consisting of N cases, each of which consists of p covariates and a single outcome. Let y i {\displaystyle y_{i}} be the outcome and x i := ( x 1 , x 2 , … , x p ) i ⊺ {\displaystyle x_{i}:=(x_{1},x_{2},\ldots ,x_{p})_{i}^{\intercal }} be the covariate vector for the i th case. Then the objective of lasso is to solve:

min β 0 , β { ∑ i = 1 N ( y i − β 0 − x i ⊺ β ) 2 } {\displaystyle \min _{\beta _{0},\beta }{\biggl \{}\sum _{i=1}^{N}{\bigl (}y_{i}-\beta _{0}-x_{i}^{\intercal }\beta {\bigr )}^{2}{\biggr \}}}

subject to

∑ j = 1 p | β j | ≤ t . {\displaystyle \sum _{j=1}^{p}|\beta _{j}|\leq t.}

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