By Hastie, Trevor

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We are using the spelling in the original paper of Breiman (1995). 6 Breiman’s paper was the inspiration for Tibshirani’s 1996 lasso paper. © 2015 by Taylor & Francis Group, LLC THE NONNEGATIVE GARROTE 21 Lin 2007c, Zou 2006) have shown that the nonnegative garrote has attractive properties when we use other initial estimators such as the lasso, ridge regression or the elastic net. 5 Comparison of the shrinkage behavior of the lasso and the nonnegative garrote for a single variable. Since their λs are on different scales, we used 2 for the lasso and 7 for the garrote to make them somewhat comparable.

On the other hand, the convergence of the parameter estimates from the nonnegative garrote tends to be slower than that of the initial estimate. 21) in the case of an orthonormal model matrix X. 9 q q Estimator 0 Best subset 1 Lasso 2 Ridge Formula √ β˜j · I[|β˜j | > 2λ] sign(β˜j )(|β˜j | − λ)+ β˜j /(1 + λ) Penalties and Bayes Estimates For a fixed real number q ≥ 0, consider the criterion p p 1 N 2 q minimize (y − x β ) + λ |β | . 21) This is the lasso for q = 1 and ridge regression for q = 2.

Best-subset selection applies the√hard thresholding operator: it leaves the coefficient alone if it is bigger than 2λ, and otherwise sets it to zero. The lasso is special in that the choice q = 1 is the smallest value of q (closest to best-subset) that leads to a convex constraint region and hence a © 2015 by Taylor & Francis Group, LLC SOME PERSPECTIVE 23 convex optimization problem. In this sense, it is the closest convex relaxation of the best-subset selection problem. There is also a Bayesian view of these estimators.