information:highlighting, structural:summarization
Making λ easier to interpret with an accuracy-simplicity tradeoff

The lasso can be rescaled so that it becomes easy to anticipate and influence the degree of shrinkage associated with a given value of λ {\displaystyle \lambda }. It is assumed that X {\displaystyle X} is standardized with z-scores and that y {\displaystyle y} is centered (zero mean). Let β 0 {\displaystyle \beta _{0}} represent the hypothesized regression coefficients and let b OLS {\displaystyle b_{\text{OLS}}} refer to the data-optimized ordinary least squares solutions. We can then define the Lagrangian as a tradeoff between the in-sample accuracy of the data-optimized solutions and the simplicity of sticking to the hypothesized values. This results in min β ∈ R p { (y − X β) ′ (y − X β) (y − X β 0) ′ (y − X β 0) + 2 λ ∑ i = 1 p | β i − β 0, i | q i } {\displaystyle \min _{\beta \in \mathbb {R} ^{p}}\left\{{\frac {(y-X\beta)'(y-X\beta)}{(y-X\beta _{0})'(y-X\beta _{0})}}+2\lambda \sum _{i=1}^{p}{\frac {|\beta _{i}-\beta _{0,i}|}{q_{i}}}\right\}} where q i {\displaystyle q_{i}} is specified below and the "prime" symbol stands for transpose. The first fraction represents relative accuracy, the second fraction relative simplicity, and λ {\displaystyle \lambda } balances between the two.