Algorithm 1. Undersmoothing procedure

$$\begin{array}{cc}\hfill & \mathrm{Require}:\hfill \\ \hfill & \text{Observed outcome}{y}_k\text{from data},k=1,2,\dots ,n.\hfill \\ \hfill & \text{Penalty parameter}{\lambda }_{cv},\text{basis functions}{\varphi }_{s,l},\text{and predictions}{\widehat{y}}_{{\lambda }_{cv}}\text{from the initial HAL fit}.\hfill \\ \hfill & \text{Let}S\text{denote the set of basis functions}{\varphi }_{s,l}\text{with nonzero coefficients from the initial HAL fit}.\hfill \\ \hfill & \text{Let}ϵ\text{be a small positive number}.\hfill \\ \hfill & {\sigma }_n=\sqrt{\frac{1}{n}\sum \begin{array}{c}n\hfill \\ k=1\hfill \end{array}[(y_k-{\widehat{y}}_{{\lambda }_{cv}}){\varphi }_{s,l}-\frac{1}{n}\sum \begin{array}{c}n\hfill \\ k=1\hfill \end{array}(y_k-{\widehat{y}}_{{\lambda }_{cv}}){\varphi }_{s,l}{]}^2}\forall {\varphi }_{s,l}\in S\hfill \\ \hfill & \lambda ={\lambda }_{cv}\hfill \\ \hfill & \mathbf{while}{\max }_S\mid \frac{1}{n}\sum \begin{array}{c}n\hfill \\ k=1\hfill \end{array}[(y_k-{\widehat{y}}_{\lambda }){\varphi }_{s,l}]\mid ∕{\sigma }_n>\frac{1}{\sqrt{n}\log (n)}\mathbf{do}\hfill \\ \hfill & \lambda =\lambda -ϵ\hfill \\ \hfill & \text{Refit HAL with}\lambda \hfill \\ \hfill & \text{Obtain new predictions}{\widehat{y}}_{\lambda }\text{frome the update HAL fit}\hfill \\ \hfill & \mathbf{end}\mathbf{while}\hfill \end{array}$$