Table 3.

Tuning parameter settings

Lasso	10-fold cross-validation is used with α = 1 for the lasso penalty. We use lambda:min and lambda:1se for λ.
Boosting	A total number of 1000 trees are fit. Testing error is calculated for every 20 trees. n:minobsinnode = 2, n1/3, 10. learning rate shrinkage = 0.001, 0.01, 0.1, interaction:depth = 1, 3, 5.
BART	A total of 18 settings: ntrees = 50 or 200; Sigma prior: (3, 0.90), (3, 0.99), (10, 0.75); μ prior: 2, 3, 5.
RF	A total of 36 settings: ntrees = 500, 1000; $\mathit{mtry}=\sqrt{p}$, p/3, p; nodesize = 2, n1/3. Bootstrap sample ratio = 1, 0.8, 2/3.
$\text{RF-}\sqrt{p}$	Select the top $\sqrt{p}$ important variables from each RF model and refit with the same settings as RF (with mtry recalculated accordingly).
RF-log(p)	Similar as $\text{RF-}\sqrt{p}$, however with top log(p) variables selected.
ET	ntrees = 500, 1000; $\mathit{mtry}=\sqrt{p}$, p/3, p; nodesize = 2, n1/3; numRandomCuts = 1, 5.
RLT-naive	ntrees = 1000; nodesize = 2, n1/3; muting rate = 0%, 50%, 80%. Bootstrap sample ratio = 1, 0.8, 2/3. number of random splits = 10 or all possible splits.
RLT	M = 100 trees with nmin = n1/3 are fit to each RLT model. We consider a total of 9 settings: k = 1, 2, 5, with no muting (pd = 0), moderate muting ( $p_d=50\%·\mid \mathcal{P}\backslash {\mathcal{P}}_A^d\mid$), and aggressive muting ( $p_d=80\%·\mid \mathcal{P}\backslash {\mathcal{P}}_A^d\mid$) as discussed in Remark 2.3. We set the number of protected variables p0 = log(p) to be on par with RF-log(p). Note that when pd = 0, all variables are considered at each internal node, hence no protection is needed. This is on par with RF.