  Random forests have become an important tool for improving accuracy in
regression and classification problems since their inception by Leo Breiman in
2001. In this paper, we revisit a historically important random forest model
originally proposed by Breiman in 2004 and later studied by G\'erard Biau in
2012, where a feature is selected at random and the splits occurs at the
midpoint of the node along the chosen feature. If the regression function is
Lipschitz and depends only on a small subset of $ S $ out of $ d $ features, we
show that, given access to $ n $ observations and properly tuned split
probabilities, the mean-squared prediction error is $ O((n(\log
n)^{(S-1)/2})^{-\frac{1}{S\log2+1}}) $. This positively answers an outstanding
question of Biau about whether the rate of convergence for this random forest
model could be improved. Furthermore, by a refined analysis of the
approximation and estimation errors for linear models, we show that this rate
cannot be improved in general. Finally, we generalize our analysis and improve
extant prediction error bounds for another random forest model in which each
tree is constructed from subsampled data and the splits are performed at the
empirical median along a chosen feature.
