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Random forests have become an important tool for improving accuracy inregression and classification problems since their inception by Leo Breiman in2001. In this paper, we revisit a historically important random forest modeloriginally proposed by Breiman in 2004 and later studied by G\'erard Biau in2012, where a feature is selected at random and the splits occurs at themidpoint of the node along the chosen feature. If the regression function isLipschitz and depends only on a small subset of $ S $ out of $ d $ features, weshow that, given access to $ n $ observations and properly tuned splitprobabilities, the mean-squared prediction error is $ O((n(\logn)^{(S-1)/2})^{-\frac{1}{S\log2+1}}) $. This positively answers an outstandingquestion of Biau about whether the rate of convergence for this random forestmodel could be improved. Furthermore, by a refined analysis of theapproximation and estimation errors for linear models, we show that this ratecannot be improved in general. Finally, we generalize our analysis and improveextant prediction error bounds for another random forest model in which eachtree is constructed from subsampled data and the splits are performed at theempirical median along a chosen feature.

Sharp Analysis of a Simple Model for Random Forests