  As a flexible nonparametric learning tool, the random forests algorithm has
been widely applied to various real applications with appealing empirical
performance, even in the presence of high-dimensional feature space. Unveiling
the underlying mechanisms has led to some important recent theoretical results
on the consistency of the random forests algorithm and its variants. However,
to our knowledge, almost all existing works concerning random forests
consistency in high dimensional setting were established for various modified
random forests models where the splitting rules are independent of the
response; a few exceptions assume simple data generating models with binary
features. In light of this, in this paper we derive the consistency rates for
the random forests algorithm associated with the sample CART splitting
criterion, which is the one used in the original version of the algorithm, in a
general high-dimensional nonparametric regression setting through a
bias-variance decomposition analysis. Our new theoretical results show that
random forests can indeed adapt to high dimensionality and allow for
discontinuous regression function. Our bias analysis characterizes explicitly
how the random forests bias depends on the sample size, tree height, and column
subsampling parameter. Some limitations on our current results are also
discussed.
