ples of size n from D and ﬁts a binary decision tree for each,
in which observations are recursively partitioned according
to some optimization target (e.g., Gini index) evaluated on
a random subset of features at each node. The size of this
set at (cid:98)√
subset is controlled by the mtry parameter, conventionally
d(cid:99) for classiﬁcation. Resulting splits are literals
of the form Xj (cid:46)(cid:47) x for some Xj, j ∈ [d] = {1, . . . , d},
and value x ∈ Xj, where (cid:46)(cid:47) ∈ {=, <} (the former for cat-
egorical, the latter for continuous variables). Data pass to
left or right child nodes depending on whether they satisfy
the literal. Splits continue until some stopping criterion is
met (e.g., purity). Terminal nodes, a.k.a. leaves, describe
hyperrectangles in feature space with boundaries given by
the learned splits. These disjoint cells collectively cover
all of X . Each leaf is associated with a label ˆy ∈ [0, 1],
Adversarial Random Forests for Density Estimation and Generative Modeling