vary with from one round to the next, but it does not
increase faster than 1/ max(cid:96),b

(A3) Trees satisfy the following conditions: (i) training
data for each tree is split into two subsets: one
to learn split parameters, the other to assign leaf
labels; (ii) trees are grown on subsamples rather
than bootstraps, with subsample size nb satisfying
nb → ∞, nb/n → 0 as n → ∞; (iii) at each in-
ternal node, the probability that a tree splits on any
given Xj is bounded from below by some π > 0;
(iv) every split puts at least a fraction γ ∈ (0, 0.5] of
the available observations into each child node; (v)
for each tree b, the total number of leaves Lb satisﬁes
Lb → ∞, Lb/n → 0 as n → ∞; and (vi) predic-
tions are made with soft labels both within leaves
and across trees, i.e. by averaging rather than voting.
Adversarial Random Forests for Density Estimation and Generative Modeling