To secure the result, we must show that (a) the discriminator reliably converges on the Bayes risk at each iteration t; and (b)
the generator’s sampling strategy drives original and synthetic data closer together, ultimately taking the Bayes risk to 1/2
as n, t → ∞. (For the purposes of this proof, we set the tolerance parameter δ to 0.)
Take (a) ﬁrst. This amounts to a consistency requirement for RFs. The consistency of RF classiﬁers has been demonstrated
under various assumptions about splitting rules and stopping criteria (Breiman, 2004; Biau et al., 2008; Biau and Devroye,
2010; Gao et al., 2022), but these results generally require trees to be grown to purity or even completion (i.e., n(cid:96)
b = 1 for all
(cid:96), b). However, this would turn the generator’s sampling strategy into a simple copy-paste operation and make intra-leaf
density estimation impossible. We therefore follow Malley et al. (2012) in observing that regression procedures constitute
Adversarial Random Forests for Density Estimation and Generative Modeling