  This article addresses the problem of testing the conditional independence of
two generic random vectors $X$ and $Y$ given a third random vector $Z$, which
plays an important role in statistical and machine learning applications. We
propose a new non-parametric testing procedure that avoids explicitly
estimating any conditional distributions but instead requires sampling from the
two marginal conditional distributions of $X$ given $Z$ and $Y$ given $Z$. We
further propose using a generative neural network (GNN) framework to sample
from these approximated marginal conditional distributions, which tends to
mitigate the curse of dimensionality due to its adaptivity to any
low-dimensional structures and smoothness underlying the data. Theoretically,
our test statistic is shown to enjoy a doubly robust property against GNN
approximation errors, meaning that the test statistic retains all desirable
properties of the oracle test statistic utilizing the true marginal conditional
distributions, as long as the product of the two approximation errors decays to
zero faster than the parametric rate. Asymptotic properties of our statistic
and the consistency of a bootstrap procedure are derived under both null and
local alternatives. Extensive numerical experiments and real data analysis
illustrate the effectiveness and broad applicability of our proposed test.
