  In this article, we study the problem of high-dimensional conditional
independence testing, a key building block in statistics and machine learning.
We propose an inferential procedure based on double generative adversarial
networks (GANs). Specifically, we first introduce a double GANs framework to
learn two generators of the conditional distributions. We then integrate the
two generators to construct a test statistic, which takes the form of the
maximum of generalized covariance measures of multiple transformation
functions. We also employ data-splitting and cross-fitting to minimize the
conditions on the generators to achieve the desired asymptotic properties, and
employ multiplier bootstrap to obtain the corresponding $p$-value. We show that
the constructed test statistic is doubly robust, and the resulting test both
controls type-I error and has the power approaching one asymptotically. Also
notably, we establish those theoretical guarantees under much weaker and
practically more feasible conditions compared to the existing tests, and our
proposal gives a concrete example of how to utilize some state-of-the-art deep
learning tools, such as GANs, to help address a classical but challenging
statistical problem. We demonstrate the efficacy of our test through both
simulations and an application to an anti-cancer drug dataset. A Python
implementation of the proposed procedure is available at
https://github.com/tianlinxu312/dgcit.
