  Generative adversarial networks (GANs) have been widely used and have
achieved competitive results in semi-supervised learning. This paper
theoretically analyzes how GAN-based semi-supervised learning (GAN-SSL) works.
We first prove that, given a fixed generator, optimizing the discriminator of
GAN-SSL is equivalent to optimizing that of supervised learning. Thus, the
optimal discriminator in GAN-SSL is expected to be perfect on labeled data.
Then, if the perfect discriminator can further cause the optimization objective
to reach its theoretical maximum, the optimal generator will match the true
data distribution. Since it is impossible to reach the theoretical maximum in
practice, one cannot expect to obtain a perfect generator for generating data,
which is apparently different from the objective of GANs. Furthermore, if the
labeled data can traverse all connected subdomains of the data manifold, which
is reasonable in semi-supervised classification, we additionally expect the
optimal discriminator in GAN-SSL to also be perfect on unlabeled data. In
conclusion, the minimax optimization in GAN-SSL will theoretically output a
perfect discriminator on both labeled and unlabeled data by unexpectedly
learning an imperfect generator, i.e., GAN-SSL can effectively improve the
generalization ability of the discriminator by leveraging unlabeled
information.
