  We deconstruct the performance of GANs into three components:
  1. Formulation: we propose a perturbation view of the population target of
GANs. Building on this interpretation, we show that GANs can be viewed as a
generalization of the robust statistics framework, and propose a novel GAN
architecture, termed as Cascade GANs, to provably recover meaningful
low-dimensional generator approximations when the real distribution is
high-dimensional and corrupted by outliers.
  2. Generalization: given a population target of GANs, we design a systematic
principle, projection under admissible distance, to design GANs to meet the
population requirement using finite samples. We implement our principle in
three cases to achieve polynomial and sometimes near-optimal sample
complexities: (1) learning an arbitrary generator under an arbitrary
pseudonorm; (2) learning a Gaussian location family under TV distance, where we
utilize our principle provide a new proof for the optimality of Tukey median
viewed as GANs; (3) learning a low-dimensional Gaussian approximation of a
high-dimensional arbitrary distribution under Wasserstein distance. We
demonstrate a fundamental trade-off in the approximation error and statistical
error in GANs, and show how to apply our principle with empirical samples to
predict how many samples are sufficient for GANs in order not to suffer from
the discriminator winning problem.
  3. Optimization: we demonstrate alternating gradient descent is provably not
locally asymptotically stable in optimizing the GAN formulation of PCA. We
diagnose the problem as the minimax duality gap being non-zero, and propose a
new GAN architecture whose duality gap is zero, where the value of the game is
equal to the previous minimax value (not the maximin value). We prove the new
GAN architecture is globally asymptotically stable in optimization under
alternating gradient descent.
