  Generative adversarial networks (GANs) are notoriously difficult to train and
the reasons underlying their (non-)convergence behaviors are still not
completely understood. By first considering a simple yet representative GAN
example, we mathematically analyze its local convergence behavior in a
non-asymptotic way. Furthermore, the analysis is extended to general GANs under
certain assumptions. We find that in order to ensure a good convergence rate,
two factors of the Jacobian in the GAN training dynamics should be
simultaneously avoided, which are (i) the Phase Factor, i.e., the Jacobian has
complex eigenvalues with a large imaginary-to-real ratio, and (ii) the
Conditioning Factor, i.e., the Jacobian is ill-conditioned. Previous methods of
regularizing the Jacobian can only alleviate one of these two factors, while
making the other more severe. Thus we propose a new JAcobian REgularization
(JARE) for GANs, which simultaneously addresses both factors by construction.
Finally, we conduct experiments that confirm our theoretical analysis and
demonstrate the advantages of JARE over previous methods in stabilizing GANs.
