  Nonconvex minimax problems appear frequently in emerging machine learning
applications, such as generative adversarial networks and adversarial learning.
Simple algorithms such as the gradient descent ascent (GDA) are the common
practice for solving these nonconvex games and receive lots of empirical
success. Yet, it is known that these vanilla GDA algorithms with constant step
size can potentially diverge even in the convex setting. In this work, we show
that for a subclass of nonconvex-nonconcave objectives satisfying a so-called
two-sided Polyak-{\L}ojasiewicz inequality, the alternating gradient descent
ascent (AGDA) algorithm converges globally at a linear rate and the stochastic
AGDA achieves a sublinear rate. We further develop a variance reduced algorithm
that attains a provably faster rate than AGDA when the problem has the
finite-sum structure.
