  Nonconvex-concave min-max problem arises in many machine learning
applications including minimizing a pointwise maximum of a set of nonconvex
functions and robust adversarial training of neural networks. A popular
approach to solve this problem is the gradient descent-ascent (GDA) algorithm
which unfortunately can exhibit oscillation in case of nonconvexity. In this
paper, we introduce a "smoothing" scheme which can be combined with GDA to
stabilize the oscillation and ensure convergence to a stationary solution. We
prove that the stabilized GDA algorithm can achieve an $O(1/\epsilon^2)$
iteration complexity for minimizing the pointwise maximum of a finite
collection of nonconvex functions. Moreover, the smoothed GDA algorithm
achieves an $O(1/\epsilon^4)$ iteration complexity for general
nonconvex-concave problems. Extensions of this stabilized GDA algorithm to
multi-block cases are presented. To the best of our knowledge, this is the
first algorithm to achieve $O(1/\epsilon^2)$ for a class of nonconvex-concave
problem. We illustrate the practical efficiency of the stabilized GDA algorithm
on robust training.
