  Recent applications that arise in machine learning have surged significant
interest in solving min-max saddle point games. This problem has been
extensively studied in the convex-concave regime for which a global equilibrium
solution can be computed efficiently. In this paper, we study the problem in
the non-convex regime and show that an \varepsilon--first order stationary
point of the game can be computed when one of the player's objective can be
optimized to global optimality efficiently. In particular, we first consider
the case where the objective of one of the players satisfies the
Polyak-{\L}ojasiewicz (PL) condition. For such a game, we show that a simple
multi-step gradient descent-ascent algorithm finds an \varepsilon--first order
stationary point of the problem in \widetilde{\mathcal{O}}(\varepsilon^{-2})
iterations. Then we show that our framework can also be applied to the case
where the objective of the "max-player" is concave. In this case, we propose a
multi-step gradient descent-ascent algorithm that finds an \varepsilon--first
order stationary point of the game in \widetilde{\cal O}(\varepsilon^{-3.5})
iterations, which is the best known rate in the literature. We applied our
algorithm to a fair classification problem of Fashion-MNIST dataset and
observed that the proposed algorithm results in smoother training and better
generalization.

Optimization and Control;Machine Learning;Machine Learning