  Min-max saddle point games appear in a wide range of applications in machine
leaning and signal processing. Despite their wide applicability, theoretical
studies are mostly limited to the special convex-concave structure. While some
recent works generalized these results to special smooth non-convex cases, our
understanding of non-smooth scenarios is still limited. In this work, we study
special form of non-smooth min-max games when the objective function is
(strongly) convex with respect to one of the player's decision variable. We
show that a simple multi-step proximal gradient descent-ascent algorithm
converges to $\epsilon$-first-order Nash equilibrium of the min-max game with
the number of gradient evaluations being polynomial in $1/\epsilon$. We will
also show that our notion of stationarity is stronger than existing ones in the
literature. Finally, we evaluate the performance of the proposed algorithm
through adversarial attack on a LASSO estimator.
