  Min-max saddle point games have recently been intensely studied, due to their
wide range of applications, including training Generative Adversarial Networks
(GANs). However, most of the recent efforts for solving them are limited to
special regimes such as convex-concave games. Further, it is customarily
assumed that the underlying optimization problem is solved either by a single
machine or in the case of multiple machines connected in centralized fashion,
wherein each one communicates with a central node. The latter approach becomes
challenging, when the underlying communications network has low bandwidth. In
addition, privacy considerations may dictate that certain nodes can communicate
with a subset of other nodes. Hence, it is of interest to develop methods that
solve min-max games in a decentralized manner. To that end, we develop a
decentralized adaptive momentum (ADAM)-type algorithm for solving min-max
optimization problem under the condition that the objective function satisfies
a Minty Variational Inequality condition, which is a generalization to
convex-concave case. The proposed method overcomes shortcomings of recent
non-adaptive gradient-based decentralized algorithms for min-max optimization
problems that do not perform well in practice and require careful tuning. In
this paper, we obtain non-asymptotic rates of convergence of the proposed
algorithm (coined DADAM$^3$) for finding a (stochastic) first-order Nash
equilibrium point and subsequently evaluate its performance on training GANs.
The extensive empirical evaluation shows that DADAM$^3$ outperforms recently
developed methods, including decentralized optimistic stochastic gradient for
solving such min-max problems.
