  In this paper, we focus on solving a class of constrained non-convex
non-concave saddle point problems in a decentralized manner by a group of nodes
in a network. Specifically, we assume that each node has access to a summand of
a global objective function and nodes are allowed to exchange information only
with their neighboring nodes. We propose a decentralized variant of the
proximal point method for solving this problem. We show that when the objective
function is $\rho$-weakly convex-weakly concave the iterates converge to
approximate stationarity with a rate of $\mathcal{O}(1/\sqrt{T})$ where the
approximation error depends linearly on $\sqrt{\rho}$. We further show that
when the objective function satisfies the Minty VI condition (which generalizes
the convex-concave case) we obtain convergence to stationarity with a rate of
$\mathcal{O}(1/\sqrt{T})$. To the best of our knowledge, our proposed method is
the first decentralized algorithm with theoretical guarantees for solving a
non-convex non-concave decentralized saddle point problem. Our numerical
results for training a general adversarial network (GAN) in a decentralized
manner match our theoretical guarantees.
