  Two of the most prominent algorithms for solving unconstrained smooth games
are the classical stochastic gradient descent-ascent (SGDA) and the recently
introduced stochastic consensus optimization (SCO) [Mescheder et al., 2017].
SGDA is known to converge to a stationary point for specific classes of games,
but current convergence analyses require a bounded variance assumption. SCO is
used successfully for solving large-scale adversarial problems, but its
convergence guarantees are limited to its deterministic variant. In this work,
we introduce the expected co-coercivity condition, explain its benefits, and
provide the first last-iterate convergence guarantees of SGDA and SCO under
this condition for solving a class of stochastic variational inequality
problems that are potentially non-monotone. We prove linear convergence of both
methods to a neighborhood of the solution when they use constant step-size, and
we propose insightful stepsize-switching rules to guarantee convergence to the
exact solution. In addition, our convergence guarantees hold under the
arbitrary sampling paradigm, and as such, we give insights into the complexity
of minibatching.
