  Variational inequalities have recently attracted considerable interest in
machine learning as a flexible paradigm for models that go beyond ordinary loss
function minimization (such as generative adversarial networks and related deep
learning systems). In this setting, the optimal $\mathcal{O}(1/t)$ convergence
rate for solving smooth monotone variational inequalities is achieved by the
Extra-Gradient (EG) algorithm and its variants. Aiming to alleviate the cost of
an extra gradient step per iteration (which can become quite substantial in
deep learning applications), several algorithms have been proposed as
surrogates to Extra-Gradient with a \emph{single} oracle call per iteration. In
this paper, we develop a synthetic view of such algorithms, and we complement
the existing literature by showing that they retain a $\mathcal{O}(1/t)$
ergodic convergence rate in smooth, deterministic problems. Subsequently,
beyond the monotone deterministic case, we also show that the last iterate of
single-call, \emph{stochastic} extra-gradient methods still enjoys a
$\mathcal{O}(1/t)$ local convergence rate to solutions of \emph{non-monotone}
variational inequalities that satisfy a second-order sufficient condition.
