  We consider the smooth convex-concave bilinearly-coupled saddle-point
problem, $\min_{\mathbf{x}}\max_{\mathbf{y}}~F(\mathbf{x}) +
H(\mathbf{x},\mathbf{y}) - G(\mathbf{y})$, where one has access to stochastic
first-order oracles for $F$, $G$ as well as the bilinear coupling function $H$.
Building upon standard stochastic extragradient analysis for variational
inequalities, we present a stochastic \emph{accelerated gradient-extragradient
(AG-EG)} descent-ascent algorithm that combines extragradient and Nesterov's
acceleration in general stochastic settings. This algorithm leverages scheduled
restarting to admit a fine-grained nonasymptotic convergence rate that matches
known lower bounds by both \citet{ibrahim2020linear} and \citet{zhang2021lower}
in their corresponding settings, plus an additional statistical error term for
bounded stochastic noise that is optimal up to a constant prefactor. This is
the first result that achieves such a relatively mature characterization of
optimality in saddle-point optimization.
