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.

翻译：我们考虑的是平滑的 convex- concove 双线双向双向双向双向马鞍点问题, $\\ mathbf{x{max{max{maxbf{maxf{y}}F(\\\mathbf{x})+H(\\mathbfff{x},\mathbff{y}}- G(mathbff{y}}) - G(\\mathbbf- 双线双线双线马鞍点问题, $$G$, 以及双线性联结函数$H$。 在对差异性不平等的标准超直径分析的基础上, 我们展示了一种随机直观的梯度外向梯度梯度递增增缩度算法( AG- EG) 和 Nesterov 的加速度算法。 这种算法拉力将重新启用以接受一个精细的不折合的直径直线不平面的双向较下框框框框框的直的双交会, 20) 最优化的双向最优化的马路段的马路面的马匹比的马匹比, 最优的马匹比的马匹比的马匹比的马匹比的马匹比。