Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization
This work provides an optimal algorithm for saddle-point optimization, which is incremental as it builds upon existing extragradient methods to achieve theoretical lower bounds in stochastic settings.
The paper tackles the problem of smooth convex-concave bilinearly-coupled saddle-point optimization by proposing a stochastic accelerated gradient-extragradient algorithm that combines extragradient and Nesterov's acceleration, achieving a nonasymptotic convergence rate that matches known lower bounds and includes an optimal statistical error term for bounded stochastic noise.
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.