A Unifying Framework of Accelerated First-Order Approach to Strongly Monotone Variational Inequalities

In this paper, we propose a unifying framework incorporating several momentum-related search directions for solving strongly monotone variational inequalities. The specific combinations of the search directions in the framework are made to guarantee the optimal iteration complexity bound of $\mathcal{O}\left(\kappa\ln(1/\epsilon)\right)$ to reach an $\epsilon$-solution, where $\kappa$ is the condition number. This framework provides the flexibility for algorithm designers to train -- among different parameter combinations -- the one that best suits the structure of the problem class at hand. The proposed framework includes the following iterative points and directions as its constituents: the extra-gradient, the optimistic gradient descent ascent (OGDA) direction (aka "optimism"), the "heavy-ball" direction, and Nesterov's extrapolation points. As a result, all the afore-mentioned methods become the special cases under the general scheme of extra points. We also specialize this approach to strongly convex minimization, and show that a similar extra-point approach achieves the optimal iteration complexity bound of $\mathcal{O}(\sqrt{\kappa}\ln(1/\epsilon))$ for this class of problems.