Accelerating Block-Decomposition First-Order Methods for Solving Composite Saddle-Point and Two-Player Nash Equilibrium Problems

This article considers the two-player composite Nash equilibrium (CNE) problem with a separable non-smooth part, which is known to include the composite saddle-point (CSP) problem as a special case. Due to its two-block structure, this problem can be solved by any algorithm belonging to the block-decomposition hybrid proximal-extragradient (BD-HPE) framework proposed in [13]. The framework consists of a family of inexact proximal point methods for solving a more general two-block structured monotone inclusion problem which, at every iteration, solves two prox sub-inclusions according to a certain relative error criterion. By exploiting the fact that the two prox sub-inclusions in the context of the NE problem are equivalent to two composite convex programs, this article proposes a new instance of the BD-HPE framework that approximately solves them using an accelerated gradient method. It is shown that the new instance is able to take significantly larger prox stepsizes than other instances from this framework that perform single composite gradient steps for solving the sub-inclusions. As a result, it is shown that the first instance has better iteration-complexity than the latter ones. Finally, it is also shown that the new accelerated BD-HPE instance computationally outperforms several state-of-the-art algorithms on many relevant classes of CSP and CNE instances.