Last-Iterate Convergence of Anchored Gradient Descent

Abstract

We study the monotone inclusion problem $0\in F(z)+A(z)$, where $F$ is monotone and Lipschitz, and $A$ is maximally monotone, a framework that encompasses monotone variational inequalities and convex-concave saddle-point problems with constraints or regularization. It is well known that vanilla gradient descent diverges for this problem, whereas optimism-based methods such as Extragradient and accelerated methods that combine both optimism and anchoring, such as Extra Anchored Gradient, achieve last-iterate convergence. However, the anchoring-only method, anchored gradient descent, has been studied only in the unconstrained setting [RYY19, SST+26]. In this note, we extend the anchored gradient descent method to the monotone inclusion problem and prove a last-iterate convergence rate of $O(1/\sqrt{T})$ in terms of the tangent residual. We build on the recent proof in the unconstrained setting [SST+26] and use techniques from [COZ24] to extend it to the general inclusion setting.