Extragradient Method: $O(1/K)$ Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity
This provides a theoretical advance for optimization and machine learning practitioners working on saddle point problems, though it is incremental relative to existing literature.
The paper resolves an open question by deriving the first last-iterate O(1/K) convergence rate for the Extragradient method in monotone and Lipschitz variational inequalities without additional assumptions, improving upon prior work that required Lipschitzness of the Jacobian. It also establishes results on the (non-)cocoercivity of update operators for related methods.
Extragradient method (EG) (Korpelevich, 1976) is one of the most popular methods for solving saddle point and variational inequalities problems (VIP). Despite its long history and significant attention in the optimization community, there remain important open questions about convergence of EG. In this paper, we resolve one of such questions and derive the first last-iterate $O(1/K)$ convergence rate for EG for monotone and Lipschitz VIP without any additional assumptions on the operator unlike the only known result of this type (Golowich et al., 2020) that relies on the Lipschitzness of the Jacobian of the operator. The rate is given in terms of reducing the squared norm of the operator. Moreover, we establish several results on the (non-)cocoercivity of the update operators of EG, Optimistic Gradient Method, and Hamiltonian Gradient Method, when the original operator is monotone and Lipschitz.