Fast Cyclic Coordinate Dual Averaging with Extrapolation for Generalized Variational Inequalities

We propose the \emph{Cyclic cOordinate Dual avEraging with extRapolation (CODER)} method for generalized variational inequality problems. Such problems are fairly general and include composite convex minimization and min-max optimization as special cases. CODER is the first cyclic block coordinate method whose convergence rate is independent of the number of blocks, which fills the significant gap between cyclic coordinate methods and randomized ones that remained open for many years. Moreover, CODER provides the first theoretical guarantee for cyclic coordinate methods for solving generalized variational inequality problems under only monotonicity and Lipschitz continuity assumptions. To remove the dependence on the number of blocks, the analysis of CODER is based on a novel Lipschitz condition with respect to a Mahalanobis norm rather than the commonly used coordinate-wise Lipschitz condition; to be applicable to general variational inequalities, CODER leverages an extrapolation strategy inspired by the recent developments in primal-dual methods. Our theoretical results are complemented by numerical experiments, which demonstrate competitive performance of CODER compared to other coordinate methods.