Adaptive and Universal Algorithms for Variational Inequalities with Optimal Convergence
This work addresses variational inequalities, which are foundational for convex optimization and saddle point problems, offering adaptive and universal algorithms with optimal convergence, though it is incremental in improving upon prior adaptive methods.
The authors tackled the problem of variational inequalities with monotone operators by developing adaptive algorithms that automatically adjust to unknown parameters and achieve optimal convergence rates across non-smooth, smooth, and stochastic settings, improving over existing methods by a factor of Ω(√ln T) and extending to unbounded domains.
We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods by a $Ω(\sqrt{\ln T})$ factor, matching the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings.