  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
$\Omega(\sqrt{\ln T})$ 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.
