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We consider variational inequalities coming from monotone operators, asetting that includes convex minimization and convex-concave saddle-pointproblems. We assume an access to potentially noisy unbiased values of themonotone operators and assess convergence through a compatible gap functionwhich corresponds to the standard optimality criteria in the aforementionedsubcases. We present a universal algorithm for these inequalities based on theMirror-Prox algorithm. Concretely, our algorithm simultaneously achieves theoptimal rates for the smooth/non-smooth, and noisy/noiseless settings. This isdone without any prior knowledge of these properties, and in the general set-upof arbitrary norms and compatible Bregman divergences. For convex minimizationand convex-concave saddle-point problems, this leads to new adaptivealgorithms. Our method relies on a novel yet simple adaptive choice of thestep-size, which can be seen as the appropriate extension of AdaGrad to handleconstrained problems.

A Universal Algorithm for Variational Inequalities Adaptive to Smoothness and Noise