  In this paper, we propose universal proximal mirror methods to solve the
variational inequality problem with Holder continuous operators in both
deterministic and stochastic settings. The proposed methods automatically adapt
not only to the oracle's noise (in the stochastic setting of the problem) but
also to the Holder continuity of the operator without having prior knowledge of
either the problem class or the nature of the operator information. We analyzed
the proposed algorithms in both deterministic and stochastic settings and
obtained estimates for the required number of iterations to achieve a given
quality of a solution to the variational inequality. We showed that, without
knowing the Holder exponent and Holder constant of the operators, the proposed
algorithms have the least possible in the worst case sense complexity for the
considered class of variational inequalities. We also compared the resulting
stochastic algorithm with other popular optimizers for the task of image
classification.
