  In this paper, we analyze the local convergence rate of optimistic mirror
descent methods in stochastic variational inequalities, a class of optimization
problems with important applications to learning theory and machine learning.
Our analysis reveals an intricate relation between the algorithm's rate of
convergence and the local geometry induced by the method's underlying Bregman
function. We quantify this relation by means of the Legendre exponent, a notion
that we introduce to measure the growth rate of the Bregman divergence relative
to the ambient norm near a solution. We show that this exponent determines both
the optimal step-size policy of the algorithm and the optimal rates attained,
explaining in this way the differences observed for some popular Bregman
functions (Euclidean projection, negative entropy, fractional power, etc.).
