  In this paper, we derive a randomized version of the Mirror-Prox method for
solving some structured matrix saddle-point problems, such as the maximal
eigenvalue minimization problem. Deterministic first-order schemes, such as
Nesterov's Smoothing Techniques or standard Mirror-Prox methods, require the
exact computation of a matrix exponential at every iteration, limiting the size
of the problems they can solve. Our method allows us to use stochastic
approximations of matrix exponentials. We prove that our randomized scheme
decreases significantly the complexity of its deterministic counterpart for
large-scale matrix saddle-point problems. Numerical experiments illustrate and
confirm our theoretical results.
