  An open problem in optimization with noisy information is the computation of
an exact minimizer that is independent of the amount of noise. A standard
practice in stochastic approximation algorithms is to use a decreasing
step-size. This however leads to a slower convergence. A second alternative is
to use a fixed step-size and run independent replicas of the algorithm and
average these. A third option is to run replicas of the algorithm and allow
them to interact. It is unclear which of these options works best. To address
this question, we reduce the problem of the computation of an exact minimizer
with noisy gradient information to the study of stochastic mirror descent with
interacting particles. We study the convergence of stochastic mirror descent
and make explicit the tradeoffs between communication and variance reduction.
We provide theoretical and numerical evidence to suggest that interaction helps
to improve convergence and reduce the variance of the estimate.
