  The mirror descent algorithm is known to be effective in situations where it
is beneficial to adapt the mirror map to the underlying geometry of the
optimization model. However, the effect of mirror maps on the geometry of
distributed optimization problems has not been previously addressed. In this
paper we study an exact distributed mirror descent algorithm in continuous-time
under additive noise. We establish a linear convergence rate of the proposed
dynamics for the setting of convex optimization. Our analysis draws motivation
from the Augmented Lagrangian and its relation to gradient tracking. To further
explore the benefits of mirror maps in a distributed setting we present a
preconditioned variant of our algorithm with an additional mirror map over the
Lagrangian dual variables. This allows our method to adapt to both the geometry
of the primal variables, as well as to the geometry of the consensus
constraint. We also propose a Gauss-Seidel type discretization scheme for the
proposed method and establish its linear convergence rate. For certain classes
of problems we identify mirror maps that mitigate the effect of the graph's
spectral properties on the convergence rate of the algorithm. Using numerical
experiments we demonstrate the efficiency of the methodology on convex models,
both with and without constraints. Our findings show that the proposed method
outperforms other methods, especially in scenarios where the model's geometry
is not captured by the standard Euclidean norm
