  We introduce Wasserstein consensus alternating direction method of
multipliers (ADMM) and its entropic-regularized version: Sinkhorn consensus
ADMM, to solve measure-valued optimization problems with convex additive
objectives. Several problems of interest in stochastic prediction and learning
can be cast in this form of measure-valued convex additive optimization. The
proposed algorithm generalizes a variant of the standard Euclidean ADMM to the
space of probability measures but departs significantly from its Euclidean
counterpart. In particular, we derive a two layer ADMM algorithm wherein the
outer layer is a variant of consensus ADMM on the space of probability measures
while the inner layer is a variant of Euclidean ADMM. The resulting
computational framework is particularly suitable for solving Wasserstein
gradient flows via distributed computation. We demonstrate the proposed
framework using illustrative numerical examples.
