  We consider decentralized consensus optimization when workers sample data
from non-identical distributions and perform variable amounts of work due to
slow nodes known as stragglers. The problem of non-identical distributions and
the problem of variable amount of work have been previously studied separately.
In our work we analyze them together under a unified system model. We study the
convergence of the optimization algorithm when combining worker outputs under
two heuristic methods: (1) weighting equally, and (2) weighting by the amount
of work completed by each. We prove convergence of the two methods under
perfect consensus, assuming straggler statistics are independent and identical
across all workers for all iterations. Our numerical results show that under
approximate consensus the second method outperforms the first method for both
convex and non-convex objective functions. We make use of the theory on minimum
variance unbiased estimator (MVUE) to evaluate the existence of an optimal
method for combining worker outputs. While we conclude that neither of the two
heuristic methods are optimal, we also show that an optimal method does not
exist.
