Similarly to what has been done with Global Constraints in Constraint Programming, different results have been recently published on Global Cost Functions in weighted CSPs, defining the premises of a Cost Function Programming paradigm. In this paper, in the spirit of Berge-acyclic decompositions of global constraints such as Regular, we explore the possibility of decomposing Global Cost Functions in such a way that enforcing soft local consistencies on the decomposed cost function offers guarantees on the level of consistency enforced on the original global cost function. We show that an extension of Directional Arc Consistency to arbitrary arities and Virtual Arc Consistency offer specific guarantees.We conclude by preliminary experiments on Weighted Regular decompositions that show that decompositions may be very useful to easily integrate global cost functions in existing solvers with good efficiency.