  This work derives upper bounds on the convergence rate of the
moment-sum-of-squares hierarchy with correlative sparsity for global
minimization of polynomials on compact basic semialgebraic sets. The main
conclusion is that both sparse hierarchies based on the Schm\"udgen and Putinar
Positivstellens\"atze enjoy a polynomial rate of convergence that depends on
the size of the largest clique in the sparsity graph but not on the ambient
dimension. Interestingly, the sparse bounds outperform the best currently
available bounds for the dense hierarchy when the maximum clique size is
sufficiently small compared to the ambient dimension and the performance is
measured by the running time of an interior point method required to obtain a
bound on the global minimum of a given accuracy.
