  An Euler discretization of the Langevin diffusion is known to converge to the
global minimizers of certain convex and non-convex optimization problems. We
show that this property holds for any suitably smooth diffusion and that
different diffusions are suitable for optimizing different classes of convex
and non-convex functions. This allows us to design diffusions suitable for
globally optimizing convex and non-convex functions not covered by the existing
Langevin theory. Our non-asymptotic analysis delivers computable optimization
and integration error bounds based on easily accessed properties of the
objective and chosen diffusion. Central to our approach are new explicit Stein
factor bounds on the solutions of Poisson equations. We complement these
results with improved optimization guarantees for targets other than the
standard Gibbs measure.
