  Stein's method has been widely used for probability approximations. However,
in the multi-dimensional setting, most of the results are for multivariate
normal approximation or for test functions with bounded second- or higher-order
derivatives. For a class of multivariate limiting distributions, we use
Bismut's formula in Malliavin calculus to control the derivatives of the Stein
equation solutions by the first derivative of the test function. Combined with
Stein's exchangeable pair approach, we obtain a general theorem for
multivariate approximations with near optimal error bounds on the Wasserstein
distance.We apply the theorem to the unadjusted Langevin algorithm.
